439 research outputs found
On the logical definability of certain graph and poset languages
We show that it is equivalent, for certain sets of finite graphs, to be
definable in CMS (counting monadic second-order logic, a natural extension of
monadic second-order logic), and to be recognizable in an algebraic framework
induced by the notion of modular decomposition of a finite graph. More
precisely, we consider the set of composition operations on graphs
which occur in the modular decomposition of finite graphs. If is a subset
of , we say that a graph is an \calF-graph if it can be
decomposed using only operations in . A set of -graphs is recognizable if
it is a union of classes in a finite-index equivalence relation which is
preserved by the operations in . We show that if is finite and its
elements enjoy only a limited amount of commutativity -- a property which we
call weak rigidity, then recognizability is equivalent to CMS-definability.
This requirement is weak enough to be satisfied whenever all -graphs are
posets, that is, transitive dags. In particular, our result generalizes Kuske's
recent result on series-parallel poset languages
The FO^2 alternation hierarchy is decidable
We consider the two-variable fragment FO^2[<] of first-order logic over
finite words. Numerous characterizations of this class are known. Th\'erien and
Wilke have shown that it is decidable whether a given regular language is
definable in FO^2[<]. From a practical point of view, as shown by Weis, FO^2[<]
is interesting since its satisfiability problem is in NP. Restricting the
number of quantifier alternations yields an infinite hierarchy inside the class
of FO^2[<]-definable languages. We show that each level of this hierarchy is
decidable. For this purpose, we relate each level of the hierarchy with a
decidable variety of finite monoids. Our result implies that there are many
different ways of climbing up the FO^2[<]-quantifier alternation hierarchy:
deterministic and co-deterministic products, Mal'cev products with definite and
reverse definite semigroups, iterated block products with J-trivial monoids,
and some inductively defined omega-term identities. A combinatorial tool in the
process of ascension is that of condensed rankers, a refinement of the rankers
of Weis and Immerman and the turtle programs of Schwentick, Th\'erien, and
Vollmer
An introduction to finite automata and their connection to logic
This is a tutorial on finite automata. We present the standard material on
determinization and minimization, as well as an account of the equivalence of
finite automata and monadic second-order logic. We conclude with an
introduction to the syntactic monoid, and as an application give a proof of the
equivalence of first-order definability and aperiodicity
On an algorithm to decide whether a free group is a free factor of another
We revisit the problem of deciding whether a finitely generated subgroup H is
a free factor of a given free group F. Known algorithms solve this problem in
time polynomial in the sum of the lengths of the generators of H and
exponential in the rank of F. We show that the latter dependency can be made
exponential in the rank difference rank(F) - rank(H), which often makes a
significant change.Comment: 20 page
Algebraic recognizability of regular tree languages
We propose a new algebraic framework to discuss and classify recognizable
tree languages, and to characterize interesting classes of such languages. Our
algebraic tool, called preclones, encompasses the classical notion of syntactic
Sigma-algebra or minimal tree automaton, but adds new expressivity to it. The
main result in this paper is a variety theorem \`{a} la Eilenberg, but we also
discuss important examples of logically defined classes of recognizable tree
languages, whose characterization and decidability was established in recent
papers (by Benedikt and S\'{e}goufin, and by Bojanczyk and Walukiewicz) and can
be naturally formulated in terms of pseudovarieties of preclones. Finally, this
paper constitutes the foundation for another paper by the same authors, where
first-order definable tree languages receive an algebraic characterization
On the complexity of the Whitehead minimization problem
The Whitehead minimization problem consists in finding a minimum size element
in the automorphic orbit of a word, a cyclic word or a finitely generated
subgroup in a finite rank free group. We give the first fully polynomial
algorithm to solve this problem, that is, an algorithm that is polynomial both
in the length of the input word and in the rank of the free group. Earlier
algorithms had an exponential dependency in the rank of the free group. It
follows that the primitivity problem -- to decide whether a word is an element
of some basis of the free group -- and the free factor problem can also be
solved in polynomial time.Comment: v.2: Corrected minor typos and mistakes, improved the proof of the
main technical lemma (Statement 2.4); added a section of open problems. 30
page
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