177 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 shuffle ideals of general algebras
We extend a word language concept called shuffle ideal to general algebras. For this purpose, we introduce the relation SH and show that there exists a natural connection between this relation and the homeomorphic embedding order on trees. We establish connections between shuffle ideals, monotonically ordered algebras and automata, and piecewise testable tree languages
Piecewise testable tree languages
This paper presents a decidable characterization of tree languages that can
be defined by a boolean combination of Sigma_1 sentences. This is a tree
extension of the Simon theorem, which says that a string language can be
defined by a boolean combination of Sigma_1 sentences if and only if its
syntactic monoid is J-trivial
Varieties of tree languages definable by syntactic monoids
An algebraic characterization of the families of tree languages definable by syntactic monoids is presented. This settles a question raised by several authors
Regular tree languages and quasi orders
Regular languages were characterized as sets closed with respect to monotone well-quasi orders. A similar result is proved here for tree languages. Moreover, families of quasi orders that correspond to positive varieties of tree languages and varieties of finite ordered algebras are characterized
A Categorical Approach to Syntactic Monoids
The syntactic monoid of a language is generalized to the level of a symmetric
monoidal closed category . This allows for a uniform treatment of
several notions of syntactic algebras known in the literature, including the
syntactic monoids of Rabin and Scott ( sets), the syntactic
ordered monoids of Pin ( posets), the syntactic semirings of
Pol\'ak ( semilattices), and the syntactic associative algebras of
Reutenauer ( = vector spaces). Assuming that is a
commutative variety of algebras or ordered algebras, we prove that the
syntactic -monoid of a language can be constructed as a
quotient of a free -monoid modulo the syntactic congruence of ,
and that it is isomorphic to the transition -monoid of the minimal
automaton for in . Furthermore, in the case where the variety
is locally finite, we characterize the regular languages as
precisely the languages with finite syntactic -monoids.Comment: arXiv admin note: substantial text overlap with arXiv:1504.0269
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