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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf,len=970
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='13800v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='LO]  31 Jan 2023  A monotone connection between model class size  and description length  Reijo Jaakkola  Tampere University  Finland  Antti Kuusisto  Tampere University  University of Helsinki  Finland  Miikka Vilander  Tampere University  Finland  Abstract  This paper links sizes of model classes to the minimum lengths of their  defining formulas, that is, to their description complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Limiting  to models with a fixed domain of size n, we study description com-  plexities with respect to the extension of propositional logic with the  ability to count assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' This logic, called GMLU, can alternati-  vely be conceived as graded modal logic over Kripke models with the  universal accessibility relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' While GMLU is expressively complete  for defining multisets of assignments, we also investigate its fragments  GMLU(d) that can count only up to the integer threshold d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We fo-  cus in particular on description complexities of equivalence classes of  GMLU(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We show that, in restriction to a poset of type realiza-  tions, the order of the equivalence classes based on size is identical to  the order based on description complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' This also demonstrates  a monotone connection between Boltzmann entropies of model classes  and description complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Furthermore, we characterize how the  relation between domain size n and counting threshold d determines  whether or not there exists a dominating class, which essentially means  a model class with limit probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' To obtain our results, we  prove new estimates on r-associated Stirling numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' As another cru-  cial tool, we show that model classes split into two distinct cases in  relation to their description complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  1  Introduction  This paper investigates how sizes of model classes are linked to the minimum  lengths of formulas needed to define the classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' In the scenarios we consider,  we first fix a class M of models that share a domain of the same finite size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  The model classes M ⊆ M are then studied with respect to the extension  of propositional logic with the ability to count propositional assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We call this logic GMLU, as it can alternatively be defined as graded modal  1  logic over Kripke models with the universal relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' In order to obtain more  fine grained results, we parameterize GMLU and study also its fragments  GMLUd that can count only up to the threshold d ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' This also enables us  to demonstrate how the relationship between minimum formula lengths and  model class sizes develops when we gradually increase the expressive power  of the logic used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For a model class M ⊆ M, the description complexity of  M with respect to GMLUd is simply the minimum length of a formula of  GMLUd needed to define M, if such a formula exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  In this paper we are particularly interested in the description complexi-  ties Cd(M) of the logical equivalence classes M determined by GMLUd over  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let us write M ≡d M′ if the models M, M′ ∈ M satisfy the same set of  formulas of GMLUd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Note that Cd(M) of an equivalence class M of ≡d can  also be regarded as the description complexity of each model M ∈ M, as the  expressive power of GMLUd suffices precisely to describe M up to the equi-  valence ≡d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' From this perspective, description complexity is analogous to  Kolmogorov complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' There exist well known links between Kolmogorov  complexity and Shannon entropy, see for example [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The recent work in  [8],[7] demonstrates a way to conceive related results also in the scenario  where relational structures are classified via logics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' In particular, it is shown  that the expected Boltzmann entropy of the equivalence classes of GMLU  is asymptotically equivalent to the expected description complexity (with  respect to GMLU) times the size of the vocabulary considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' It is also  shown that for d = 1, the greatest equivalence class of GMLUd has maxi-  mum description complexity among the classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' This paper builds on those  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Firstly, as a crucial tool for our proofs, we establish a classification of  description complexities into two distinct classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  This division is based  on the numbers ni of elements realizing different propositional types i ∈ I  in models of the described model class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' here I is just an index set for the  types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The division is then determined by whether or not ni = d for at least  two different types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Using this, we establish a strong connection between  model class sizes and description complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For each model M ∈ M, let  nM denote the tuple (ni)i∈I that gives the numbers ni of points realizing  propositional types in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Furthermore, instead of recording numbers ni  greater than the counting threshold d, simply put d in nM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We define a  poset (M, ⪯τ) over the models, where τ is the vocabulary and the order  ⪯τ is based on comparing the tuples nM coordinatewise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The order ⪯τ is  directly inherited also by the classes of ≡d such that M ⪯τ M′ if and only  if for some (or equivalently, all) models M and M′ in the respective classes,  we have M ⪯τ M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We will prove that for all classes M and M′ of ≡d such  that M and M′ are ⪯τ-comparable, we have  |M| < |M′| ⇔ Cd(M) < Cd(M′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  2  In other words, over ⪯τ, the ordering of model classes according to size is  identical to the ordering based on description complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' This is an intimate  link between syntax and semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' As a corollary, we obtain a correspon-  ding relationship between Boltzmann entropies and description complexities  of model classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We then investigate how the classes of ≡d behave when we alter the  domain size n and counting threshold d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Note that increasing d corresponds  to moving to more and more expressive logics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' First we observe that for  thresholds d and d′ > d and the corresponding Shannon entropies HS(≡d)  and HS(≡d′) of the model class distributions given by ≡d and ≡d′, we have  HS(≡d) < HS(≡d′) when d′ is at most n/2, and  HS(≡d) = H(≡d′) when d is at least n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  A similar result also follows for expected Boltzmann entropies HB(≡d) and  HB(≡d′), but with the orders reversed, that is HB(≡d) > HB(≡d′) for d′ at  most n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  To get a better sense of the relative sizes of the classes when n and d are  altered, we prove an asymptotic characterization of the class distributions  as n → ∞ and d is a function of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let us say that ≡d(n) has a dominating  class if with limit probability one, a random model of size n belongs to a  maximum size class in ≡d(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Similarly, all classes in ≡d(n) are vanishing if  with limit probability zero, a random model of size n belongs to a maximum  size class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Then the following results hold as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  If d(n) ≤ n/2|τ| − f(n) where f(n) = ω(√n), then ≡d(n) has a domi-  nating class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  If d(n) ≥ n/2|τ| − f(n) where f(n) = o(√n), then ≡d(n) has no domi-  nating class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  If d(n) ≥ n/2|τ| + f(n) where f(n) = ω(√n), then every class in ≡d(n)  is vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  One corollary of these results is that for d(n) ≤ n/t−f(n), if f(n) = ω(√n),  then with limit probability one, two random models of size n cannot be  separated in GMLUd(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Finally, we also give a non-asymptotic variant  of the characterization of the class distributions for ≡d including explicit  bounds on d for separating the cases where ≡d has a majority class or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  By a majority class, we mean a class containing more than half of all models  in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Concerning related work, as already mentioned, it is well known that  entropy and Kolmogorov complexity are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Indeed, for computable  distributions, Shannon entropy links to Kolmogorov complexity to within  a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' This result is discussed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=', in [11], [4], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' However, it is  shown in [15] that the general link fails for R´enyi and Tsallis entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' See  3  for example [4], [10], [15] for R´enyi and Tsallis entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The first connec-  tion between logical formula length and entropy has—to our knowledge—  been obtained in [8], [7], where expected Boltzmann entropy is shown to be  asymptotically equivalent to description complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Concerning further related work, we will next discuss the proof tech-  niques used in the current paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For proving bounds on formula sizes, we  use formula size games for the logics GMLUd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Indeed, variants of stan-  dard Ehrenfeucht-Fra¨ıss´e games and (graded) bisimulation games would not  suffice, as we need to deal with formula length, and thereby with all logi-  cal operators, including connectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The formula size games for the logics  GMLUd will be developed below based on a similar game used in [8], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  That game builds on the game for standard modal logic ML used and devel-  oped in [5] for proving a nonelementary succinctness gap between first-order  logic and ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The first formula size game, developed by Razborov in [13],  dealt with propositional logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' A later variant of the game was defined by  Adler and Immerman for CTL in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Designing the games for GMLUd is  relatively straightforward