25 research outputs found
A Fibrational Approach to Automata Theory
For predual categories C and D we establish isomorphisms between opfibrations
representing local varieties of languages in C, local pseudovarieties of
D-monoids, and finitely generated profinite D-monoids. The global sections of
these opfibrations are shown to correspond to varieties of languages in C,
pseudovarieties of D-monoids, and profinite equational theories of D-monoids,
respectively. As an application, we obtain a new proof of Eilenberg's variety
theorem along with several related results, covering varieties of languages and
their coalgebraic modifications, Straubing's C-varieties, fully invariant local
varieties, etc., within a single framework
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
Varieties of Languages in a Category
Eilenberg's variety theorem, a centerpiece of algebraic automata theory,
establishes a bijective correspondence between varieties of languages and
pseudovarieties of monoids. In the present paper this result is generalized to
an abstract pair of algebraic categories: we introduce varieties of languages
in a category C, and prove that they correspond to pseudovarieties of monoids
in a closed monoidal category D, provided that C and D are dual on the level of
finite objects. By suitable choices of these categories our result uniformly
covers Eilenberg's theorem and three variants due to Pin, Polak and Reutenauer,
respectively, and yields new Eilenberg-type correspondences
Varieties of Data Languages
We establish an Eilenberg-type correspondence for data languages, i.e.
languages over an infinite alphabet. More precisely, we prove that there is a
bijective correspondence between varieties of languages recognized by
orbit-finite nominal monoids and pseudovarieties of such monoids. This is the
first result of this kind for data languages. Our approach makes use of nominal
Stone duality and a recent category theoretic generalization of Birkhoff-type
HSP theorems that we instantiate here for the category of nominal sets. In
addition, we prove an axiomatic characterization of weak pseudovarieties as
those classes of orbit-finite monoids that can be specified by sequences of
nominal equations, which provides a nominal version of a classical theorem of
Eilenberg and Sch\"utzenberger
Varieties of Data Languages
We establish an Eilenberg-type correspondence for data languages, i.e.
languages over an infinite alphabet. More precisely, we prove that there is a
bijective correspondence between varieties of languages recognized by
orbit-finite nominal monoids and pseudovarieties of such monoids. This is the
first result of this kind for data languages. Our approach makes use of nominal
Stone duality and a recent category theoretic generalization of Birkhoff-type
HSP theorems that we instantiate here for the category of nominal sets. In
addition, we prove an axiomatic characterization of weak pseudovarieties as
those classes of orbit-finite monoids that can be specified by sequences of
nominal equations, which provides a nominal version of a classical theorem of
Eilenberg and Sch\"utzenberger