86 research outputs found
Solving word equations modulo partial commutations
AbstractIt is shown that it is decidable whether an equation over a free partially commutative monoid has a solution. We give a proof of this result using normal forms. Our method is a direct reduction of a trace equation system to a word equation system with regular constraints. Hereby we use the extension of Makanin's theorem on the decidability of word equations to word equations with regular constraints, which is due to Schulz
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
Alphabetical satisfiability problem for trace equations
It is known that the satisfiability problem for equations over free partially commutative monoids is decidable but computationally hard. In this paper we consider the satisfiability problem for equations over free partially commutative monoids under the constraint that the solution is a subset of the alphabet. We prove that this problem is NP-complete for quadratic equations and that its uniform version is NP-complete for linear equations
Markovian dynamics of concurrent systems
Monoid actions of trace monoids over finite sets are powerful models of
concurrent systems---for instance they encompass the class of 1-safe Petri
nets. We characterise Markov measures attached to concurrent systems by
finitely many parameters with suitable normalisation conditions. These
conditions involve polynomials related to the combinatorics of the monoid and
of the monoid action. These parameters generalise to concurrent systems the
coefficients of the transition matrix of a Markov chain.
A natural problem is the existence of the uniform measure for every
concurrent system. We prove this existence under an irreducibility condition.
The uniform measure of a concurrent system is characterised by a real number,
the characteristic root of the action, and a function of pairs of states, the
Parry cocyle. A new combinatorial inversion formula allows to identify a
polynomial of which the characteristic root is the smallest positive root.
Examples based on simple combinatorial tilings are studied.Comment: 35 pages, 6 figures, 33 reference
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
Observational Equivalence and Full Abstraction in the Symmetric Interaction Combinators
The symmetric interaction combinators are an equally expressive variant of
Lafont's interaction combinators. They are a graph-rewriting model of
deterministic computation. We define two notions of observational equivalence
for them, analogous to normal form and head normal form equivalence in the
lambda-calculus. Then, we prove a full abstraction result for each of the two
equivalences. This is obtained by interpreting nets as certain subsets of the
Cantor space, called edifices, which play the same role as Boehm trees in the
theory of the lambda-calculus
- …