and based directly on similar earlier systems, but  using them requires some nontrivial combinatorial arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  In addition to games, we also make use of a range of techniques for esti-  mating model class sizes and description complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' These include Stirling’s  approximations and Chernoff bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' In particular, to obtain our results,  we prove new estimates on r-associated Stirling numbers, which may be of  independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  As a brief summary of our paper, the main objective is to elucidate the  general picture of how description length relates to model class size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' This  also builds links between logic and notions of entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The logic GMLU is  suitable for the current study, and it even allows simple access to a chain  of increasingly expressive logics GMLUd via increasing d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The concluding  section discusses possibilities for generalizing to further logics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' While the  current paper focuses on theory, the notion of description complexity is also  relevant in a range of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For example, in some currently active  research on explainability in AI, minimal length specifications can be used  as explanations of longer formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For work on this topic see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=', [2], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  The plan of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' After the preliminaries in Section  2, we prove crucial lower bounds for description complexity in Section 3  using games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' In Section 4 we prove a monotone connection between model  class size and description complexity, and in Section 5 we investigate phase  transitions of class size distributions by varying n and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Section 6 concludes  the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  4  2  Preliminaries  We first define the logics studied in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Let τ be a finite set of  proposition symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We consider τ to be fixed throughout the entire paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  The syntax of graded universal modal logic GMLU[τ] is generated as  follows (the syntactic choices will be explained later on):  ϕ :=♦≥kψ | ■<kψ | ♦=kψ | ■̸=kψ |  ϕ ∧ ϕ | ϕ ∨ ϕ | ♦≥kϕ | ■<kϕ | ♦=kϕ | ■̸=kϕ  ψ :=p | ¬p | ψ ∧ ψ | ψ ∨ ψ  Here p ∈ τ and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Note that the formulas of GMLU[τ] have proposition  symbols only in the scope of modal operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Furthermore, all formulas are  given in negation normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' In the current paper, ¬ϕ will always mean  a formula where ¬ has been pushed all the way to the level literals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Let M be a Kripke model with domain W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' In this paper, modal logics  will always have a unary vocabulary, so therefore Kripke models will not be  associated with a binary accessibility relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We define the semantics of  the global graded modalities as follows: (M, w) ⊨ ♦≥kϕ ⇔ there exist at  least d elements v ∈ W such that (M, v) ⊨ ϕ and (M, w) ⊨ ♦=kϕ ⇔ there  exist exactly d elements v ∈ W such that (M, v) ⊨ ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Additionally, (M, w) ⊨  ■<kϕ ⇔ (M, w) ⊨ ¬♦≥k¬ϕ and (M, w) ⊨ ■̸=kϕ ⇔ (M, w) ⊨ ¬♦=k¬ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The  semantics of the Boolean connectives ¬, ∧, ∨ is defined in the usual way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Notice that ♦≥k and ■<k as well as ♦=k and ■̸=k are dual to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Thus the modalities of GMLU are the diamonds ♦≥k, ♦=k and their duals  ■<k, ■̸=k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Intuitively ■<k (respectively, ■̸=k) means that all points satisfy  ϕ, except for some number m < k (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' m ̸= k) of exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Let M be a Kripke model over τ and ϕ ∈ GMLU[τ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The point-free truth  relation is defined such that M ⊨ ϕ if and only if M, w ⊨ ϕ for all w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  As all proposition symbols occur in the scope of a global modality, M ⊨  ϕ if and only if there exists some w ∈ W such that M, w ⊨ ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Clearly  truth of GMLU-formulas does not depend on the evaluation point w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' This  independence property is the reason behind the definition of the syntax of  GMLU such that proposition symbols are guaranteed to be in the scope of  modal operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  For a set M of pointed models, we denote M ⊨ ϕ ⇔  (M, w) ⊨ ϕ for every (M, w) ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  A propositional type π over τ is a maximally consistent set of literals  (that is, proposition symbols and negated proposition symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Therefore  π has exactly one of p, ¬p for each symbol p ∈ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We henceforth refer to  propositional types as just types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The number of types over τ is denoted by  t = 2|τ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  The counting depth of a formula ϕ ∈ GMLU[τ], denoted depth(ϕ), is  defined as follows:  depth(α) = 0 for any literal α,  5  depth(ϕ ∧ ψ) = depth(ϕ ∨ ψ)  = max(depth(ϕ), depth(ψ)),  depth(♦≥kϕ) = depth(■<kϕ) = k,  depth(♦=kϕ) = depth(■̸=kϕ) = k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We denote by GMLUd[τ] the counting depth d fragment of GMLU[τ],  where the counting depth of formulas is restricted to at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The re-  sults of this paper are formulated for the logics GMLUd[τ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Note that  depth(♦=kϕ) = k + 1 while depth(♦≥kϕ) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We give some intuition  to explain this choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' First of all, ♦=d−1ϕ ≡ ♦≥d−1ϕ ∧ ¬♦≥dϕ, so we see  that when the counting depth of formulas is restricted to d, the allowed  “exact counting” diamonds ♦=k always have k ≤ d − 1 and thus they add  no expressive power over “threshold counting” diamonds ♦≥k with k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Moreover, k + 1 also corresponds to the number of quantifiers required to  express “exact counting” of k elements in monadic first-order logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  The size of a formula ϕ ∈ GMLU[τ], denoted size(ϕ), is defined as  follows:  size(α) = 1 for any literal α,  size(ϕ ∧ ψ) = size(ϕ ∨ ψ) = size(ϕ) + size(ψ) + 1,  size(♦≥kϕ) = size(■<kϕ) = size(ϕ) + k,  size(♦=kϕ) = size(■̸=kϕ) = size(φ) + k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Note that all literals have the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' This is because we wish to consider  negative (that is, negated) information and positive (that is, non-negated)  information as equal in relation to formula size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' This idea explains why we  defined GMLU so that formulas are in negation normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Let M be the set of all τ-models with the fixed domain W = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  When M is clear from the context, a formula ϕ ∈ GMLUd is said to define  a set M ⊆ M if for every M ∈ M, we have M ⊨ ϕ if and only if M ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  The set M is then called GMLUd-definable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  The GMLUd-description  complexity C(M) of a GMLUd-definable set M is the minimum size of a  formula ϕ ∈ GMLUd which defines M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We may write M ≡d N if the models M and N satisfy exactly the same  GMLUd-formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The relation ≡d is clearly an equivalence relation and  defines a natural related partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For an example of description complexity,  consider GMLU1[τ] for the case with the singleton alphabet τ = {p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The  model where p is true in every point constitutes a singleton class in the  partition of models defined by ≡1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The description complexity of this class  is 2 as a minimum size formula that defines the class is ■<1p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Let I := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , t} and fix an enumeration (πi)i∈I of all the types over  the propositional vocabulary τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Now let M be an equivalence class of ≡d  6  over the set M of models of size n with domain W = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For a type  πi, all models in the class either have exactly ni points of type πi for some  ni < d, or all the models in M have at least d points of type πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' In the latter  case we define ni := d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We thus get a characterization of the classes of ≡d  in terms of t-tuples n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' A t-tuple n = (ni)i∈I is called (n, d)-admissible, if  ni ≤ d for every i ∈ I, �  i∈I ni ≤ n and either there is at least one i ∈ I  such that ni = d or �  i∈I ni = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Note that (n, d)-admissible tuples n and  equivalence classes of ≡d are in one-to-one correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Thus we can  write Mn for the class corresponding to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Given an (n, d)-admissible tuple n and a type πi that has precisely the  same number k of realizing points in every model M ∈ Mn, we denote this  number k by |πi|n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' (Note that k can be greater than d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=') We will often omit  the tuple n in the subscript of |πi|n when it is clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Following [7], we define the Boltzmann entropy of a class M as  HB(M) := log(|M|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' As discussed in [7], this terminology comes from sta-  tistical mechanics, where Boltzmann entropy measures the randomness of a  macrostate via the number of microstates that correspond to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The idea  is that a larger macrostate is “more random” (or “less specific”) since it  is more likely to be hit by a random selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  In statistical mechanics,  the formula for Boltzmann entropy is kB ln Ω, where Ω is the number of  microstates and kB the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' In our definition, we use the  binary logarithm (and do not use kB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' As a general intuition, it is natural  to associate a formula ϕ (or the class it defines) with a macrostate, while  the models of ϕ are then the corresponding microstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Consider now the following natural probability distribution over the equi-  valence classes of ≡d: p≡d(M) = |M|/|M| for each class M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We again refer  to this distribution