84 research outputs found
Recognisable languages over monads
The principle behind algebraic language theory for various kinds of
structures, such as words or trees, is to use a compositional function from the
structures into a finite set. To talk about compositionality, one needs some
way of composing structures into bigger structures. It so happens that category
theory has an abstract concept for this, namely a monad. The goal of this paper
is to propose monads as a unifying framework for discussing existing algebras
and designing new algebras
Equational theories of profinite structures
In this paper we consider a general way of constructing profinite struc-
tures based on a given framework - a countable family of objects and a
countable family of recognisers (e.g. formulas). The main theorem states:
A subset of a family of recognisable sets is a lattice if and only if it is
definable by a family of profinite equations.
This result extends Theorem 5.2 from [GGEP08] expressed only for finite words
and morphisms to finite monoids. One of the applications of our theorem is the
situation where objects are finite relational structures and recognisers are
first order sentences. In that setting a simple characterisation of lattices of
first order formulas arise
Automata and rational expressions
This text is an extended version of the chapter 'Automata and rational
expressions' in the AutoMathA Handbook that will appear soon, published by the
European Science Foundation and edited by JeanEricPin
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
Syntactic Monoids in a Category
The syntactic monoid of a language is generalized to the level of a symmetric
monoidal closed category D. This allows for a uniform treatment of several
notions of syntactic algebras known in the literature, including the syntactic
monoids of Rabin and Scott (D = sets), the syntactic semirings of Polak (D =
semilattices), and the syntactic associative algebras of Reutenauer (D = vector
spaces). Assuming that D is an entropic variety of algebras, we prove that the
syntactic D-monoid of a language L can be constructed as a quotient of a free
D-monoid modulo the syntactic congruence of L, and that it is isomorphic to the
transition D-monoid of the minimal automaton for L in D. Furthermore, in case
the variety D is locally finite, we characterize the regular languages as
precisely the languages with finite syntactic D-monoids
Modal logics on rational Kripke structures
This dissertation is a contribution to the study of infinite graphs which can be
presented in a finitary way. In particular, the class of rational graphs is studied. The
vertices of a rational graph are labeled by a regular language in some finite alphabet
and the set of edges of a rational graph is a rational relation on that language. While
the first-order logics of these graphs are generally not decidable, the basic modal and
tense logics are.
A survey on the class of rational graphs is done, whereafter rational Kripke models
are studied. These models have rational graphs as underlying frames and are equipped
with rational valuations. A rational valuation assigns a regular language to each propositional
variable. I investigate modal languages with decidable model checking on rational
Kripke models. This leads me to consider regularity preserving relations to see if
the class can be generalised even further. Then the concept of a graph being rationally
presentable is examined - this is analogous to a graph being automatically presentable.
Furthermore, some model theoretic properties of rational Kripke models are examined.
In particular, bisimulation equivalences between rational Kripke models are studied.
I study three subclasses of rational Kripke models. I give a summary of the results
that have been obtained for these classes, look at examples (and non-examples in the
case of automatic Kripke frames) and of particular interest is finding extensions of the
basic tense logic with decidable model checking on these subclasses.
An extension of rational Kripke models is considered next: omega-rational Kripke
models. Some of their properties are examined, and again I am particularly interested
in finding modal languages with decidable model checking on these classes.
Finally I discuss some applications, for example bounded model checking on rational
Kripke models, and mention possible directions for further research
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