Generic properties in some classes of automaton groups
Thibault Godin


Diversity with Cooperation: Ensemble Methods for Few-Shot Classification
    Nikita Dvornik,  Cordelia Schmid,  Julien Mairal



Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, 38000
Grenoble, France

firstname.lastname@inria.fr

    December 30, 2023
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	We prove, for various important classes of Mealy automata, that almost all generated groups have an element of infinite order. In certain cases, it also implies other results such as exponential growth.


















§ INTRODUCTION




The class of groups generated by Mealy automata presents a considerable variety of behaviours and has been widely used since the eighties as a powerful source of interesting groups <cit.>. It seems natural to try to produce new examples of groups to be studied by picking a random Mealy automaton and considering the group it generates, or to try  to get an interesting distribution over some class of groups starting from a distribution over some class of Mealy automata <cit.>. This approach also raises a natural question: "how does a typical automaton group look like?". In this paper, we tackle this problem and give partial answers for several important and well-studied classes, by proving that automata belonging to the class of reversible, reset, or polynomial activity automata generate with great probability a group having at least one element of infinite order. In particular, it means that these groups are generically infinite and not Burnside.


Another motivation for this paper is that the —how to decide whether an element generates an infinite group—was recently proven undecidable among automaton groups <cit.>, while the –how to decide whether the whole group is infinite—is known to be undecidable for automaton semigroups but remains open for automata groups <cit.>. On the other hand, some classes of automaton (semi)groups are known to have decidable  <cit.>. Our results provide statistical answers for these problems.

Depending on the class, we also get stronger or additional statements, among others, the groups generated by reversible or reset Mealy automata have generically exponential growth.

The proposed proofs vary strongly with the considered class and rely on the structural properties of the automata. In particular, the case  of general  invertible Mealy automata remains open. 




In order to simplify the statements, we will use the informal "let  be a random automaton in 𝒞" instead of the  formal "let  be a random variable uniformly distributed over the set 𝒞". All probabilistic statements should be understood accordingly.

















§ AUTOMATON GROUPS


We recall that the order of an element g of a group G is the least (strictly positive) integer α such that g^α = 𝕀. If such an integer does not exist, we say that g has infinite order. Equivalently, the order of g is the cardinal of the subgroup it generates, hence having an element of infinite order implies the infiniteness of the whole group.



If X is a finite set then X^k denotes the set of words of length k, and X^* ( X^+) the set of words of arbitrary (positive) length. We take as a convention that elements of X^ℓ,  ℓ>1 are represented with a bold font. 






 §.§ Mealy automata and automaton (semi)groups



A Mealy automaton is  a 4-tuple = where  and  are finite sets, called the stateset and the alphabet, δ = δ_x : →_x ∈ is a set of functions called transition functions, and ρ = ρ_q : →_q ∈ is a set of functions called production functions. Examples of such  automata are presented on Figure <ref>, and we refer the reader to <cit.> for a more complete introduction.





The map ρ_q extends to a length-preserving map on words ρ_q^* →^* by the recursive definition:

    ∀ i ∈,  ∀s∈^*,     ρ_q(is) = ρ_q(i)ρ_δ_i(q)(s)  .

We can also extend the set of maps ρ to words of states u∈^* by composing the production functions associated with the letters of u:

    ∀ q ∈,  ∀u∈^*,     ρ_qu = ρ_u∘ρ_q .


Likewise, we extend the functions δ to words of state and words via

    ∀ i ∈,  ∀s∈^*,  ∀ q ∈,  ∀u∈^*,   δ_i(qu) = δ_i(q)δ_ρ_q(i)(u)   and δ_is = δ_i ∘δ_s .


[t]0.46   
For each automaton transition q  δ_x(q), we associate the cross-transition depicted in the following way:

[t]0.55   



    [             x       ;      q        δ_x(q);        ρ_q(x)        ]




The production functions ρ_q : ^* →^* of an automaton  generate a semigroup = {ρ_u : ^* →^* | u∈^+}.



A Mealy automaton is invertible when the functions ρ are permutations of . When a Mealy automaton is invertible one can define its inverse ^-1 by 
    p  q  ∈⇔ p^-1 q^-1∈^-1 .
 