with the symbol ≡d (with slight abuse of notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We  define the Boltzmann entropy of the distribution ≡d as the expected  value of HB over the distribution ≡d and we denote it by HB(≡d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' In other  words, we define HB(≡d) as �  M∈M/≡d p≡d(M)HB(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Roughly speaking,  HB(≡d) is large when ≡d is far from the uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  The Boltzmann entropy of the distribution ≡d is closely related to the  Shannon entropy HS(≡d) of ≡d, which we define as the expected value of  − log(p≡d(M)) over the distribution ≡d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' More explicitly, we define HS(≡d)  as − �  M∈M/≡d p≡d(M) log(p≡d(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  In contrast to Boltzmann entropy,  Shannon entropy measures randomness of ≡d by looking at how uniform the  distribution is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Indeed, if ≡d contains a very large class, then its Shannon  entropy is small, while its Boltzmann entropy is relatively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The follow-  ing result, which was shown in [7] in a more general setting (with slightly  different notation), formally establishes that the two notions of entropy are  complementary in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' HS(≡d) + HB(≡d) = |τ|n  7  3  Description complexity  In this section we investigate the GMLUd-description complexity of equi-  valence classes in the partition ≡d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We will utilize a formula size game for  GMLUd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Let I = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , t} and let (πi)i∈I be an enumeration of the types of  the set τ of proposition symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let d ≤ n be the counting depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We  consider the partition induced by GMLUd for models of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For each  admissible tuple n = (ni)i∈I there is an equivalence class Mn, where each  type πi realized ni times, with ni = d meaning the type πi is realized at least  d times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We denote the number of occurrences of d in n by kd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Such a class Mn can be defined via the following GMLUd formula:  ϕ(n) :=  �  ni<d  ♦=niψ(πi) ∧  �  ni=d  ♦≥dψ(πi)  The size of the formula ϕ(n) is �  i∈I ni + t(2|τ| + 1) − kd − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  If ni = d for at most one i ∈ I, then Mn can be defined by a smaller  formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let πj be a type with maximal nj in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Now Mn is also defined  by the formula  ϕ′(n) :=  �  i̸=j  ♦=niψ(πi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  To see this, recall that we restrict to models of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' As all other types  have been exactly specified, the only option for the remaining points is the  type πj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The size of ϕ′(n) is n−|πj|+(t−1)(2|τ|+1)−2, where |πj| denotes  the number of points in models of Mn with type πj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We define the constant cτ := t(2|τ| + 1) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' By the formulas above we  see that C(Mn) ≤ �  i∈I ni+cτ if kd ≥ 2 and C(Mn) ≤ n−|πj|+cτ if kd ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We will use the formula size game for GMLUd to show that these bounds  are optimal up to the constant cτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Let r0 ∈ N and let A0, B0 be sets of τ-models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The GMLUd-formula size  game GAMEd(r0, A0, B0) has two players, S and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Positions of the game  are of the form P = (r, A, B) and the starting position is P0 = (r0, A0, B0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  In a position P, if r = 0, then D wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Otherwise S chooses between the  following moves:  p-move: S chooses a τ-literal α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The game ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' If A ⊨ α and B ⊨ ¬α, then  S wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Otherwise D wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' S cannot make this move if he has not made a  modal move so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  ∨-move: S chooses A1, A2 ⊆ A such that A1 ∪ A2 = A and r1, r2 ≥ 1 such  that r1 + r2 + 1 = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' D chooses whether the next position is (r1, A1, B) or  (r2, A2, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  ∧-move: The same as a ∨-move with the roles of A and B switched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  8  ♦≥d-move: S chooses a number k ∈ N with k ≤ d and k < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For every  (M, w) ∈ A, S chooses k different points v ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let A′ be the set of models  (M, v) chosen this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For every (M, w) ∈ B, S chooses n − k + 1 different  points v ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let B′ again be the set of models chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The next position  of the game is (r − k, A′, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  ■<d-move: The same as a ♦≥d-move with the roles of A and B switched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  ♦=d-move: S chooses a number k ∈ N with k < d and k < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For every  (M, w) ∈ A, S chooses a set PM,w of k different points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Let NM,w :=  W \\ PM,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For every (M, w) ∈ B, S chooses either a set PM,w of k + 1  different points or a set NM,w of |M| − k + 1 different points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Finally we let  A′ := {(M, v) | (M, w) ∈ A ∪ B, v ∈ PM,w} and B′ := {(M, v) | (M, w) ∈  A ∪ B, v ∈ NM,w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The next position of the game is (r − k − 1, A′, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  ■̸=d-move: The same as a ♦=d-move with the roles of A and B switched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  The formula size game characterizes the size of formulas that separate  model classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' This is formalized in the following theorem:  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The following statements are equivalent:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' S has a winning strategy in the game GAMEd(r, A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' There is ϕ ∈ GMLUd[τ] with size at most r such that A ⊨ ϕ and  B ⊨ ¬ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Easy proof by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' See [5] for a version of the proof for basic  modal logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  It will be useful for the proofs below to note that if (essentially) the same  model is on both sides of the game, then D has an easy winning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let P = (r, A, B) be a position of a game GAMEd(r0, A0, B0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Let there be propositionally equivalent versions (M, w) ∈ A and (M, v) ∈ B  of the same model M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Now D has a winning strategy from position P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' It is easy to see that the pair of propositionally equivalent versions  of the same model is maintained through any modal move of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For ∨-moves  and ∧-moves one of the two possible following positions will always have  such a pair of models and the strategy of D is to choose this position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Let n be an admissible tuple and assume that n1 is one of the largest  coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We first define the sets An and Bn of models for the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' All  models have the same universe W = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We denote supp(n) := {i ∈  I | ni > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We first define a model M0 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For each i ∈ I, i ̸= 1, the  type πi is realized precisely ni times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The type π1 is realized n − �  i̸=1 ni  times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Additionally, the point 1 is of type π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Intuitively the model M0 is a  model of the class Mn, where all points not fixed by the tuple n are of type  π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We set An := {(M0, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  9  For the set Bn we define models Mi→j for some pairs (i, j) ∈ supp(n) ×  supp(n) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For i ̸= 1 and any j ∈ supp(n), the model Mi→j is  obtained from the model M0 by changing one point of type πi to type πj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  If i = 1 and nj = d, then the model Mi→j is obtained from M0 by changing  |π1|M0 − d + 1 points w ̸= 1 of type π1 to type πj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' If i = 1 and nj < d,  no model is defined for the pair (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We set Bn := {(Mi→j, 1) | i, j ∈  supp(n)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  In terms of their tuples n′, the models Mi→j have n′  i = ni − 1 and  n′  j = nj + 1, except if nj = d, in which case n′  j = nj = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For all other  indices ℓ, n′  ℓ = nℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' All models (M0, 1) and (Mi→j, 1) are propositionally  equivalent as they realize the type π1 in the point 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Let P = (r, A, B) be a position of the game GAMEd(r, An, Bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We  define a directed graph G(A, B) := (V, E), where V = supp(n) and (i, j) ∈ E  if there are propositionally equivalent (M0, w) ∈ A and (Mi→j, v) ∈ B, or  vice versa with respect to A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We call a set C ⊆ supp(n) a cover of  G(A, B) if for every (i, j) ∈ E, we have that either i ∈ C or both j ∈ C and  nj < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The cost r(C) of a cover C is  r(C) :=  �  i∈C  ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Intuitively, the definition of a cover means that including an index i ∈ I  generally covers all edges to and from i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The exception is that when ni = d,  incoming edges are not covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  In the proof of the following lemma we use the notation Md(A) = {M |  (M, w) ∈ A, w ∈ W} for the set of models that have at least one pointed  version in the set A of pointed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We also denote by tp(A) the set of  propositional types realized in the set A of points in a model M that will  below be clear from the set of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let P = (r, A, B) be a position of the game GAMEd(r, An, Bn)  and let  R(P) := min{r(C) | C is a cover of G(A, B)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  If r < R(P), then D has a winning strategy from position P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We show for all possible moves of S that either the condition r <  R(P) is maintained or D has a winning strategy for some other reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Since