In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.

Ex Input:
LTL synthesis – the construction of a function to satisfy a logical specification formulated in Linear Temporal Logic – is a 2EXPTIME-complete problem with relevant applications in controller synthesis and a myriad of artificial intelligence applications. In this research note we consider De Giacomo and Vardi’s variant of the synthesis problem for LTL formulas interpreted over finite rather than infinite traces. Rather surprisingly, given the existing claims on complexity, we establish that LTL synthesis is EXPTIME-complete for the finite interpretation, and not 2EXPTIME-complete as previously reported. Our result coincides nicely with the planning perspective where non-deterministic planning with full observability is EXPTIME-complete and partial observability increases the complexity to 2EXPTIME-complete; a recent related result for LTL synthesis shows that in the finite case with partial observability, the problem is 2EXPTIME-complete.

Ex Output:
Finite LTL Synthesis is EXPTIME-complete


Ex Input:
We consider relative error low rank approximation of tensors with respect to the Frobenius norm. Namely, given an order-q tensor A ∈ R ∏q i=1 ni , output a rank-k tensor B for which ‖A − B‖F ≤ (1 + ) OPT, where OPT = infrank-k A′ ‖A − A‖F . Despite much success on obtaining relative error low rank approximations for matrices, no such results were known for tensors. One structural issue is that there may be no rank-k tensor Ak achieving the above infinum. Another, computational issue, is that an efficient relative error low rank approximation algorithm for tensors would allow one to compute the rank of a tensor, which is NP-hard. We bypass these two issues via (1) bicriteria and (2) parameterized complexity solutions: 1. We give an algorithm which outputs a rank k′ = O((k/ )q−1) tensor B for which ‖A − B‖F ≤ (1+ ) OPT in nnz(A)+n ·poly(k/ ) time in the real RAM model, whenever either Ak exists or OPT > 0. Here nnz(A) denotes the number of non-zero entries in A. If both Ak does not exist and OPT = 0, then B instead satisfies ‖A − B‖F < γ, where γ is any positive, arbitrarily small function of n. 2. We give an algorithm for any δ > 0 which outputs a rank k tensor B for which ‖A−B‖F ≤ (1+ ) OPT and runs in (nnz(A)+n poly(k/ )+exp(k/ )) ·nδ time in the unit cost RAM model, whenever OPT > 2−O(n ) and there is a rank-k tensor B = ∑k i=1 ui ⊗ vi ⊗ wi for which ‖A − B‖F ≤ (1 + /2) OPT and ‖ui‖2, ‖vi‖2, ‖wi‖2 ≤ 2 δ). If OPT ≤ 2−Ω(nδ), then B instead satisfies ‖A−B‖F ≤ 2−Ω(n δ). Our first result is polynomial time, and in fact input sparsity time, in n, k, and 1/ , for any k ≥ 1 and any 0 < < 1, while our second result is fixed parameter tractable in k and 1/ . For outputting a rank-k tensor, or even a bicriteria solution with rank-Ck for a certain constant C > 1, we show a 2 1−o(1)) time lower bound under the Exponential Time Hypothesis. Our results are based on an “iterative existential argument”, and give the first relative error low rank approximations for tensors for a large number of error measures for which nothing was known. In particular, we give the first relative error approximation algorithms on tensors for: column row and tube subset selection, entrywise `p-low rank approximation for 1 ≤ p < 2, low rank approximation with respect to sum of Euclidean norms of faces or tubes, weighted low rank approximation, and low rank approximation in distributed and streaming models. We also obtain several new results for matrices, such as nnz(A)-time CUR decompositions, improving the previous nnz(A) log n-time CUR decompositions, which may be of independent interest. ∗Work done while visiting IBM Almaden, and supported in part by UTCS TAship (CS361 Spring 17 Introduction to Computer Security). †Supported in part by Simons Foundation, and NSF CCF-1617955. ar X iv :1 70 4. 08 24 6v 1 [ cs .D S] 2 6 A pr 2 01 7

Ex Output:
Relative Error Tensor Low Rank Approximation


Ex Input:
Without discourse connectives, classifying implicit discourse relations is a challenging task and a bottleneck for building a practical discourse parser. Previous research usually makes use of one kind of discourse framework such as PDTB or RST to improve the classification performance on discourse relations. Actually, under different discourse annotation frameworks, there exist multiple corpora which have internal connections. To exploit the combination of different discourse corpora, we design related discourse classification tasks specific to a corpus, and propose a novel Convolutional Neural Network embedded multi-task learning system to synthesize these tasks by learning both unique and shared representations for each task. The experimental results on the PDTB implicit discourse relation classification task demonstrate that our model achieves significant gains over baseline systems.

Ex Output:

Implicit Discourse Relation Classification via Multi-Task Neural Networks


