Word Equations in Nondeterministic Linear Space

Satisfiability of word equations is an important problem in the intersection of formal languages and algebra: Given two sequences consisting of letters and variables we are to decide whether there is a substitution for the variables that turns this equation into true equality of strings. The computational complexity of this problem remains unknown, with the best lower and upper bounds being, respectively, NP and PSPACE. Recently, the novel technique of recompression was applied to this problem, simplifying the known proofs and lowering the space complexity to (nondeterministic) O(n log n). In this paper we show that satisfiability of word equations is in nondeterministic linear space, thus the language of satisfiable word equations is context-sensitive. We use the known recompression-based algorithm and additionally employ Huffman coding for letters. The proof, however, uses analysis of how the fragments of the equation depend on each other as well as a new strategy for nondeterministic choices of the algorithm, which uses several new ideas to limit the space occupied by the letters

Hardness Results for Constant-Free Pattern Languages and Word Equations

We study constant-free versions of the inclusion problem of pattern languages and the satisfiability problem of word equations. The inclusion problem of pattern languages is known to be undecidable for both erasing and nonerasing pattern languages, but decidable for constant-free erasing pattern languages. We prove that it is undecidable for constant-free nonerasing pattern languages. The satisfiability problem of word equations is known to be in PSPACE and NP-hard. We prove that the nonperiodic satisfiability problem of constant-free word equations is NP-hard. Additionally, we prove a polynomial-time reduction from the satisfiability problem of word equations to the problem of deciding whether a given constant-free equation has a solution morphism ? such that ?(xy) ? ?(yx) for given variables x and y

Solutions of twisted word equations, EDT0L languages, and context-free groups

© Volker Diekert and Murray Elder; 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, F.4.2 Grammars and Other Rewriting Systems, F.4.3 Formal Languages. We prove that the full solution set of a twisted word equation with regular constraints is an EDT0L language. It follows that the set of solutions to equations with rational constraints in a contextfree group (= finitely generated virtually free group) in reduced normal forms is EDT0L. We can also decide whether or not the solution set is finite, which was an open problem. Moreover, this can all be done in PSPACE. Our results generalize the work by Lohrey and Sénizergues (ICALP 2006) and Dahmani and Guirardel (J. of Topology 2010) with respect to complexity and with respect to expressive power. Both papers show that satisfiability is decidable, but neither gave any concrete complexity bound. Our results concern all solutions, and give, in some sense, the "optimal" formal language characterization

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

On Solving Word Equations Using SAT

We present Woorpje, a string solver for bounded word equations (i.e., equations where the length of each variable is upper bounded by a given integer). Our algorithm works by reformulating the satisfiability of bounded word equations as a reachability problem for nondeterministic finite automata, and then carefully encoding this as a propositional satisfiability problem, which we then solve using the well-known Glucose SAT-solver. This approach has the advantage of allowing for the natural inclusion of additional linear length constraints. Our solver obtains reliable and competitive results and, remarkably, discovered several cases where state-of-the-art solvers exhibit a faulty behaviour

Equivalence of finite-valued streaming string transducers is decidable

In this paper we provide a positive answer to a question left open by Alur and and Deshmukh in 2011 by showing that equivalence of finite-valued copyless streaming string transducers is decidable

47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

We study constant-free versions of the inclusion problem of pattern languages and the satisfiability problem of word equations. The inclusion problem of pattern languages is known to be undecidable for both erasing and nonerasing pattern languages, but decidable for constant-free erasing pattern languages. We prove that it is undecidable for constant-free nonerasing pattern languages. The satisfiability problem of word equations is known to be in PSPACE and NP-hard. We prove that the nonperiodic satisfiability problem of constant-free word equations is NP-hard. Additionally, we prove a polynomial-time reduction from the satisfiability problem of word equations to the problem of deciding whether a given constant-free equation has a solution morphism α such that α(xy) ≠ α(yx) for given variables x and y. </p

An Optimal Bound on the Solution Sets of One-Variable Word Equations and its Consequences

We solve two long-standing open problems on word equations. Firstly, we prove that a one-variable word equation with constants has either at most three or an infinite number of solutions. The existence of such a bound had been conjectured, and the bound three is optimal. Secondly, we consider independent systems of three-variable word equations without constants. If such a system has a nonperiodic solution, then this system has at most 17 equations. Although probably not optimal, this is the first finite bound found. However, the conjecture of that bound being actually two still remains open

Light On String Solving: Approaches to Efficiently and Correctly Solving String Constraints

Widespread use of string solvers in formal analysis of string-heavy programs has led to a growing demand for more efficient and reliable techniques which can be applied in this context, especially for real-world cases. Designing an algorithm for the (generally undecidable) satisfiability problem for systems of string constraints requires a thorough understanding of the structure of constraints present in the targeted cases. We target the aforementioned case in different perspectives: We present an algorithm which works by reformulating the satisfiability of bounded word equations as a reachability problem for non-deterministic finite automata. Secondly, we present a transformation-system-based technique to solving string constraints. Thirdly, we investigate benchmarks presented in the literature containing regular expression membership predicates and design a decission procedure for a PSPACE-complete sub-theory. Additionally, we introduce a new benchmarking framework for string solvers and use it to showcase the power of our algorithms via an extensive empirical evaluation over a diverse set of benchmarks