235 research outputs found
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
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
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
Register automata with linear arithmetic
We propose a novel automata model over the alphabet of rational numbers,
which we call register automata over the rationals (RA-Q). It reads a sequence
of rational numbers and outputs another rational number. RA-Q is an extension
of the well-known register automata (RA) over infinite alphabets, which are
finite automata equipped with a finite number of registers/variables for
storing values. Like in the standard RA, the RA-Q model allows both equality
and ordering tests between values. It, moreover, allows to perform linear
arithmetic between certain variables. The model is quite expressive: in
addition to the standard RA, it also generalizes other well-known models such
as affine programs and arithmetic circuits.
The main feature of RA-Q is that despite the use of linear arithmetic, the
so-called invariant problem---a generalization of the standard non-emptiness
problem---is decidable. We also investigate other natural decision problems,
namely, commutativity, equivalence, and reachability. For deterministic RA-Q,
commutativity and equivalence are polynomial-time inter-reducible with the
invariant problem
A Logic for Document Spanners
Document spanners are a formal framework for information extraction that was introduced by [Fagin, Kimelfeld, Reiss, and Vansummeren, J.ACM, 2015]. One of the central models in this framework are core spanners, which are based on regular expressions with variables that are then extended with an algebra. As shown by [Freydenberger and Holldack, ICDT, 2016], there is a connection between core spanners and EC^{reg}, the existential theory of concatenation with regular constraints. The present paper further develops this connection by defining SpLog, a fragment of EC^{reg} that has the same expressive power as core spanners. This equivalence extends beyond equivalence of expressive power, as we show the existence of polynomial time conversions between this fragment and core spanners. This even holds for variants of core spanners that are based on automata instead of regular expressions. Applications of this approach include an alternative way of defining relations for spanners, insights into the relative succinctness of various classes of spanner representations, and a pumping lemma for core spanners
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