More Than 1700 Years of Word Equations

Geometry and Diophantine equations have been ever-present in mathematics. Diophantus of Alexandria was born in the 3rd century (as far as we know), but a systematic mathematical study of word equations began only in the 20th century. So, the title of the present article does not seem to be justified at all. However, a linear Diophantine equation can be viewed as a special case of a system of word equations over a unary alphabet, and, more importantly, a word equation can be viewed as a special case of a Diophantine equation. Hence, the problem WordEquations: "Is a given word equation solvable?" is intimately related to Hilbert's 10th problem on the solvability of Diophantine equations. This became clear to the Russian school of mathematics at the latest in the mid 1960s, after which a systematic study of that relation began. Here, we review some recent developments which led to an amazingly simple decision procedure for WordEquations, and to the description of the set of all solutions as an EDT0L language.Comment: The paper will appear as an invited address in the LNCS proceedings of CAI 2015, Stuttgart, Germany, September 1 - 4, 201

Equations over free inverse monoids with idempotent variables

We introduce the notion of idempotent variables for studying equations in inverse monoids. It is proved that it is decidable in singly exponential time (DEXPTIME) whether a system of equations in idempotent variables over a free inverse monoid has a solution. The result is proved by a direct reduction to solve language equations with one-sided concatenation and a known complexity result by Baader and Narendran: Unification of concept terms in description logics, 2001. We also show that the problem becomes DEXPTIME hard , as soon as the quotient group of the free inverse monoid has rank at least two. Decidability for systems of typed equations over a free inverse monoid with one irreducible variable and at least one unbalanced equation is proved with the same complexity for the upper bound. Our results improve known complexity bounds by Deis, Meakin, and Senizergues: Equations in free inverse monoids, 2007. Our results also apply to larger families of equations where no decidability has been previously known.Comment: 28 pages. The conference version of this paper appeared in the proceedings of 10th International Computer Science Symposium in Russia, CSR 2015, Listvyanka, Russia, July 13-17, 2015. Springer LNCS 9139, pp. 173-188 (2015

Finding All Solutions of Equations in Free Groups and Monoids with Involution

The aim of this paper is to present a PSPACE algorithm which yields a finite graph of exponential size and which describes the set of all solutions of equations in free groups as well as the set of all solutions of equations in free monoids with involution in the presence of rational constraints. This became possible due to the recently invented emph{recompression} technique of the second author. He successfully applied the recompression technique for pure word equations without involution or rational constraints. In particular, his method could not be used as a black box for free groups (even without rational constraints). Actually, the presence of an involution (inverse elements) and rational constraints complicates the situation and some additional analysis is necessary. Still, the recompression technique is general enough to accommodate both extensions. In the end, it simplifies proofs that solving word equations is in PSPACE (Plandowski 1999) and the corresponding result for equations in free groups with rational constraints (Diekert, Hagenah and Gutierrez 2001). As a byproduct we obtain a direct proof that it is decidable in PSPACE whether or not the solution set is finite.Comment: A preliminary version of this paper was presented as an invited talk at CSR 2014 in Moscow, June 7 - 11, 201

Logics with rigidly guarded data tests

The notion of orbit finite data monoid was recently introduced by Bojanczyk as an algebraic object for defining recognizable languages of data words. Following Buchi's approach, we introduce a variant of monadic second-order logic with data equality tests that captures precisely the data languages recognizable by orbit finite data monoids. We also establish, following this time the approach of Schutzenberger, McNaughton and Papert, that the first-order fragment of this logic defines exactly the data languages recognizable by aperiodic orbit finite data monoids. Finally, we consider another variant of the logic that can be interpreted over generic structures with data. The data languages defined in this variant are also recognized by unambiguous finite memory automata

Solutions of Word Equations over Partially Commutative Structures

We give NSPACE(n log n) algorithms solving the following decision problems. Satisfiability: Is the given equation over a free partially commutative monoid with involution (resp. a free partially commutative group) solvable? Finiteness: Are there only finitely many solutions of such an equation? PSPACE algorithms with worse complexities for the first problem are known, but so far, a PSPACE algorithm for the second problem was out of reach. Our results are much stronger: Given such an equation, its solutions form an EDT0L language effectively representable in NSPACE(n log n). In particular, we give an effective description of the set of all solutions for equations with constraints in free partially commutative monoids and groups

The Existential Theory of Equations with Rational Constraints in Free Groups is PSPACE-Complete

It is known that the existential theory of equations in free groups is decidable. This is a famous result of Makanin. On the other hand it has been shown that the scheme of his algorithm is not primitive recursive. In this paper we present an algorithm that works in polynomial space, even in the more general setting where each variable has a rational constraint, that is, the solution has to respect a specification given by a regular word language. Our main result states that the existential theory of equations in free groups with rational constraints is PSPACE-complete. We obtain this result as a corollary of the corresponding statement about free monoids with involution.Comment: 45 pages. LaTeX sourc

Complex Algebras of Arithmetic

An 'arithmetic circuit' is a labeled, acyclic directed graph specifying a sequence of arithmetic and logical operations to be performed on sets of natural numbers. Arithmetic circuits can also be viewed as the elements of the smallest subalgebra of the complex algebra of the semiring of natural numbers. In the present paper, we investigate the algebraic structure of complex algebras of natural numbers, and make some observations regarding the complexity of various theories of such algebras