the definition of the graph G(A, B) is symmetrical with respect to A  and B, we only need to handle one of each pair of dual moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  p-move: Since 0 < r < R(P), we have propositionally equivalent models on  both sides of the game and thus D wins if S makes any p-move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  ∨-move: Let A1, A2 ⊆ A and r1, r2 ≥ 1 be the choices of S and let P1 =  (r1, A1, B) and P2 = (r2, A2, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  For each (i, j) ∈ E there is a pair of  10  propositionally equivalent models (M, w) ∈ A and (M′, v) ∈ B as witnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Since A1∪A2 = A, each model (M, w) ∈ A is in A1 or A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The set B remains  unchanged so for each (i, j) ∈ E we have (i, j) ∈ E1 or (i, j) ∈ E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Now if  r1 ≥ R(P1) and r2 ≥ R(P2), then there is a minimal cover C1 of position P1  with r(C1) ≤ r1 and the same for P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Now the set C := C1 ∪ C2 is a cover  of position P with r(C) ≤ r1 + r2 < r, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Therefore  we have r1 < R(P1) or r2 < R(P2) and D can maintain the condition by  choosing such a position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  ♦≥d-move: Let k ≤ d be the number chosen by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The following position  is P ′ = (r − k, A′, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We first note that if M0 ∈ Md(A) ∩ Md(B), then S  must choose k points from the version in A and n − k + 1 points from the  version in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Thus the next position P ′ will have propositionally equivalent  versions (M0, w) ∈ A′ and (M0, v) ∈ B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='2 this gives D a  winning strategy so we assume M0 is only present on one side of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Case A: M0 ∈ Md(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For each (M0, w) ∈ A, S chooses a set PM0,w of k  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let PM0 := �  w∈W PM0,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We consider the following cases:  1) We have �  πi∈tp(PM0) ni > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let (i, j) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Since the model Mi→j  only differs from M0 by n′  i = ni − 1 and n′  j ≥ nj, the model Mi→j has  at least k points with types from tp(PM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Thus, when S chooses the set  NMi→j of n − k + 1 points, it contains at least one point with a type from  tp(PM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Thus the pair of propositionally equivalent models is maintained  and (i, j) ∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  2) We have �  πi∈tp(PM0) ni = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let (i, j) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Assume πi /∈ tp(PM0) or  πj ∈ tp(PM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Now the model Mi→j has at least k points with types from  tp(PM0) and (i, j) ∈ E′ as in case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Assume then that πi ∈ tp(PM0) and πj /∈ tp(PM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' All edges of this kind  being eliminated is an acceptable worst case, so we assume that (i, j) /∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Summing up Case A, the worst case is that S chooses a set tp(PM0)  of types with �  πi∈tp(PM0) ni = k ≤ d and eliminates all edges with πi ∈  tp(PM0) and πj /∈ tp(PM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Case B: M0 ∈ Md(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For each (M0, w) ∈ B, S chooses a set NM0,w of  n − k + 1 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let NM0 := �  w∈W NM0,w and let Π be the complement of  tp(NM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We note that ni < d for each πi ∈ Π since M0 only has at most  k − 1 points with types from Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Let (i, j) ∈ E and assume πi ∈ Π or πj /∈ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Now πi /∈ tp(NM0)  or πj ∈ tp(NM0) so the model Mi→j has at least n − k + 1 points with  types from tp(NM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Thus any set PMi→j,v of k points contains at least one  point with a type from tp(NM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Thus the pair of propositionally equivalent  models is maintained and (i, j) ∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Now assume πi /∈ Π and πj ∈ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' If Mi→j has at least n − k + 1 points  with types from tp(NM0), then (i, j) ∈ E′ as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We thus assume that  Mi→j has less than n − k + 1 points with types from tp(NM0), meaning it  11  has at least k points with types from Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Since nj < d and Mi→j differs from  M0 only by n′  i = ni − 1 and n′  j = nj + 1, the only remaining option is that  Mi→j has exactly k points with types from Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Thus by the same reasoning  M0 has exactly k − 1 points with types from Π and since k − 1 < d, we have  �  i∈Π ni = k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We again accept S eliminating these edges as a worst case  and assume (i, j) /∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Summing up Case B, the worst case is that S chooses a set Π of types  with �  πi∈Π ni = k −1 < d and eliminates all edges with πi /∈ Π and πj ∈ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We now consider the condition r − k < R(P ′) in the following position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  From the above arguments we see that the only way for S to eliminate  edges moving from G(A, B) to G(A′, B′) is to choose in each model (M0, w)  or (Mi→j, v) in A exactly all points that satisfy some set Π of types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' If  M0 ∈ Md(A), we have �  πi∈Π ni = k ≤ d and S can eliminate all edges from  types in Π to other types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' If M0 ∈ Md(B), we have �  πi∈Π ni = k − 1 and  ni < d for all πi ∈ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' In this case S can eliminate all edges from other types  to types in Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' In both cases, the set C = {i ∈ supp(n) | πi ∈ Π} covers all  eliminated edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The cost of C is r(C) = �  i∈C ni ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let C′ be a cover  of P ′ with minimal cost so r(C′) = R(P ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Now C ∪ C′ is a cover of P so  r < R(P) ≤ R(P ′) + r(C) ≤ R(P ′) + k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Thus r − k < R(P ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  ♦=d-move: Let k < d be the number chosen by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The following position  is P ′ = (r − k − 1, A′, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' As for the ♦≥d-move, we may assume that M0 is  only present on one side of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Case A: M0 ∈ Md(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For each (M0, w) ∈ A, S chooses a k point set  PM0,w and NM0,w = W \\ PM0,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We denote PM0 := �  w∈W PM0,w and  NM0 := �  w∈W NM0,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We consider the following cases:  1) We have �  πi∈tp(PM0) ni > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Thus there are propositionally equivalent  w ∈ PM0 and v ∈ NM0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' This means that in the following position P ′ there  are propositionally equivalent versions of the model M0 on both sides of the  game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' D has a winning strategy from P ′ by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  2) We have �  πi∈tp(PM0) ni = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Note that since k < d, also ni < d for all  πi ∈ tp(PM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let (i, j) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Assume πi, πj /∈ tp(PM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Now the model Mi→j has exactly k points  with types from tp(PM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Thus if S chooses a k + 1 point set PMi→j, then  one of those points v′ has a type πℓ /∈ tp(PM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' By the definition of the  models, nℓ ̸= 0 so there is a point w′ in the model M0 of type πℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Now  there are propositionally equivalent (Mi→j, v′) ∈ A′ and (M0, w′) ∈ B′ so  (i, j) ∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Similarly, if S chooses for Mi→j a set NMi→j of n − k + 1 points, then  this set contains at least one point v′ of a type πℓ ∈ tp(PM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Now there  is a point w′ ∈ PM0 of type πℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Thus there are propositionally equivalent  (M0, w′) ∈ A′ and (Mi→j, v′) ∈ B′ so (i, j) ∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  If πi ∈ tp(PM0) or πj ∈ tp(PM0), we assume (i, j) /∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' As for the  12  ♦≥d-move, this is an acceptable worst case for our lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Summing up Case A, the worst case is that S chooses a set tp(PM0) of  types with �  πi∈tp(PM0) ni = k < d and eliminates all edges (i, j) ∈ E with  πi ∈ tp(PM0) or πj ∈ tp(PM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Case B: M0 ∈ Md(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For each (M0, w) ∈ B, S chooses either a set PM0,w  of k + 1 points or a set NM0,w of n − k + 1 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' There are three cases:  1) Assume first that there are (M0, w), (M0, w′) ∈ B with sets PM0,w and  NM0,w′ chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Since k + 1 + n − k + 1 > n, there is v ∈ PM0,w ∩ NM0,w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' In  the following position P ′ the model (M0, v) is on both sides of the game so  by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='2 D has a winning strategy from P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  2) Assume then that S chooses a set PM0,w of k+1 points for each (M0, w) ∈  B and let PM0 = �  w∈W PM0,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let (i, j) ∈ E and let (Mi→j, v) ∈ A and  (M0, w) ∈ B be the corresponding models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Assume πi /∈ tp(PM0) or πj ∈ tp(PM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Now the model Mi→j has at  least k+1 points with types from tp(PM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Thus the set NMi→j,v of size n−k  has at least one point with a type from tp(PM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Thus the propositionally  equivalent pair of models is maintained and (i, j) ∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Assume πi ∈ tp(PM0) and πj /∈ tp(PM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' If Mi→j has at least k+1 points  with types from tp(PM0), the above argument again works and (i, j) ∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Since k < d and Mi→j only differs from M0 by n′  i = ni − 1 and n′  j ≤ nj + 1,  the only remaining option is that Mi→j has exactly k points with types from  tp(PM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We obtain �  i∈tp(PM0) n′  i = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We again assume as a worst case  that (i, j) /∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  3) Finally assume that S chooses a set NM0,w of n − k + 1 points for each  (M0, w) ∈ B and let NM0 = �  w∈W NM0,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let