331 research outputs found
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
Varieties of Cost Functions.
Regular cost functions were introduced as a quantitative generalisation of regular languages, retaining many of their equivalent characterisations and decidability properties. For instance, stabilisation monoids play the same role for cost functions as monoids do for regular languages. The purpose of this article is to further extend this algebraic approach by generalising two results on regular languages to cost functions: Eilenberg's varieties theorem and profinite equational characterisations of lattices of regular languages. This opens interesting new perspectives, but the specificities of cost functions introduce difficulties that prevent these generalisations to be straightforward. In contrast, although syntactic algebras can be defined for formal power series over a commutative ring, no such notion is known for series over semirings and in particular over the tropical semiring
Application of verification techniques to inverse monoids
The word problem for inverse monoids generated by
a set subject to relations of the form , where and
are both idempotents in the free inverse monoid generated by ,
is investigated. It is
shown that for every fixed monoid of this form the word problem
can be solved in polynomial time which solves an open problem of
Margolis and Meakin. For the uniform word problem, where the presentation is
part of the input, EXPTIME-completeness is shown.
For the Cayley-graphs of these
monoids, it is shown that the first-order theory with regular path
predicates is decidable. Regular path predicates allow to state
that there is a path from a node to a node that is labeled
with a word from some regular language. As a corollary, the decidability
of the generalized word problem is deduced. Finally, some results
on free partially commutative inverse monoids are presented
Adding modular predicates to first-order fragments
We investigate the decidability of the definability problem for fragments of
first order logic over finite words enriched with modular predicates. Our
approach aims toward the most generic statements that we could achieve, which
successfully covers the quantifier alternation hierarchy of first order logic
and some of its fragments. We obtain that deciding this problem for each level
of the alternation hierarchy of both first order logic and its two-variable
fragment when equipped with all regular numerical predicates is not harder than
deciding it for the corresponding level equipped with only the linear order and
the successor. For two-variable fragments we also treat the case of the
signature containing only the order and modular predicates.Relying on some
recent results, this proves the decidability for each level of the alternation
hierarchy of the two-variable first order fragmentwhile in the case of the
first order logic the question remains open for levels greater than two.The
main ingredients of the proofs are syntactic transformations of first order
formulas as well as the algebraic framework of finite categories
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