590 research outputs found
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
On Varieties of Automata Enriched with an Algebraic Structure (Extended Abstract)
Eilenberg correspondence, based on the concept of syntactic monoids, relates
varieties of regular languages with pseudovarieties of finite monoids. Various
modifications of this correspondence related more general classes of regular
languages with classes of more complex algebraic objects. Such generalized
varieties also have natural counterparts formed by classes of finite automata
equipped with a certain additional algebraic structure. In this survey, we
overview several variants of such varieties of enriched automata.Comment: In Proceedings AFL 2014, arXiv:1405.527
Varieties of Cost Functions.
Regular cost functions were introduced as a quantitative generalisation of regular languages, retaining many of their equivalent characterisations and decidability properties. For instance, stabilisation monoids play the same role for cost functions as monoids do for regular languages. The purpose of this article is to further extend this algebraic approach by generalising two results on regular languages to cost functions: Eilenberg's varieties theorem and profinite equational characterisations of lattices of regular languages. This opens interesting new perspectives, but the specificities of cost functions introduce difficulties that prevent these generalisations to be straightforward. In contrast, although syntactic algebras can be defined for formal power series over a commutative ring, no such notion is known for series over semirings and in particular over the tropical semiring
Church-Rosser Systems, Codes with Bounded Synchronization Delay and Local Rees Extensions
What is the common link, if there is any, between Church-Rosser systems,
prefix codes with bounded synchronization delay, and local Rees extensions? The
first obvious answer is that each of these notions relates to topics of
interest for WORDS: Church-Rosser systems are certain rewriting systems over
words, codes are given by sets of words which form a basis of a free submonoid
in the free monoid of all words (over a given alphabet) and local Rees
extensions provide structural insight into regular languages over words. So, it
seems to be a legitimate title for an extended abstract presented at the
conference WORDS 2017. However, this work is more ambitious, it outlines some
less obvious but much more interesting link between these topics. This link is
based on a structure theory of finite monoids with varieties of groups and the
concept of local divisors playing a prominent role. Parts of this work appeared
in a similar form in conference proceedings where proofs and further material
can be found.Comment: Extended abstract of an invited talk given at WORDS 201
Ehrenfeucht-Fraisse Games on Omega-Terms
Fragments of first-order logic over words can often be characterized in terms
of finite monoids or finite semigroups. Usually these algebraic descriptions
yield decidability of the question whether a given regular language is
definable in a particular fragment. An effective algebraic characterization can
be obtained from identities of so-called omega-terms. In order to show that a
given fragment satisfies some identity of omega-terms, one can use
Ehrenfeucht-Fraisse games on word instances of the omega-terms. The resulting
proofs often require a significant amount of book-keeping with respect to the
constants involved. In this paper we introduce Ehrenfeucht-Fraisse games on
omega-terms. To this end we assign a labeled linear order to every omega-term.
Our main theorem shows that a given fragment satisfies some identity of
omega-terms if and only if Duplicator has a winning strategy for the game on
the resulting linear orders. This allows to avoid the book-keeping. As an
application of our main result, we show that one can decide in exponential time
whether all aperiodic monoids satisfy some given identity of omega-terms,
thereby improving a result of McCammond (Int. J. Algebra Comput., 2001)
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