Π be the complement of  tp(NM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Assume πi ∈ Π or πj /∈ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Now πi /∈ tp(NM0) or πj ∈ tp(NM0) so the  model Mi→j has at least n−k+1 points with types from tp(NM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Thus the  set PMi→j,v of size k has at least one point with a type from tp(NM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Thus  the propositionally equivalent pair of models is maintained and (i, j) ∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Assume πi /∈ Π and πj ∈ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' If Mi→j has at least n − k + 1 points with  types from tp(NM0), the above argument again works and (i, j) ∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We assume Mi→j has less than n−k+1 points with types from tp(NM0),  meaning it has at least k points with types from Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Since k < d and Mi→j  only differs from M0 by n′  i = ni − 1 and n′  j ≤ nj + 1, the only remaining  option is that Mi→j has exactly k points with types from Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We obtain  �  i∈Π n′  i = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We again assume as a worst case that (i, j) /∈ E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Edges  eliminated this way are ones from other types to types in Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' In particular,  we note that for any edge (i, j) eliminated this way, nj < d since M0 has at  most n − (n − k + 1) = k − 1 points of type πj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Summing up Case B, the worst case is that S chooses a set Π of types  and eliminates either all edges (i, j) ∈ E with πi ∈ Π or πj /∈ Π or all edges  13  (i, j) ∈ E with πi /∈ Π or πj ∈ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' In both cases �  i∈Π n′  i = k for the model  Mi→j and for the latter case nj < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We now consider the condition r−k−1 < R(P ′) in the following position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  From the above arguments, we see that the only way for S to eliminate  edges moving from G(A, B) to G(A′, B′) is to choose in each model (M0, w)  or (Mi→j, v) in A exactly all points that satisfy some set Π of types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' If  M0 ∈ Md(A), we have �  πi∈Π ni = k < d and S can eliminate all edges to  and from the set Π of types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' If M0 ∈ Md(B), we have �  πi∈Π n′  i = k < d and  S can eliminate either all edges to or all edges from the set Π of types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' In  particular, if nj = d for πj ∈ Π, then only outgoing edges can be eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  In all cases, the set C = {i ∈ supp(n) | πi ∈ Π} covers all eliminated edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  The cost of C is r(C) = �  i∈C ni ≤ k + 1 since �  i∈C ni ≤ �  i∈C n′  i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Let C′ be a cover of P ′ with minimal cost so r(C′) = R(P ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Now C ∪ C′  is a cover of P so r < R(P) ≤ R(P ′) + r(C) ≤ R(P ′) + k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Thus  r − k − 1 < R(P ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  A lower bound for the description complexity of an arbitrary class Mn  can now be obtained by simply calculating the minimum cost of a cover of  G(An, Bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let Mn be a class with ni = d for at least two different i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Then C(Mn) ≥ �  i∈I  ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We assume that n1 = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We consider the graph G(An, Bn) = {(i, j) ∈  supp(n) | i ̸= 1 or nj = d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' This graph has edges from all types πi with  ni ̸= 0 to each other, with the exception that there are no edges from π1 to  πj with nj < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We first note that C = supp(n) is a cover of G(An, Bn) with cost �  i∈I ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  For C′ ̸= C if i /∈ C′ with i ̸= 1, then by the definition of a cover, the edge  (i, 1) ∈ E is not covered, since i /∈ C and n1 = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' If 1 /∈ C′, then let j ∈ I  be the other index with nj = d besides 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Now the edge (1, d) ∈ E is not  covered since 1 /∈ C and nj = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We see that C is a minimal cost cover and  by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='3 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='1 the claim holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let Mn be a class with ni = d for at most one i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let πj  be one of the types with the most realizing points in Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Then C(Mn) ≥  �  i∈I\\{j} ni = n − |πj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We assume n1 = max{ni | i ∈ I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Thus π1 is one of the types with  the most realizing points and if n1 < d, then ni < d for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We consider the graph G(An, Bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' From the definition we get G(An, Bn) =  {(i, j) ∈ supp(n) | i ̸= 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' This graph has edges from all types πi with ni ̸= 0  to each other, with the exception that there are no edges originating from  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  14  We first note that C = supp(n)\\{1} is a cover of G(An, Bn) with r(C) =  �  i∈I\\{1} ni = n − |π1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Clearly supp(n) is a cover with higher cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Let C′ ⊂ supp(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' If there are i, j ∈ supp(n) with i, j /∈ C′, then the  edge (i, j) or (j, i) is in E and is not covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' If there is only one i ∈ supp(n)  with i /∈ C′ and i ̸= 1, then r(C′) = �  i∈I\\{j} ni ≥ r(C) since π1 is one of  the types with the largest ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We see that C is a minimal cost cover and by  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='3 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='1 the claim holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We sum up the above lemmas into the Theorem below:  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let n be an (n, d)-admissible tuple and let Mn be the cor-  responding equivalence class of ≡d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' If ni = d for at least two i ∈ I, then  C(Mn) ≥ �  i∈I ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Otherwise C(Mn) ≥ �  i∈I\\{j} ni = n − |πj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  4  A monotone connection  Consider the following strict partial orders on the set of all (n, d)-admissible  tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' n <s n′ iff |Mn| < |Mn′|  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' n <c n′ iff C(Mn) < C(Mn′)  Note that neither |Mn| = |Mn′| nor C(Mn) = C(Mn′) necessarily entails  that n = n′, as demonstrated by the tuples (1, 2, d) and (2, 1, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We let ≤s  and ≤c denote the partial orders obtained by adding loops to <s and <c  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The main purpose of this section is to show that there exists  a natural and non-trivial partial order which is contained both in ≤s and  in ≤c (provided that n is sufficiently large w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' d), which then gives us a  monotone connection between sizes of model classes and their description  complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' To establish this result, we will use the heavy machinery devel-  oped in the previous section together with further combinatorial arguments  that involve r-associated Stirling numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='1  r-associated Stirling numbers  We will use r-associated Stirling numbers to count the number of models in  a given model class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Given positive integers n, m, r such that n ≥ mr, we  define  �n  m  �  ≥r  to be the number of partitions of [n] which partition [n] into m, each set  having size at least r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' When r = 1 these numbers are also known as the  Stirling numbers of the second kind and they simply count the number of  partitions of [n] into m sets [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  15  The following lemma will play the key role when we estimate the sizes  of the model classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Even though the proof is simple and elementary, we  were not able to find these estimates in the existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let n, m, r ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Suppose that n ≥ mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Then  mn  mmr ≤  �n  m  �  ≥r  ≤ mn  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For the upper bound note that  �n  m  �  ≥r  ≤  �n  m  �  ≥1  ≤ mn  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  The last inequality follows from the fact that mn counts the number of  mappings f : [n] → [m], while m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' ·  � n  m  �  ≥1 only counts those f that are  surjections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For the lower bound, we use the fact that m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='·  � n  m  �  ≥r counts the  number of mappings f : [n] → [m] which have the property that for every  1 ≤ k ≤ m we have |f −1({k})| ≥ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' To give a lower bound on the number  of such functions, we count the number of mappings f : [n] → [m] which  have the property that the sets {(k − 1) · r + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , k · r}, where 1 ≤ k ≤ m,  are mapped to distinct elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Since the remaining n − mr elements can  be mapped arbitrarily, the number of such mappings is simply m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' · mn−mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Thus  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' · mn−mr ≤ m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' ·  �n  m  �  ≥r  ,  giving the wanted lower bound after dividing by m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='.  We note that if m and r are much smaller than n — as they will be in  our applications — the estimates of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='1 are (perhaps surprisingly)  quite sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  One important consequence of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='1 is the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Again,  we emphasize that we were unable to find an estimate of this form in the  existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let n, m, r ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' If n ≥ mr + 1, then  �n  m  �  ≥r  ≤ mmr+1  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  �n − 1  m  �  ≥r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Follows immediately from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  If m and r are fixed, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='2 bounds the growth rate of r-associated  Stirling numbers as n increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  16  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='2  Combinatorics of model classes  In this subsection we use the above results on r-associated Stirling numbers  to investigate the sizes of model classes in terms of their (n, d)-admissible  tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We begin with the following lemma gives a simple and closed formula  for the size of a model class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let n be an (n, d)-admissible tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , ik ∈ I be the  indices for which niℓ < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We then have that  |Mn| =  �  n  ni1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , nik, m  �  kd!