Whenever a Mealy automaton is invertible we can consider the group  it generates:

    =  ρ_q  |  q ∈ = ρ^± 1_u|u∈^* .
 

A group (a semigroup) is an automaton group (semigroup) if it can be generated by some Mealy automaton.



Given a Mealy automaton =, its dual is the Mealy automaton = (,,ρ,δ) where the roles of the stateset and of the alphabet are exchanged. Its ℓ-th power is the automaton (^ℓ,,δ,ρ)where the production and transition functions have been naturally extended. We define also the automaton = (,^ℓ,δ,ρ) = ()^ℓ acting on sequences of ℓ letters and remark that this operation does not change the generated semigroup,  =.


	An automaton is called minimal if each state induces a different element in the generated group. Given an automaton , one can algorithmically construct its minimization , which generates the same group <cit.>.






From an algebraic point of view, it is convenient to describe the elements of an automaton group via the so-called wreath recursions. For any g in an automaton group  on alphabet  and any word s∈^*, let gs denotes the image of s by g, and gs the unique h ∈ satisfying g(st)=(gs)ht
for all t∈^*.
The wreath recursion of g is:

    g=(gx_1,…,gx_||)σ_g,

where σ_g ∈Σ denotes the permutation on  induces by g.





In what follows,  will denote, if not explicited, an invertible Mealy automaton .




 §.§ Classes of Mealy automata


We now describe several important classes of (invertible) Mealy automata.


An automaton = is called reversible when the functions δ  are permutations of . If an automaton is invertible then its dual is reversible. A Mealy automaton is bireversible if it is invertible and reversible, and so is its inverse. 


Another, somewhat opposite, restriction on the transition function leads to the class of reset automata, studied in <cit.>. An automaton  is called reset is there exists a function ϕ: → such that ∀ x, ∀ q, δ_x(q) = ϕ(x). In other words,  all the arrows labelled by an input letter x lead to the same state ϕ(x). Up to renaming the states and pruning the automaton of its vertices without ingoing edges (which does not change the finiteness of the generated group nor the existence of element of infinite order), we may assume that all studied reset automata are unfolded, that = and ϕ=𝕀.




Another class of Mealy automata linked to the cycle structure is defined in <cit.>, via the activity. Assume that there is a unique state inducing the identity in the group, denoted 𝕀. The activity of an automorphism t ∈ is defined as the function

    α_t : ℓ↦ x ∈^ℓ,  tx≠𝕀 .

It is known that the activity α_t is polynomial if and only if there is not two nontrivial simple cycles accessible one from another in the automaton , and that in this case the degree of the polynomial is the maximal number of nontrivial cycles that can be reached along of a simple cycle minus one. For a fixed alphabet , we denote  ( (d)) the set of all Mealy automata with polynomial activity (with activity bounded by a polynomial of degree d), and in particular we call bounded (finitary) the set (0) ( (-1) =  t,   α_t(ℓ) → 0). Notice that every finitary automaton generates a finite group <cit.>.



In <cit.> is defined a tool to understand the orbits of elements of ^* under the action of an automorphism described by a Mealy automaton. Given =, t ∈ and x∈^* define _t(x)  = min_α>0t^αx=x the size of the orbit of x under the action of t. The Orbit Signalizer is the graph Γ_t whose vertices  are the t^_t(x) x,  x∈^*  and edges from t^_t(x)x to t^_t(xy)xy, x∈^*, y ∈ with label _t^_t(x)x(y).

The Orbit signalizer is used in <cit.> to solve the . Indeed the order of t is the lowest common multiple of all labels along paths starting from vertex t in Γ_t. In particular if the orbit signalizer is finite then the is decidable for t, as it reduces to checking if cycles have labels all 1.















§ REVERSIBLE MEALY AUTOMATA

We show that groups generated by (invertible) reversible Mealy automata have an element of infinite order with high probability. In fact we are going to prove a stronger result by showing that almost all invertible reversible automata are not bireversible, then using known results from <cit.>, we obtain that the generated semigroups are almost surely  torsion-free.

Since a Mealy automaton is completely defined by its transition and production functions, an invertible reversible Mealy automaton can be understood as  permutations in  and  permutations in , thus the uniform distribution on the set of invertible reversible Mealy automata with stateset  and alphabet  is the uniform distribution on ^×^.



