137 research outputs found
On the rational subset problem for groups
We use language theory to study the rational subset problem for groups and
monoids. We show that the decidability of this problem is preserved under graph
of groups constructions with finite edge groups. In particular, it passes
through free products amalgamated over finite subgroups and HNN extensions with
finite associated subgroups. We provide a simple proof of a result of
Grunschlag showing that the decidability of this problem is a virtual property.
We prove further that the problem is decidable for a direct product of a group
G with a monoid M if and only if membership is uniformly decidable for
G-automata subsets of M. It follows that a direct product of a free group with
any abelian group or commutative monoid has decidable rational subset
membership.Comment: 19 page
The Self-Similarity of Free Semigroups and Groups (Logic, Algebraic system, Language and Related Areas in Computer Science)
We give a survey on results regarding self-similar and automaton presentations of free groups and semigroups and related products. Furthermore, we discuss open problems and results with respect to algebraic decision problems in this area
Logic Meets Algebra: the Case of Regular Languages
The study of finite automata and regular languages is a privileged meeting
point of algebra and logic. Since the work of Buchi, regular languages have
been classified according to their descriptive complexity, i.e. the type of
logical formalism required to define them. The algebraic point of view on
automata is an essential complement of this classification: by providing
alternative, algebraic characterizations for the classes, it often yields the
only opportunity for the design of algorithms that decide expressibility in
some logical fragment.
We survey the existing results relating the expressibility of regular
languages in logical fragments of MSO[S] with algebraic properties of their
minimal automata. In particular, we show that many of the best known results in
this area share the same underlying mechanics and rely on a very strong
relation between logical substitutions and block-products of pseudovarieties of
monoid. We also explain the impact of these connections on circuit complexity
theory.Comment: 37 page
Deciding FO-definability of regular languages
We prove that, similarly to known PSpace-completeness of recognising FO(<)-definability of the language L(A) of a DFA A,
deciding bothFO(<,equiv)- and FO(<,MOD)-definability
(corresponding to circuit complexity in AC0 and ACC0) are PSpace-complete. We obtain these results by first showing that known algebraic characterisations of FO-definability
of L(A) can be captured by `localisable' properties of the transition monoid of A.
Using our criterion, we then generalise the known proof of PSpace-hardness of FO(<)-definability, and establish the upper bounds not only for arbitrary DFAs but also for 2NFAs
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