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' ·  �m  kd  �  ≥d  ,  where m := n − �  i̸∈{i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=',ik} ni and kd := t − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Each model in Mn can be constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  (1) We first pick k subsets of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , n} of sizes ni1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , nik and define that  each element in the iℓth set realizes the iℓth type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The number of ways this  can be done is given by  �  n  ni1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=',nik,m  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  (2) We then partition the remaining subset of size m to kd pieces, each piece  having size at least d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The number of ways this can be done is given by the  associated Stirling number  �m  kd  �  ≥d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  (3) For each piece we select a unique type from the remaining types — the  number of which is kd — and define that each element in a piece realizes  the type associated with that piece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The number of ways this can be done  is given by kd!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='.  Multiplying the above factors gives us the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  The following lemmas establish how the size of a model class changes  when we modify its admissible tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Fix d ∈ Z+ and let n be an (n, d)-admissible tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Suppose  that n is sufficiently large with respect to d and |τ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Then for every i ∈ I  such that ni < d − 1 we have that  |Mn| < |Mn′|,  where n′  i = ni + 1 and n′  j = nj, for every j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For notational simplicity we assume that nℓ < d iff ℓ ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' By Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='3 the inequality that we need to establish is  �  n  n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , nk, m  �  kd!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' ·  �m  kd  �  ≥d  <  �  n  n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , ni + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , nk, m − 1  �  kd!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' ·  �m − 1  kd  �  ≥d  ,  17  where m = n − �k  ℓ=1 nℓ and kd = t − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Note that since n is large enough  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' d, kd ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Simplifying this gives us the equivalent inequality  �m  kd  �  ≥d  <  m  ni + 1  �m − 1  kd  �  ≥d  ,  which follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='2, as long as  m  ni + 1 > kkdd+1  d  kd!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  ⇔ n > (ni + 1)kkdd+1  d  kd!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  +  k  �  ℓ=1  nℓ  Using nℓ < d, which holds for every 1 ≤ ℓ ≤ k, and 1 ≤ kd ≤ t gives us the  desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Fix d ∈ Z+ and let n be an (n, d)-admissible tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Suppose  that n is sufficiently large with respect to d and |τ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Then for every i ∈ I  such that ni < d we have that  |Mn| < |Mn′|,  where n′  j = d, when j = i, and n′  j = nj otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For notational simplicity we assume that nℓ < d iff ℓ ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' By Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='3 the inequality that we need to establish is  �  n  n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , ni, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , nk, m  �  kd!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' ·  �m  kd  �  ≥d  <  �  n  n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , ni−1, ni+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , nk, m + ni  �  (kd + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' ·  �m + ni  kd + 1  �  ≥d  ,  where m = n − �k  ℓ=1 nℓ and kd = t − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Note that since n is large enough  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' d, kd ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Simplifying this gives us the equivalent inequality  �m  kd  �  ≥d  <  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='ni!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  (m + ni)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' · (kd + 1) ·  �m + ni  kd + 1  �  ≥d  It follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='1 that  �m + ni  kd + 1  �  ≥d  ≥ (kd + 1)m+ni  (kd + 1)(kd+1)d  and  �m  kd  �  ≥d  ≤ km  d  kd!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  18  Hence we only have to show that  km  d  kd!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' <  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='ni!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  (m + ni)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' · (kd + 1) · (kd + 1)m+ni  (kd + 1)(kd+1)d  or equivalently that  (kd + 1)(kd+1)d  (kd + 1)ni+1kd!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='ni!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' < m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' (kd + 1)m  km  d (m + ni)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  =  (kd + 1)m  km  d  �ni−1  j=0 (m + ni − j)  =  � kd + 1  kd  � �� �  >1  �m� ni−1  �  j=0  (m + ni − j)  Recall 0 ≤ ni < d and 1 ≤ kd ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Hence in the above inequality the left  hand side is a constant, and thus it suffices to show that the right hand side  formula tends to infinity as n grows (recall m = n−�k  ℓ=1 nℓ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=', m is just n  minus a constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' However, this is clear, since the right hand side is of the  form f(n)/g(n), where f grows exponentially w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' n while g grows only  polynomially w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='3  Connecting size and description complexity  Let Pn,d denote the set of all (n, d)-admissible tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We define a natural  partial order ⪯ on Pn,d as follows: n ⪯ n′ if and only if ni ≤ n′  i, for every  i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' By writing n ≺ n′ we mean that n ⪯ n′ and n ̸= n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Suppose that n is sufficiently large with respect to d and |τ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Let n and n′ be (n, d)-admissible tuples such that n ≺ n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Then  |Mn| < |Mn′|  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' It suffices to show that if n′ is an immediate successor of n, then  |Mn| < |Mn′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Now, there must exist exactly one i ∈ I such that n′  i = ni + 1  and n′  j = nj for every j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' If n′  i < d, then the claim follows from Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='4, while if n′  i = d, then the claim follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Recall the constant cτ = 2|τ|(2|τ| + 1) − 1 from the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We  define yet another partial order ⪯τ on Pn,d as follows: n ⪯τ n′ if and only  if the following two conditions hold:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For every i ∈ I we have that ni ≤ n′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Either n = n′ or �  i∈I(n′  i − ni) > cτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  19  Roughly speaking n ⪯τ n′ means that if the tuples are distinct, then the  distance between them w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' to the order ⪯ is more than cτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Again, by  writing n ≺τ n′ we mean that n ⪯τ n′ and n ̸= n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For example (1, d) ≺τ  (2 + cτ, d), but (1, d) ̸⪯τ (1 + cτ, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Suppose that n is sufficiently large with respect to d and |τ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Let n and n′ be (n, d)-admissible tuples such that n ≺τ n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Then we have  that  C(Mn) < C(Mn′)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Suppose first that d occurs at least twice in n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='6 entails  that C(Mn′) ≥ �  i∈I n′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We now have two cases based on whether or not d  occurs at least twice in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Suppose first that d occurs exactly once in n say,  nj = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Then Mn can be defined by a formula of size  cτ +  �  i∈I−{j}  ni <  �  i∈I−{j}  n′  i ≤ C(Mn′)  and hence C(Mn) < C(Mn′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' On the other hand, if d occurs at least twice  in n, then Mn can be defined by a formula of size  cτ +  �  i∈I  ni <  �  i∈I  n′  i ≤ C(Mn′)  and hence C(Mn) < C(Mn′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Suppose then that d occurs exactly once in n′, say n′  j = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='6  entails that C(Mn′) ≥ �  i∈I−{j} n′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Since n ≺τ n′, we have that ni < d, for  every i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Since d must occur at least once in n — as n is (n, d)-admissible  — we have that nj = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Now Mn can be defined by a formula of size  cτ +  �  i∈I−{j}  ni <  �  i∈I−{j}  n′  i ≤ C(Mn′)  and hence C(Mn) < C(Mn′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Recall the partial orderings ≤s and ≤c on Pn,d, which were introduced at  the beginning of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The following theorem formalizes a connection  between sizes of model classes and their description complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' If n is sufficiently large with respect to d, then  ⪯τ ⊆ ≤s ∩ ≤c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  In particular, if n and n′ are two distinct ⪯τ-comparable tuples, then  |Mn| < |Mn′| ⇔ C(Mn) < C(Mn′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  20  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Suppose that n ≺τ n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='7 guarantee that if n is  large enough w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' d, then |Mn| < |Mn′| and C(Mn) < C(Mn′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Hence  n ≤s n′ and n ≤c n′, which proves the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The second claim follows  directly from the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  The intuitive content of the above theorem is that the partial order ⪯τ  approximates both ≤s and ≤c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Hence ⪯τ can be viewed as a highly non-  trivial monotone connection between ≤s and ≤c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We note that the above theorem also works for Boltzmann entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' This  is because the