An invertible reversible automaton is bireversible if and only if each output letter induces a permutation of the stateset. In particular, for bireversible automata, we have that : 
    ∀ (p,i), (q,j) ∈×, (p,i)≠ (q,j),  δ_i(p) = δ_j(q) ⇒ρ_p(i)≠ρ_q(j) .



	(see also Fig. <ref>)
comment



Define, for r∈, the set 𝒪_r= j ∈, ∃ (p,i) ∈×,  p  r ∈ of output letters that lead to r. An invertible reversible automaton is bireversible if and only if, for all states r, the set 𝒪_r is the whole alphabet.


	Consider the  automaton Fig. <ref>. We have 𝒪_a = 1,2,3 =𝒪_c but 𝒪_b = 1,3, hence the automaton is not bireversible.





The probability that a random  invertible reversible automaton  with k letters and n states  is bireversible is less than 
    1  n^k-1 +  1 k



Let =. For r ∈, we denote _r=p ∈| ∃ i ∈, δ_i(p)=r the size of the set of predecessor of r. 
Let us fix a state r. We have:


    (∈BIR)    = (∀ q ∈, 𝒪_q=) 
       ≤(𝒪_r = ) .
    From the law of total probability we get:(∈BIR)   ≤(𝒪_r = |_r = 1 )( _r = 1 )    
              +   (𝒪_r = |_r ≥ 2 )(  _r ≥ 2) 
       ≤( _r = 1 ) + (𝒪_r = |_r ≥ 2)
     The probability that  r has exactly one predecessor can be seen as fixing  δ_i^-1(r) for some reference letter i ∈ and requiring that the k-1 other δ_j^-1(r),   j ∈∖{ i} are equal to   δ_i^-1(r), hence: (∈BIR)   ≤1  n^k-1 + (𝒪_r = |_r  ≥ 2)
    For the second term, let us consider a predecessor p of r and let λ be the number of input letters leading from p to r. We have 1≤λ≤k-1. To enforce bireversibility, we have to  avoid that p outputs a letter that is already leading to r (ρ_p(i) ≠ρ_q(j) for δ_p(i)=δ_q(j)=r). Assume that  the set 𝒪_r^(p)= j ∈| ∃ q ≠ p ∈,  q  r ∈ of output letters leading to r from a state q different from p is of maximal size k-λ. Since ρ_p is random and independent from the others ρ_q we can bound this probability from above: having the letters leading from p to r produce the λ out of k required letters is k λ^-1. Hence:(∈BIR)   ≤1  n^k-1 +  k λ^-1
       ≤1  n^k-1 +  1  k .







The obtained bound goes to 0 as k goes to infinity, and the same reasoning applied to the dual automaton allows us to conclude that the proportion of bireversible automata among the invertible reversible automata goes to 0 as the number of letters or states grows.

It is proven in <cit.>
that an invertible reversible Mealy automaton without bireversible component generates a torsion-free semigroup. Whence our theorem:
 
The probability that an invertible reversible Mealy automaton taken uniformly at random generates a  torsion-free semigroup goes to 1 as the size of  the alphabet grows. Moreover, the probability for the group to have an element of infinite order also goes to 1  as  the stateset or  the alphabet grows.


	It is known that, with great probability, two random permutations on a large set generate a transitive group <cit.>, in terms of a graph, it means that a typical reversible Mealy automaton on a large alphabet is (strongly) connected, and is not bireversible by Proposition <ref> whence the first part of the result. The second part comes from the remark on the dual. 



From <cit.>, where it is shown that having an element of infinite order implies exponential growth among groups generated by invertible reversible automata, we obtain:

The probability that an invertible reversible Mealy automaton taken uniformly at random generates a group with exponential growth goes to 1 as the size of the stateset or of the alphabet grows.



	Notice that Theorem <ref> is not a priori a consequence of Theorem <ref>: there exists infinite Burnside group with exponential growth (<cit.>). However, no example of such a group is known within the class of automaton groups.




It is worthwhile noting that it is unknown whether the is decidable within the class of (semi)groups generated by reversible Mealy automata. 












§ RESET MEALY AUTOMATA

The class of reset automata is of particular interest since it is linked to one-way cellular automata, and  were used by Gillibert to prove the undecidability of the for automata semigroups <cit.>. For groups generated by (invertible) reset Mealy automata the remains open <cit.>. 

