Quadratic Word Equations with Length Constraints, Counter Systems, and Presburger Arithmetic with Divisibility

Abstract

Word equations are a crucial element in the theoretical foundation of constraint solving over strings, which have received a lot of attention in recent years. A word equation relates two words over string variables and constants. Its solution amounts to a function mapping variables to constant strings that equate the left and right hand sides of the equation. While the problem of solving word equations is decidable, the decidability of the problem of solving a word equation with a length constraint (i.e., a constraint relating the lengths of words in the word equation) has remained a long-standing open problem. In this paper, we focus on the subclass of quadratic word equations, i.e., in which each variable occurs at most twice. We first show that the length abstractions of solutions to quadratic word equations are in general not Presburger-definable. We then describe a class of counter systems with Presburger transition relations which capture the length abstraction of a quadratic word equation with regular constraints. We provide an encoding of the effect of a simple loop of the counter systems in the theory of existential Presburger Arithmetic with divisibility (PAD). Since PAD is decidable, we get a decision procedure for quadratic words equations with length constraints for which the associated counter system is flat (i.e., all nodes belong to at most one cycle). We show a decidability result (in fact, also an NP algorithm with a PAD oracle) for a recently proposed NP-complete fragment of word equations called regular-oriented word equations, together with length constraints. Decidability holds when the constraints are additionally extended with regular constraints with a 1-weak control structure.

Program Specialization as a Tool for Solving Word Equations

“…For every quadratic equation, the solution graph constructed with the use of the algorithm AlgWE Base is finite. This fact is well-known due to the works by Matiyasevich [18,19,21,26]. Thus, specialization of WIBase L w.r.t.…”

“…[2,10,11,13,23,29,40]. On the other hand, the number of works on analysis of string manipulating programs and string constraint solvers is rapidly growing during last years [1,4,6,8,19,20,21,24,32,36,41]. As far as we know, there are few interactions between the two research areas, although some of their methods exploit similar concepts.…”

An SMT Solver for Regular Expressions and Linear Arithmetic over String Length

“…On the other hand, if we only consider regular expressions without stacked complements, then the decision problems for the considered theories are PSPACE-complete. Indeed, the automata-based approach described above can be implemented to work in nondeterministic polynomial space; strongly related complexity results are obtained in [26,27]. At the opposite end of the spectrum is the theory T LRE,n,c , which is undecidable.…”

Unification Modulo Lists with Reverse Relation with Certain Word Equations

“…We will again reduce the problem to solving some word equations. It seems appropriate here to quote [17]: "The problem of solving a word equation with a length constraint (i.e., a constraint relating the lengths of words in the word equation) has remained a long-standing open problem". However, thanks to the special form of the word equations we deal with, we will be able to provide a decision algorithm in our case.…”

“…In Section 4 we show how the considerations of length of words can be built into the unification algorithms for the theories rcons and reverse. These could be of use in formal techniques based on word constraints (e.g., [2,11,17,16]) or in constraint programming [7]. Several examples are given in Section 5, to illustrate how the method we have developed in this paper operates.…”