order ≤s is the same as the order based on Boltzmann entropy,  since logarithm is an increasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We formulate the latter of the two  claims as a corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Assume n is sufficiently large with respect to d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' If n and n′  are two distinct ⪯τ-comparable tuples, then  HB(Mn) < HB(Mn′) ⇔ C(Mn) < C(Mn′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  5  The phase transitions of class size distributions  In this section we move our attention from single classes to the entire pro-  bability distribution given by ≡d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' By allowing d to depend on n, we obtain  qualitative results which link the growth rate of d with the emergence of a  dominating class, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=', a class which contains almost all the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For fixed  n, we also obtain quantitative results on how the relationship between d and  n determines whether there exists a class in ≡d which contains majority of  all the models of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We will also point out consequences of these results  on the “average-case” expressive power of GMLUd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Throughout this section we continue to use our previous convention that  t = 2|τ| denotes the number of types π of the alphabet τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We start with the  following observation, which gives a sense of what happens in the distribu-  tion, when the counting depth is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Suppose that d < d′ ≤ n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Then  HS(≡d) < HS(≡d′) and HB(≡d) > HB(≡d′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Furthermore, for every n/2 ≤ d ≤ d′ we have that  HS(≡d) = HS(≡d′) and HB(≡d) = HB(≡d′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='1 HS(≡d) + HB(≡d) = |τ|n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Hence it suffices to  establish the claims in the case of Boltzmann entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We first note that if  n/2 ≤ d ≤ d′, then HB(≡d) = HB(≡d′), since the logic GMLUd can already  specify each structure up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  21  Suppose then that d < d′ ≤ n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Consider the class Mn, where n =  (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , 0, d, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' As the depth is increased to d′ > d, this class is divided to  at least two smaller classes with tuples (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , 0, d, d′) and (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , 0, d′, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Meanwhile, clearly no class increases in size so we see that HB(≡d) >  HB(≡d′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We take this opportunity to point out an easy corollary of the previous  result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' In [7] it was established that HB(≡n) ∼ |τ|n, by which we mean  that limn→∞ HB(≡n)/|τ|n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' When combined with Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='1, this  result yields quite directly the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For any counting depth d(n), we have  HB(≡d(n)) ∼ |τ|n  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' we assume that d(n) ≤ n, for every n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Since log(|M|) ≤ |τ|n,  for any class M, we have that HB(≡d(n)) ≤ |τ|n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Since HB(≡n) ∼ |τ|n, for  every ε > 0 we have that if n is large enough, then HB(≡n) ≥ (1 − ε)|τ|n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Since HB(≡d(n)) ≥ HB(≡n), for every ε > 0 we have that HB(≡d(n)) ≥  (1 − ε)|τ|n, provided that n is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Combining these bounds  yields the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Thus, from an asymptotic point of view, the dependence of the counting  depth d on n has no effect on the Boltzmann entropy of ≡d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' By virtue of  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='1, the same is true for the Shannon entropy of ≡d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We proceed now to further analyze the effect of counting depth on the  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' To formulate some of our results, we will use standard asymp-  totic notation, which we recall here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let f, g : N → R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We use f = o(g)  to denote that limn→∞ f(n)/g(n) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Furthermore, we use f = ω(g) to  denote that limn→∞ f(n)/g(n) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We first show that if the counting depth grows slowly enough with respect  to n, then the distribution contains a class which contains almost all of the  models of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' More formally, fix a function d : Z+ → N and thereby  a sequence ≡d(n) of equivalence relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' A class sequence M(n) with  respect to the sequence ≡d(n) is a function that outputs a single class for each  individual equivalence relation in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Each class sequence M(n)  is naturally associated with the corresponding probability sequence  pn := p≡d(n)(M(n)) = |M(n)|  2n|τ| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We say that ≡d(n) has a dominating class if there exists a class sequence  M(n) such that pn → 1 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Then the class sequence M(n) is said to  dominate ≡d(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Intuitively, a class (sequence) is dominating if a random  model of size n belongs to it with limit probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Our proof uses the well-known Chernoff bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The following lemma  will require the lower-tail estimate, while the upper-tail estimate will be  used later in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  22  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='3 (Chernoff bounds [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let X := �n  i=1 Xi be a sum of  independent 0-1-valued random variables, where Xi = 1 with probability p  and Xi = 0 with probability 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Then for every δ ≥ 0 we have that  (Lower tail) Pr[X ≤ (1 − δ)np] ≤ e−δ2 np  2  (Upper tail) Pr[X ≥ (1 + δ)np] ≤ e−δ2 np  2+δ  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For any f(n) = ω(√n), if the counting depth is d(n) ≤ n/t −  f(n), then the class sequence Mn, where n = (d(n), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , d(n)), dominates  ≡d(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We will show that for each f(n), such that f(n) = ω(√n) and f(n) ≤  n/t, we have that with limit probability one a random model realizes each  type more than n/t − f(n) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Given a type π, we let Xπ denote a  random variable which counts the number of times π is realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Now Xπ :=  �  1≤i≤n Xπ,i, where Xπ,i is an indicator random variable for the event that  the element i realizes the type π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Since the success probability of Xπ,i is  t−1, we have that E(Xπ) = n/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Now, Chernoff bound give us that for every n and for every 0 ≤ δ we  have  Pr  �  Xπ ≤ (1 − δ)n  t  �  ≤ e−δ2 n  2t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Note that δ can indeed depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Setting δ(n) :=  �  2tg(n)/n, for any  g(n) = ω(1), we obtain that  e−δ2 n  2t = e−g(n) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Furthermore  δ(n)n  t =  √  2t · 1  t  � �� �  =:C �  g(n) · √n = C  �  ng(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Hence for any function g(n) = ω(1) we have that with high probability the  type π is realized more than n/t − C  �  ng(n) many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Since π was  arbitrary, it follows from the union bound that with high probability every  type π is realized more than n/t − C  �  ng(n) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Now, if f(n) = ω(√n),  then by setting g(n) = (f(n)/C√n)2 we obtain that every type is realized  more than n/t − f(n) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We showed that when the counting depth is low enough, the distribution  has a dominating class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Next we will show that for a high enough counting  depth, there is no dominating class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For this, we will utilize the following  inequality version of Stirling’s approximation, due to Robbins [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  23  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='5 (Stirling’s approximation [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For all n ∈ N, n > 0, we  have  √  2πn  �n  e  �n  e  1  12n+1 < n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' <  √  2πn  �n  e  �n  e  1  12n  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For any f(n) = o(√n), if the counting depth is d(n) ≥ n/t −  f(n), then ≡d(n) has no dominating class as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let M(n) be an arbitrary sequence of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We will show that  there is some n0 such that for every n ≥ n0 the probability of a model  belonging to class M(n) is at most half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Let Mn be a class of the sequence M(n) with i, j ≤ t such that ni ̸= nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Let n′ be obtained from n by switching the numbers ni and nj in the tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We clearly have |Mn′| = |Mn| by symmetry so the probability of a model  belonging to Mn is at most half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  It remains to show that classes of the sequence M(n) with tuples re-  peating only one number have probability at most half, when n is large  enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let d ≥ n/t − f(n), where f(n) = o(√n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Assume d < n/t and let  M = Mn, where n = (d, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Note that this is the only (n, d)-admissible  tuple that repeats only one number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We show that this class M has proba-  bility less than half if n is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' In fact, we establish the stronger  claim that the sequence of such classes with tuples (d(n), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , d(n)) has limit  probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  The size of the above class M is given by the sum  |M| =  �  n1+···+nt=n  ni≥d  �  n  n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , nt  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  First we note that a multinomial coefficient is largest, when the numbers  n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , nt are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Using this and Stirling’s approximation, we get  �  n  n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , nt  �  ≤  �  n  n  t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , n  t  �  ≤  √  2πn(n  e )ne  1  12n  (�2π n  t ( n  et)  n  t e  1  12 n  t +1)t  =  √  2πn(n  e )ne  1  12n  �2π n  t  t(n  e )n 1  tn e  t  12 n  t +1  =  �  tt  (2π)t−1 ·  1  √  nt−1 · eg(n) · tn  ≤  �  tt  (2π)t−1 ·  1  √  nt−1 · 2n|τ|  24  The exponent of e above is  g(n) =  1  12n −  t  12n  t + 