As the transition function is trivial in a (unfolded) reset Mealy automaton, the uniform distribution on the set of unfolded invertible reset Mealy automata with stateset  and alphabet  is the uniform distribution on ^.

We are going to use a result from <cit.>: let =( ,δ,ρ) be a (unfolded) reset automaton and define the transformation π_ : q ↦ρ_q^-1(q) for all q ∈.

	Let  be a reset automaton. If π_ is not a permutation then the group generated by  has an element of infinite order.

We give the proof for the sake of completeness.

	If π_ is not a permutation, then there exists x_0 which does not belong to any cycle of π_ and such that π_(x_0)=x_1 belongs to a cycle x_1 ⋯ x_ℓ x_1 of π_. Computing the orbit of x_0(x_1⋯ x_ℓ)^α under the action of any given state q ∈  gives:
	
    [           x_0           x_1           x_2             ⋯           x_ℓ           x_1      ⋯;      q           x_0           x_1           x_2      ⋯  x_ℓ-1           x_ℓ           x_1;           y_1           x_0           x_1             ⋯        x_ℓ -1           x_ℓ;      q           y_1           x_0                    ⋯  x_ℓ-2         x_ℓ-1           x_ℓ;          y'_1          y'_2           x_0             ⋯         x_ℓ-2         x_ℓ-1;      ⋮      ⋮             ⋮             ⋮                    ⋮             ⋮;        ]

Since x_1=ρ_x_0^-1(x_0) = ρ_x_ℓ^-1(x_ℓ). So q^α ix_0(x_1⋯ x_ℓ)^α = ux_0(x_1⋯ x_ℓ)^(α-i), for some u∈^iα, hence q has infinite order.



The probability that a random (unfolded) reset automaton on k	letters has an element of infinite order is at least 1-e√(k)e^-k.


	Since the ρ_q ,  q∈ are random permutations, the function q ↦ρ_q^-1(q) can be considered has a random mapping from  to , and the number of permutations among mappings is k! k^k. We conclude using Stirling's approximation and known bounds.


Using <cit.>, where Olukoya proves that groups generated by reset automata are either finite or have exponential growth, we get (see also Remark  <ref>):

The probability that a random (unfolded) reset automaton on k	letters has exponential growth is at least 1-e√(k)e^-k.



From Delacourt and Ollinger <cit.>, our result also means that permutive one way cellular automata are generically aperiodic.



	
	The Theorem<ref>  (<cit.>) remains true for a more general definition of reset automata (where a family of letter leads to a given state), and so does our results.








An unfolded reset automaton is minimal (<cit.>) if and only if each states induces a different permutation on letters. By the birthday problem, we can extend our result a bit: a random minimal unfolded reset automaton generically generates a group with exponential growth and elements of infinite order.















§ MEALY AUTOMATA WITH POLYNOMIAL ACTIVITY


The class of Mealy automata with polynomial activity is interesting as the is decidable for (semi)groups generated by automata with bounded activity but remains open for the higher levels of the hierarchy <cit.>.

Recall that 𝕀 denotes the identity state in the automaton, which is supposed to be unique.
We are going to define a normal form: let = be an automaton with polynomial activity and let ℓ be the lowest common multiple of the sizes of the (simple) cycles. Since an automaton with polynomial growth has no entangled cycles, we have that  has all cycles of length one. Now put d the maximal length of an (oriented) path between a state and a self-loop in . Then the normal form of the automaton  is the automaton dℓ and it looks as follows: it is a directed acyclic graph whose leaf induces the identity 𝕀, and where each state either has a self-loop or leads to a state with a self-loop. For instance the normal form 3G of the (invertible) Grigorchuk automaton G  (which has bounded activity) Fig. <ref> (left) is depicted Fig. <ref>


To the best of our knowledge, and even among automata under normal form, there is no easy description of the uniform distribution on the set of (invertible) Mealy automata with polynomial activity, even if one fixes the degree of the activity. To bypass this difficulty, we show that automata with finitary activity are rare even among  automata with bounded activity, and use the fact that once the transition functions are fixed, the choice of production functions does not change the activity.


The next proposition is a simple yet useful observation:




	Let = be a Mealy automaton with bounded activity under normal form. If there is some t ∈ with δ_i(t) = t and ρ_t(i)≠ i, then ρ_t has infinite order.


