The Hardness of Solving Simple Word Equations

We investigate the class of regular-ordered word equations. In such equations, each variable occurs at most once in each side and the order of the variables occurring in both left and right hand sides is preserved (the variables can be, however, separated by potentially distinct constant factors). Surprisingly, we obtain that solving such simple equations, even when the sides contain exactly the same variables, is NP-hard. By considerations regarding the combinatorial structure of the minimal solutions of the more general quadratic equations we obtain that the satisfiability problem for regular-ordered equations is in NP. The complexity of solving such word equations under regular constraints is also settled. Finally, we show that a related class of simple word equations, that generalises one-variable equations, is in P

Upper Bounds on the Length of Minimal Solutions to Certain Quadratic Word Equations

It is a long standing conjecture that the problem of deciding whether a quadratic word equation has a solution is in NP. It has also been conjectured that the length of a minimal solution to a quadratic equation is at most exponential in the length of the equation, with the latter conjecture implying the former. We show that both conjectures hold for some natural subclasses of quadratic equations, namely the classes of regular-reversed, k-ordered, and variable-sparse quadratic equations. We also discuss a connection of our techniques to the topic of unavoidable patterns, and the possibility of exploiting this connection to produce further similar results

Unary patterns under permutations

Thue characterized completely the avoidability of unary patterns. Adding function variables gives a general setting capturing avoidance of powers, avoidance of patterns with palindromes, avoidance of powers under coding, and other questions of recent interest. Unary patterns with permutations have been previously analysed only for lengths up to 3. Consider a pattern $p=\pi_{i_1}(x)\ldots \pi_{i_r}(x)$, with $r\geq 4$, $x$ a word variable over an alphabet $\Sigma$ and $\pi_{i_j}$ function variables, to be replaced by morphic or antimorphic permutations of $\Sigma$. If $|\Sigma|\ge 3$, we show the existence of an infinite word avoiding all pattern instances having $|x|\geq 2$. If $|\Sigma|=3$ and all $\pi_{i_j}$ are powers of a single morphic or antimorphic $\pi$, the length restriction is removed. For the case when $\pi$ is morphic, the length dependency can be removed also for $|\Sigma|=4$, but not for $|\Sigma|=5$, as the pattern $x\pi^2(x)\pi^{56}(x)\pi^{33}(x)$ becomes unavoidable. Thus, in general, the restriction on $x$ cannot be removed, even for powers of morphic permutations. Moreover, we show that for every positive integer $n$ there exists $N$ and a pattern $\pi^{i_1}(x)\ldots \pi^{i_n}(x)$ which is unavoidable over all alphabets $\Sigma$ with at least $N$ letters and $\pi$ morphic or antimorphic permutation

Local Patterns

A pattern is a word consisting of constants from an alphabet Sigma of terminal symbols and variables from a set X. Given a pattern alpha, the decision-problem whether a given word w may be obtained by substituting the variables in alpha for words over Sigma is called the matching problem. While this problem is, in general, NP-complete, several classes of patterns for which it can be efficiently solved are already known. We present two new classes of patterns, called k-local, and strongly-nested, and show that the respective matching problems, as well as membership can be solved efficiently for any fixed k

One-Variable Word Equations and Three-Variable Constant-Free Word Equations

We prove connections between one-variable word equations and three-variable constant-free word equations, and use them to prove that the number of equations in an independent system of three-variable constant-free equations is at most logarithmic with respect to the length of the shortest equation in the system. We also study two well-known conjectures. The first conjecture claims that there is a constant c such that every one-variable equation has either infinitely many solutions or at most c. The second conjecture claims that there is a constant c such that every independent system of three-variable constant-free equations with a nonperiodic solution is of size at most c. We prove that the first conjecture implies the second one, possibly for a different constant

Combinatorics and Algorithmics of Strings

Edited in cooperation with Robert MercaşStrings (aka sequences or words) form the most basic and natural data structure. They occur whenever information is electronically transmitted (as bit streams), when natural language text is spoken or written down (as words over, for example, the Latin alphabet), in the process of heredity transmission in living cells (through DNA sequences) or the protein synthesis (as sequence of amino acids), and in many more different contexts. Given this universal form of representing information, the need to process strings is apparent and is actually a core purpose of computer use. Algorithms to efficiently search through, analyze, (de-)compress, match, encode and decode strings are therefore of chief interest. Combinatorial problems about strings lie at the core of such algorithmic questions. Many such combinatorial problems are common in the string processing efforts in the different fields of application.http://drops.dagstuhl.de/opus/volltexte/2014/4552

k-Universality of Regular Languages

A subsequence of a word w is a word u such that u = w[i1]w[i2] . . . w[ik], for some set of indices 1 ≤ i1 < i2 < · · · < ik ≤ |w|. A word w is k-subsequence universal over an alphabet Σ if every word in Σk appears in w as a subsequence. In this paper, we study the intersection between the set of k-subsequence universal words over some alphabet Σ and regular languages over Σ. We call a regular language L k-∃-subsequence universal if there exists a k-subsequence universal word in L, and k-∀-subsequence universal if every word of L is k-subsequence universal. We give algorithms solving the problems of deciding if a given regular language, represented by a finite automaton recognising it, is k-∃-subsequence universal and, respectively, if it is k-∀-subsequence universal, for a given k. The algorithms are FPT w.r.t. the size of the input alphabet, and their run-time does not depend on k; they run in polynomial time in the number n of states of the input automaton when the size of the input alphabet is O(log n). Moreover, we show that the problem of deciding if a given regular language is k-∃-subsequence universal is NP-complete, when the language is over a large alphabet. Further, we provide algorithms for counting the number of k-subsequence universal words (paths) accepted by a given deterministic (respectively, nondeterministic) finite automaton, and ranking an input word (path) within the set of k-subsequence universal words accepted by a given finite automaton

Computer Arithmetic: Logic, Calculation, and Rewriting

. Computer arithmetic is the logical theory which formalizes the way computers manipulate integer numbers. In this paper, we describe a combined system whose components are a logical theory for the Isabelle theorem prover, a calculational engine based on rewriting techniques, and a decision procedure for an extension of quantifier-free Presburger arithmetic. The goal of this work is to provide a general and efficient tool to help proving theorems in computer arithmetic. This contribution shows how it is possible to combine different formal techniques (deductive systems, rewriting techniques, decision procedures) in order to solve a notoriously hard problem. Keywords: Computer Arithmetic, theorem proving, decision procedures, rewriting, Presburger arithmetic 1. Introduction Computer arithmetic is the mathematical theory which underlies the way calculational machines operate on integer numbers. Computers manipulate integer numbers of a finite, fixed precision, internally represented as..