A Decision Procedure for String Logic with Quadratic Equations, Regular Expressions and Length Constraints

Abstract

In this work, we consider the satisfiability problem in a logic that combines word equations over string variables denoting words of unbounded lengths, regular languages to which words belong and Presburger constraints on the length of words. We present a novel decision procedure over two decidable fragments that include quadratic word equations (i.e., each string variable occurs at most twice). The proposed procedure reduces the problem to solving the satisfiability in the Presburger arithmetic. The procedure combines two main components: (i) an algorithm to derive a complete set of all solutions of conjunctions of word equations and regular expressions; and (ii) two methods to precisely compute relational constraints over string lengths implied by the set of all solutions. We have implemented a prototype tool and evaluated it over a set of satisfiability problems in the logic. The experimental results show that the tool is effective and efficient.Concrete string models assume a finite alphabet Σ whose elements are called letters, set of finite words over Σ * including ǫ -the empty word, and a set of integer numbers Z. We

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.…”

“…3), can be solved by this method as well. In the paper [19], Nielsen's transformation is used for solving unbounded-length quadratic word equations. As in the original Matiyasevich work, the algorithm does not terminate if the input equation contains more than two occurrences of some variable.…”

“…Thus, these algorithms are not able to solve non-quadratic regular-ordered equations with repetitions, which are solvable by our specialization scheme. Advanced SMT-solvers such as CVC4 [20], Z3str3 [24], or S3P presented in [36] demonstrate a very good efficiency in many practical cases, however the paper [19] shows that their algorithms are not complete even w.r.t. the set of the quadratic equations, e.g.…”

“…[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.…”

High-Level Abstractions for Simplifying Extended String Constraints in SMT

“…The Slog set uses the replace function extensively but does not contain other extended functions. We also evaluate the impact on Aplas, a set of handcrafted benchmarks involving looping word equations [10] (string equalities whose left and right sides have variables in common).…”