Pattern matching with variables: Efficient algorithms and complexity results

A pattern α (i. e., a string of variables and terminals) matches a word w, if w can be obtained by uniformly replacing the variables of α by terminal words. The respective matching problem, i. e., deciding whether or not a given pattern matches a given word, is generally NP-complete, but can be solved in polynomial-time for restricted classes of patterns. We present efficient algorithms for the matching problem with respect to patterns with a bounded number of repeated variables and patterns with a structural restriction on the order of variables. Furthermore, we show that it is NP-complete to decide, for a given number k and a word w, whether w can be factorised into k distinct factors. As an immediate consequence of this hardness result, the injective version (i. e., different variables are replaced by different words) of the matching problem is NP-complete even for very restricted clases of patterns

Revisiting Shinohara's algorithm for computing descriptive patterns

A pattern α is a word consisting of constants and variables and it describes the pattern language L(α) of all words that can be obtained by uniformly replacing the variables with constant words. In 1982, Shinohara presents an algorithm that computes a pattern that is descriptive for a finite set S of words, i.e., its pattern language contains S in the closest possible way among all pattern languages. We generalise Shinohara’s algorithm to subclasses of patterns and characterise those subclasses for which it is applicable. Furthermore, within this set of pattern classes, we characterise those for which Shinohara’s algorithm has a polynomial running time (under the assumption P 6= N P). Moreover, we also investigate the complexity of the consistency problem of patterns, i.e., finding a pattern that separates two given finite sets of words

On the structure of solution-sets to regular word equations

For quadratic word equations, there exists an algorithm based on rewriting rules which generates a directed graph describing all solutions to the equation. For regular word equations – those for which each variable occurs at most once on each side of the equation – we investigate the properties of this graph, such as bounds on its diameter, size, and DAG-width, as well as providing some insights into symmetries in its structure. As a consequence, we obtain a combinatorial proof that the problem of deciding whether a regular word equation has a solution is in NP

On the structure of solution sets to regular word equations

© Joel D. Day and Florin Manea; licensed under Creative Commons License CC-BY 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). For quadratic word equations, there exists an algorithm based on rewriting rules which generates a directed graph describing all solutions to the equation. For regular word equations - those for which each variable occurs at most once on each side of the equation - we investigate the properties of this graph, such as bounds on its diameter, size, and DAG-width, as well as providing some insights into symmetries in its structure. As a consequence, we obtain a combinatorial proof that the problem of deciding whether a regular word equation has a solution is in NP

k-spectra of weakly-c-balanced words

A word u is a scattered factor of w if u can be obtained from w by deleting some of its letters. That is, there exist the (potentially empty) words u1, u2, ..., un, and v0, v1, .., vn such that u = u1u2...un and w = v0u1v1u2v2...unvn. We consider the set of length-k scattered factors of a given word w, called here k-spectrum and denoted ScatFactk (w). We prove a series of properties of the sets ScatFactk (w) for binary weakly-0-balanced and, respectively, weakly-c-balanced words w, i.e., words over a two-letter alphabet where the number of occurrences of each letter is the same, or, respectively, one letter has c-more occurrences than the other. In particular, we consider the question which cardinalities n = | ScatFactk (w)| are obtainable, for a positive integer k, when w is either a weakly-0-balanced binary word of length 2k, or a weakly-c-balanced binary word of length 2k − c. We also consider the problem of reconstructing words from their k-spectra

On the expressive power of string constraints

We investigate properties of strings which are expressible by canonical types of string constraints. Specifically, we consider a landscape of 20 logical theories, whose syntax is built around combinations of four common elements of string constraints: language membership (e.g. for regular languages), concatenation, equality between string terms, and equality between string-lengths. For a variable x and formula f from a given theory, we consider the set of values for which x may be substituted as part of a satisfying assignment, or in other words, the property f expresses through x. Since we consider string-based logics, this set is a formal language. We firstly consider the relative expressive power of different combinations of string constraints by comparing the classes of languages expressible in the corresponding theories, and are able to establish a mostly complete picture in this regard. Secondly, we consider the question of deciding whether the language or property expressed by a variable/formula in one theory can be expressed in another theory. We establish several negative results which are relevant to preprocessing and normalisation of string constraints in practice. Some of our results have strong connections to important open problems regarding word equations and the theory of string solving.</p

Hide and seek with repetitions

Pseudo-repetitions are a natural generalization of the classical notion of repetitions in sequences: they are the repeated concatenation of a word and its encoding under a certain morphism or antimorphism. Thus, such occurrences can be regarded as hidden repetitive structures of, or within, a word. We solve fundamental algorithmic questions on pseudo-repetitions by application of insightful combinatorial results on words. More precisely, we efficiently decide whether a word is a pseudo-repetition and find all the pseudo-repetitive factors of a word. We also approach the problem of deciding whether there exists an anti-/morphism for which a word is a pseudo-repetition. We show that some variants of this later problem are efficiently solvable, while some others are NPcomplete

Equations enforcing repetitions under permutations

© 2020 Elsevier B.V. The notion of repetition of factors in words is central to combinatorics on words. A recent generalisation of this concept considers repetitions under permutations: given an alphabet Σ and a morphism or antimorphism f on Σ∗, whose restriction to Σ is a permutation, w is an [f]-repetition if there exists γ∈Σ∗, an integer k≥2, and the positive integers i1,…,ik such that w=fi1(γ)fi2(γ)⋯fik(γ). In this paper, we extend a series of classical repetition enforcing word equations to this general setting to obtain a series of word equations whose solutions are [f]-repetitions

A closer look at the expressive power of logics based on word equations

Word equations are equations α = β where α and β are words consisting of letters from some alphabet Σ and variables from a set X. Recently, there has been substantial interest in the context of string solving in logics combining word equations with other kinds of constraints on words such as (regular) language membership (regular constraints) and arithmetic over string lengths (length constraints). We consider the expressive power of such logics by looking at the set of all values a single variable might take as part of a satisfying assignment for a given formula. Hence, each formula-variable pair defines a formal language, and each logic defines a class of formal languages. We consider logics arising from combining word equations with either length constraints, regular constraints, or both. We also consider word equations with visibly pushdown language membership constraints as a generalisation of the combination of regular and length constraints. We show that word equations with visibly pushdown membership constraints are sufficient to express all recursively enumerable languages and hence satisfiability is undecidable in this case. We then establish a strict hierarchy involving the other combinations. We also provide a complete characterisation of when a thin regular language is expressible by word equations (alone) and some further partial results for regular languages in the general case.</p

A closer look at the expressive power of logics based on word equations

Word equations are equations α = β where α and β are words consisting of letters from some alphabet Σ and variables from a set X. Recently, there has been substantial interest in the context of string solving in logics combining word equations with other kinds of constraints on words such as (regular) language membership (regular constraints) and arithmetic over string lengths (length constraints). We consider the expressive power of such logics by looking at the set of all values a single variable might take as part of a satisfying assignment for a given formula. Hence, each formula-variable pair defines a formal language, and each logic defines a class of formal languages. We consider logics arising from combining word equations with either length constraints, regular constraints, or both. We also consider word equations with visibly pushdown language membership constraints as a generalisation of the combination of regular and length constraints. We show that word equations with visibly pushdown membership constraints are sufficient to express all recursively enumerable languages and hence satisfiability is undecidable in this case. We then establish a strict hierarchy involving the other combinations. We also provide a complete characterisation of when a thin regular language is expressible by word equations (alone) and some further partial results for regular languages in the general case.</p