1 =  1  12n −  t2  12n + t  = 12n + t − 12t2n  12n(12n + t)  = (1 − t2)12n + t  12n(12n + t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Clearly g(n) < 0 for all positive n so eg(n) < 1 and the above estimate holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  By the stars and bars method, the original sum has  �tf(n) + t − 1  t − 1  �  ≤ (2t)t−1 · f(n)t−1  terms so the final estimate is  |M| ≤  �  tt  (2π)t−1 · (2t)t−1 · f(n)t−1  √  nt−1 · 2n|τ|  Recall that there are 2n|τ| models of size n in total and f(n) = o(√n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We obtain the following limit:  lim  n→∞  �  tt  (2π)t−1 · (2t)t−1 · f(n)t−1  √  nt−1 · 2n|τ|  2n|τ|  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We see that the sequence of classes Mn with n = (d, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , d) has limit pro-  bability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Thus for a single such class, the probability is certainly at most  half if n is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Now assume d ≥ n/t, and consider the sequence of classes Mn, where n =  (n/t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , n/t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' This is again the only (n, d)-admissible tuple that repeats  only one number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The size of these classes is given by  �  n  n  t ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=', n  t  �  and it is easy  to see from the above that this sequence also has limit probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Thus  a single such class has probability less than half for large enough n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Intuitively, a class (sequence) is vanishing, if a random model of size n  belongs to it with limit probability zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' To define this notion formally, fix  a sequence ≡d(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We say that all classes in ≡d(n) are vanishing as n → ∞  if for all class sequences M(n) for ≡d(n), we have pn → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We  now show that if the counting depth is high enough, then all classes in the  distribution sequence are vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' For any f(n) such that f(n) = ω(√n), if the counting depth  is d(n) ≥ n/t + f(n), then all classes in ≡d(n) are vanishing as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Fix some function f(n) such that f(n) = ω(√n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We first show  that with limit probability zero a random model realizes each type less  than n/t + f(n) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Without loss of generality we can assume that  25  f(n) ≤ n−n/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Given a type π, we let Xπ denote the same random variable  as in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Chernoff bound give us again that for every  n and for every 0 < δ ≤ 1 we have  Pr  �  Xπ ≥ (1 + δ)n  t  �  ≤ e−δ2  n  t(2+δ) ≤ e−δ2 n  3t  Note that δ can depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Setting δ(n) :=  �  3tg(n)/n, for any g(n) such  that g(n) = ω(1) and g(n) ≤ n/(3t) (to guarantee that δ(n) ≤ 1), we get  that  e−δ2 n  3t = e−g(n) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Furthermore  δ(n)n  t =  √  3t · 1  t  � �� �  =:C �  g(n) · √n = C  �  ng(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Hence for any function g(n) such that g(n) = ω(1) and g(n) ≤ n/(3t) we  have that with high probability the type π is realized less than n  t −C  �  ng(n)  many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Since π was arbitrary, it follows from the union bound that  with high probability every type π is realized less than n  t − C  �  ng(n) many  times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' By setting g(n) = (f(n)/C√n)2 we obtain that every type is realized  less than n/t − f(n) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Now consider an arbitrary class sequence M(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' We want to show that  for every ε > 0 we have that pn < ε, provided that n is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Consider a class M(n) and let n be the corresponding (n, d(n))-admissible  tuple, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=', M(n) is the class Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' If there is i ∈ I such that ni = d(n) =  n/t + f(n), then it follows from the previous result that p≡d(n)(Mn) < ε, as  long as n is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Suppose then that ni < n/t + f(n), for every  i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' In this case the class Mn is an isomorphism class, which means that  the probability that a random model belongs to Mn is simply  �  n  n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , nt  �  /2n|τ|,  which, as calculated in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='6, is at most a constant times  1/  √  nt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' This latter quantity is certainly less than ε, provided that n is  sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Hence p≡d(n)(Mn) < ε, provided that n is sufficiently  large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We gather the above results in the following theorem:  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The following statements hold for counting depth d(n) as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  If d(n) ≤ n/t−f(n) where f(n) = ω(√n), then ≡d(n) has a dominating  class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  26  If d(n) ≥ n/t − f(n) where f(n) = o(√n), then ≡d(n) has no domina-  ting class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  If d(n) ≥ n/t + f(n) where f(n) = ω(√n), then every class in ≡d(n)  is vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We point out a corollary of the above result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' When the counting depth  is too low, almost all models are in the same dominating class in terms  of GMLUd definability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Conversely, if the counting depth is high enough,  GMLUd can separate the models into classes that vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' These observations  directly give us the following result:  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let f(n) = ω(√n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  If d(n) ≤ n/t − f(n), then with  limit probability one, two random models of size n cannot be separated in  GMLUd(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' If d(n) ≥ n/t + f(n), then with limit probability one, two ran-  dom models of size n can be separated in GMLUd(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We say that a class is a majority class, if it contains more than half of  all the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' The following theorem is a quantitative version of Theorem  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  If d ≤ n/t − c1  √n, where  c1 := 1  t  �  2t ln(2t),  then the distribution ≡d for models of size n has a majority class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  If d ≥ n/t − c2  √n, where  c2 :=  �  π  2t3(4t)1/(t−1) < c1  then the distribution ≡d for models of size n does not have a majority  class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Suppose first that d ≤ n/t − c1  √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Set δ = (tc1)/√n, in which case  δ · n  t = c1  √n and δ2 · n/(2t) = (tc1)2/(2t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Applying Chernoff bound and  the union bound we obtain, in a similar manner as in the proof of Lemma  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='4, that with probability strictly greater than (1 − te−(tc1)2/(2t)) every type  is realized at least n/2 − c1  √n-times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' A quick calculation shows that this  latter probability is equal to 1/2 (hence the choice of c1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Consider then the case d ≥ n/t − c2  √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Let M = Mn, where n =  (d, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' , d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Using the estimate from the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='6 with f(n) =  c2  √n, we obtain  |M|  2n|τ| ≤  �  tt  (2π)t−1 · (2t)t−1 · (c2  √n)t−1  √  nt−1  = 1/2  27  Thus M is not a majority class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  The same reasoning as in the proof of  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='6 shows there is no other majority class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  We of course obtain a corresponding quantitative version of Corollary  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Let n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' If d ≤ n/t − c1  √n, then the probability that  two random models of size n can be separated in GMLUd is less than 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  If d ≥ n/t − c2  √n, then the probability that two random models of size n  can be separated in GMLUd is at least 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  6  Conclusion  We have established an interesting monotone connection between model  class sizes and description complexities, also obtaining related results for  entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Furthermore, we have characterized the phase transitions of model  class size when the domain size n and expressive power (in the form of count-  ing depth d) is altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' These results elucidate the interplay of class size and  formula length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' While focusing on GMLUd, the results have been intended  to give a general overview of related phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Thereby, an obvious future  direction involves investigating how these results lift into the framework of  first-order logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' There, natural parametrizations—analogous to varying the  counting depth d—can possibly be obtained by using quantifier depth and  the number of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Moving beyond first-order logic, it would be interesting to investigate  the expressively Turing-complete logic CL, or computation logic, introduced  in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Studying description complexities within that framework would lead  to an even closer link to Kolmogorov complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Antti Kuusisto and Miikka Vilander were supported  by the Academy of Finland project Explaining AI via Logic (XAILOG),  grant number 345612 (Kuusisto).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Antti Kuusisto was also supported by the  Academy of Finland project Theory of computational logics, grant numbers  324435, 328987, 352419, 352420, 352419, 353027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
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+page_content=' Texts in Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
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+page_content='  [12] Michael Mitzenmacher and Eli Upfal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
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+page_content=' A remark on stirling’s formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
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+page_content='  [15] Andreia Teixeira,  Armando Matos,  Andre Souto,  and Luis Fil-  ipe Coelho Antunes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content='  Entropy measures vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
+page_content=' Kolmogorov complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}
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+page_content='  30' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FST4oBgHgl3EQfbzgH/content/2301.13800v1.pdf'}