1,513 research outputs found
Eilenberg Theorems for Free
Eilenberg-type correspondences, relating varieties of languages (e.g. of
finite words, infinite words, or trees) to pseudovarieties of finite algebras,
form the backbone of algebraic language theory. Numerous such correspondences
are known in the literature. We demonstrate that they all arise from the same
recipe: one models languages and the algebras recognizing them by monads on an
algebraic category, and applies a Stone-type duality. Our main contribution is
a variety theorem that covers e.g. Wilke's and Pin's work on
-languages, the variety theorem for cost functions of Daviaud,
Kuperberg, and Pin, and unifies the two previous categorical approaches of
Boja\'nczyk and of Ad\'amek et al. In addition we derive a number of new
results, including an extension of the local variety theorem of Gehrke,
Grigorieff, and Pin from finite to infinite words
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
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
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