Parikh Motivated Study on Repetitions in Words

We introduce the notion of general prints of a word, which is substantialized by certain canonical decompositions, to study repetition in words. These associated decompositions, when applied recursively on a word, result in what we term as core prints of the word. The length of the path to attain a core print of a general word is scrutinized. This paper also studies the class of square-free ternary words with respect to the Parikh matrix mapping, which is an extension of the classical Parikh mapping. It is shown that there are only finitely many matrix-equivalence classes of ternary words such that all words in each class are square-free. Finally, we employ square-free morphisms to generate infinitely many pairs of square-free ternary words that share the same Parikh matrix.Comment: 15 pages, preprint submitte

Parikh Matrices and Strong M-Equivalence

Parikh matrices have been a powerful tool in arithmetizing words by numerical quantities. However, the dependence on the ordering of the alphabet is inherited by Parikh matrices. Strong M-equivalence is proposed as a canonical alternative to M-equivalence to get rid of this undesirable property. Some characterization of strong M-equivalence for a restricted class of words is obtained. Finally, the existential counterpart of strong M-equivalence is introduced as well.Comment: 10 pages. Revised version. preprin

Parikh matrices and Parikh Rewriting Systems

Since the introduction of the Parikh matrix mapping, its injectivity problem is on top of the list of open problems in this topic. In 2010 Salomaa provided a solution for the ternary alphabet in terms of a Thue system with an additional feature called counter. This paper proposes the notion of a Parikh rewriting system as a generalization and systematization of Salomaa's result. It will be shown that every Parikh rewriting system induces a Thue system without counters that serves as a feasible solution to the injectivity problem.Comment: 15 pages, preprin

M-Ambiguity Sequences for Parikh Matrices and Their Periodicity Revisited

The introduction of Parikh matrices by Mateescu et al. in 2001 has sparked numerous new investigations in the theory of formal languages by various researchers, among whom is Serbanuta. Recently, a decade-old conjecture by Serbanuta on the M-ambiguity of words was disproved, leading to new possibilities in the study of such words. In this paper, we investigate how selective repeated duplications of letters in a word affect the M-ambiguity of the resulting words. The corresponding M-ambiguity of those words are then presented in sequences, which we term as M-ambiguity sequences. We show that nearly all patterns of M-ambiguity sequences are attainable. Finally, by employing certain algebraic approach and some underlying theory in integer programming, we show that repeated periodic duplications of letters of the same type in a word results in an M-ambiguity sequence that is eventually periodic.Comment: 16 pages, submitted for publication consideratio

Parikh Word Representability of Bipartite Permutation Graphs

The class of Parikh word representable graphs were recently introduced. In this work, we further develop its general theory beyond the binary alphabet. Our main result shows that this class is equivalent to the class of bipartite permutation graphs. Furthermore, we study certain graph theoretic properties of these graphs in terms of the arity of the representing word.Comment: Preprint, 20 page

Efficient Quantile Computation in Markov Chains via Counting Problems for Parikh Images

A cost Markov chain is a Markov chain whose transitions are labelled with non-negative integer costs. A fundamental problem on this model, with applications in the verification of stochastic systems, is to compute information about the distribution of the total cost accumulated in a run. This includes the probability of large total costs, the median cost, and other quantiles. While expectations can be computed in polynomial time, previous work has demonstrated that the computation of cost quantiles is harder but can be done in PSPACE. In this paper we show that cost quantiles in cost Markov chains can be computed in the counting hierarchy, thus providing evidence that computing those quantiles is likely not PSPACE-hard. We obtain this result by exhibiting a tight link to a problem in formal language theory: counting the number of words that are both accepted by a given automaton and have a given Parikh image. Motivated by this link, we comprehensively investigate the complexity of the latter problem. Among other techniques, we rely on the so-called BEST theorem for efficiently computing the number of Eulerian circuits in a directed graph

Natural Language Inference over Interaction Space

Natural Language Inference (NLI) task requires an agent to determine the logical relationship between a natural language premise and a natural language hypothesis. We introduce Interactive Inference Network (IIN), a novel class of neural network architectures that is able to achieve high-level understanding of the sentence pair by hierarchically extracting semantic features from interaction space. We show that an interaction tensor (attention weight) contains semantic information to solve natural language inference, and a denser interaction tensor contains richer semantic information. One instance of such architecture, Densely Interactive Inference Network (DIIN), demonstrates the state-of-the-art performance on large scale NLI copora and large-scale NLI alike corpus. It's noteworthy that DIIN achieve a greater than 20% error reduction on the challenging Multi-Genre NLI (MultiNLI) dataset with respect to the strongest published system.Comment: 15 pages, 2 figures, under review as ICLR proceeding, Published at Sixth International Conference on Learning Representations, ICLR 201

Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding

We introduce a first order method for solving very large convex cone programs. The method uses an operator splitting method, the alternating directions method of multipliers, to solve the homogeneous self-dual embedding, an equivalent feasibility problem involving finding a nonzero point in the intersection of a subspace and a cone. This approach has several favorable properties. Compared to interior-point methods, first-order methods scale to very large problems, at the cost of requiring more time to reach very high accuracy. Compared to other first-order methods for cone programs, our approach finds both primal and dual solutions when available or a certificate of infeasibility or unboundedness otherwise, is parameter-free, and the per-iteration cost of the method is the same as applying a splitting method to the primal or dual alone. We discuss efficient implementation of the method in detail, including direct and indirect methods for computing projection onto the subspace, scaling the original problem data, and stopping criteria. We describe an open-source implementation, which handles the usual (symmetric) non-negative, second-order, and semidefinite cones as well as the (non-self-dual) exponential and power cones and their duals. We report numerical results that show speedups over interior-point cone solvers for large problems, and scaling to very large general cone programs.Comment: 23 pages, no figure

Computational Bounds For Photonic Design

Physical design problems, such as photonic inverse design, are typically solved using local optimization methods. These methods often produce what appear to be good or very good designs when compared to classical design methods, but it is not known how far from optimal such designs really are. We address this issue by developing methods for computing a bound on the true optimal value of a physical design problem; physical designs with objective smaller than our bound are impossible to achieve. Our bound is based on Lagrange duality and exploits the special mathematical structure of these physical design problems. For a multi-mode 2D Helmholtz resonator, numerical examples show that the bounds we compute are often close to the objective values obtained using local optimization methods, which reveals that the designs are not only good, but in fact nearly optimal. Our computational bounding method also produces, as a by-product, a reasonable starting point for local optimization methods

A Fast First-Order Optimization Approach to Elastoplastic Analysis of Skeletal Structures

It is classical that, when the small deformation is assumed, the incremental analysis problem of an elastoplastic structure with a piecewise-linear yield condition and a linear strain hardening model can be formulated as a convex quadratic programming problem. Alternatively, this paper presents a different formulation, an unconstrained nonsmooth convex optimization problem, and proposes to solve it with an accelerated gradient-like method. Specifically, we adopt an accelerated proximal gradient method, that has been developed for a regularized least squares problem. Numerical experiments show that the presented algorithm is effective for large-scale elastoplastic analysis. Also, a simple warm-start strategy can speed up the algorithm when the path-dependent incremental analysis is carried out