	Up to renaming, we can assume that i=1 and ρ_t(1) =2. We  use the orbit signalizer of t to prove that ρ_t has infinite order (see Fig. <ref>): put α_j =_t(1^j).


Since the activity is bounded, the set t^_t(1^j)1^j_j is finite, so there is a self-loop t^α_i1^i = t^α_i+j1^i+j. By putting  t^α_i1^i = ts_α we get that (ts_i)^β1^j = ts_i for some integer β=_ts_i(1^j).
Suppose that the size of the orbit is 1. We obtain:




	
    [     1^j    ;   t       t;     2^j    ; s_i     s_i;     1^j    ;     ]
     but 
    [       1    ;   t       t;       2    ; s_i       𝕀;       1    ;     ]
     so  s_i = 𝕀 and 
    [   1  ; t   t;   2  ; 𝕀   𝕀;   1  ;   ]




Hence 2=1, contradiction. The size of the orbit of ρ_t  increases strictly through the cycle 1^β, whence the order of ρ_t–the lowest common multiple of the  paths in the orbit signalizer of t (<cit.>)–is infinite.






From this result, we get:

	The probability that the group generated by an automaton with polynomial activity has an element of infinite order goes to 1 as the size of the alphabet goes to infinity.


	We first prove that groups generated by automata with polynomial non finitary activity  generically  have an element of infinite order: let = be an automaton in (d)∖(-1) and let t be a state with bounded activity on a nontrivial cycle. Since the activity does not depend on the choice of the production functions (except for the trivial state), we can consider the set 𝒞_ of automata in (d)∖(-1) with same transition functions and trivial state. Among 𝒞_, we have ρ_t(i)≠ i with probability 1-1/k, so, in the normal form, t is on a cycle labelled by ix∈^ℓ with ρ_t(ix)≠ ix. We can apply Proposition <ref>.


Now we show that the set (-1)  has measure 0 in the set (d),   d ≥ 0 . If an automaton  has polynomial activity, then there is at least one state t satisfying δ_i(t) = 𝕀 for all i. Given ∈(-1), we can build k automata _i with bounded but not finitary activity by changing for exactly one letter δ_i(t) = 𝕀 to δ_i(t) = t. If we consistently chose t to be, , the minimal among acceptable states, we can uniquely reconstruct  from these _i, whence the result.

We conclude using the law of total probability: the probability that an automaton in (d) has an element of infinite order is equal to the probability that it has an element of infinite order given it belongs to (d)∖(-1) times the probability of the later ; we showed that both goes to one, the result follows.


From the proof we extract the following:


		The probability that the group generated by a automaton in (0)  on an alphabet of size k has an element of infinite order is at least k-1k+1.




	
	
    (∋el. of inf. order )    =
    	(∋el. of inf. order |∈bound.  not finitary)( ∈bound.  not finitary)  
             + (∋el. of inf. order |∈ finitary)( ∈ finitary ) 
    	   ≥(k-1) k k  (k+1)

	




§ CONCLUSION AND FUTURE WORK

In this work, we proved, for various important classes of Mealy automata, that the generated groups have generically an element of infinite order, thus are infinite. It is natural to wonder whether other properties, such as non-amenability,  are generic and to extend these results to the full class of automaton group.

One interesting direction is to determine if generating a free or an infinitely presented group is generic in a class. These properties are mutually exclusive. Automata with polynomial activity cannot generate free groups <cit.>, while reversible ones can <cit.> ; infinitely presented groups can be found in both classes.

It would be striking to find two classes and a group property which is nontrivial in both classes yet generically true in one and generically false in the other.
One can also ask about random automaton semigroups. This seems more difficult as less is known for  random semigroups (even finite ones). However, since the infiniteness of the (semi)group generated by an automata implies the infiniteness of the semigroup generated by its dual (<cit.>), our results imply the genericity of infiniteness in some classes of automaton semigroups.
	




Acknowledgements: The author thanks Ville Salo who initiated this work and for interesting discussions, and Matthieu Picantin and Jérémie Brieussel for their comments. He was supported by Academy of Finland grant 296018 and  by the French Agence Nationale de la Recherche
through the project AGIRA.










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