540 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 Isomorphism Relation Between Tree-Automatic Structures
An -tree-automatic structure is a relational structure whose domain
and relations are accepted by Muller or Rabin tree automata. We investigate in
this paper the isomorphism problem for -tree-automatic structures. We
prove first that the isomorphism relation for -tree-automatic boolean
algebras (respectively, partial orders, rings, commutative rings, non
commutative rings, non commutative groups, nilpotent groups of class n >1) is
not determined by the axiomatic system ZFC. Then we prove that the isomorphism
problem for -tree-automatic boolean algebras (respectively, partial
orders, rings, commutative rings, non commutative rings, non commutative
groups, nilpotent groups of class n >1) is neither a -set nor a
-set
An Application of the Feferman-Vaught Theorem to Automata and Logics for<br> Words over an Infinite Alphabet
We show that a special case of the Feferman-Vaught composition theorem gives
rise to a natural notion of automata for finite words over an infinite
alphabet, with good closure and decidability properties, as well as several
logical characterizations. We also consider a slight extension of the
Feferman-Vaught formalism which allows to express more relations between
component values (such as equality), and prove related decidability results.
From this result we get new classes of decidable logics for words over an
infinite alphabet.Comment: 24 page
An Upper Bound on the Complexity of Recognizable Tree Languages
The third author noticed in his 1992 PhD Thesis [Sim92] that every regular
tree language of infinite trees is in a class
for some natural number , where is the game quantifier. We
first give a detailed exposition of this result. Next, using an embedding of
the Wadge hierarchy of non self-dual Borel subsets of the Cantor space
into the class , and the notions of Wadge degree
and Veblen function, we argue that this upper bound on the topological
complexity of regular tree languages is much better than the usual
Logics with rigidly guarded data tests
The notion of orbit finite data monoid was recently introduced by Bojanczyk
as an algebraic object for defining recognizable languages of data words.
Following Buchi's approach, we introduce a variant of monadic second-order
logic with data equality tests that captures precisely the data languages
recognizable by orbit finite data monoids. We also establish, following this
time the approach of Schutzenberger, McNaughton and Papert, that the
first-order fragment of this logic defines exactly the data languages
recognizable by aperiodic orbit finite data monoids. Finally, we consider
another variant of the logic that can be interpreted over generic structures
with data. The data languages defined in this variant are also recognized by
unambiguous finite memory automata
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