34,760 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
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
Specifying Graph Languages with Type Graphs
We investigate three formalisms to specify graph languages, i.e. sets of
graphs, based on type graphs. First, we are interested in (pure) type graphs,
where the corresponding language consists of all graphs that can be mapped
homomorphically to a given type graph. In this context, we also study languages
specified by restriction graphs and their relation to type graphs. Second, we
extend this basic approach to a type graph logic and, third, to type graphs
with annotations. We present decidability results and closure properties for
each of the formalisms.Comment: (v2): -Fixed some typos -Added more reference
On the Problem of Computing the Probability of Regular Sets of Trees
We consider the problem of computing the probability of regular languages of
infinite trees with respect to the natural coin-flipping measure. We propose an
algorithm which computes the probability of languages recognizable by
\emph{game automata}. In particular this algorithm is applicable to all
deterministic automata. We then use the algorithm to prove through examples
three properties of measure: (1) there exist regular sets having irrational
probability, (2) there exist comeager regular sets having probability and
(3) the probability of \emph{game languages} , from automata theory,
is if is odd and is otherwise
A B\"uchi-Elgot-Trakhtenbrot theorem for automata with MSO graph storage
We introduce MSO graph storage types, and call a storage type MSO-expressible
if it is isomorphic to some MSO graph storage type. An MSO graph storage type
has MSO-definable sets of graphs as storage configurations and as storage
transformations. We consider sequential automata with MSO graph storage and
associate with each such automaton a string language (in the usual way) and a
graph language; a graph is accepted by the automaton if it represents a correct
sequence of storage configurations for a given input string. For each MSO graph
storage type, we define an MSO logic which is a subset of the usual MSO logic
on graphs. We prove a B\"uchi-Elgot-Trakhtenbrot theorem, both for the string
case and the graph case. Moreover, we prove that (i) each MSO graph
transduction can be used as storage transformation in an MSO graph storage
type, (ii) every automatic storage type is MSO-expressible, and (iii) the
pushdown operator on storage types preserves the property of
MSO-expressibility. Thus, the iterated pushdown storage types are
MSO-expressible
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