651 research outputs found
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
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
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
Lambek vs. Lambek: Functorial Vector Space Semantics and String Diagrams for Lambek Calculus
The Distributional Compositional Categorical (DisCoCat) model is a
mathematical framework that provides compositional semantics for meanings of
natural language sentences. It consists of a computational procedure for
constructing meanings of sentences, given their grammatical structure in terms
of compositional type-logic, and given the empirically derived meanings of
their words. For the particular case that the meaning of words is modelled
within a distributional vector space model, its experimental predictions,
derived from real large scale data, have outperformed other empirically
validated methods that could build vectors for a full sentence. This success
can be attributed to a conceptually motivated mathematical underpinning, by
integrating qualitative compositional type-logic and quantitative modelling of
meaning within a category-theoretic mathematical framework.
The type-logic used in the DisCoCat model is Lambek's pregroup grammar.
Pregroup types form a posetal compact closed category, which can be passed, in
a functorial manner, on to the compact closed structure of vector spaces,
linear maps and tensor product. The diagrammatic versions of the equational
reasoning in compact closed categories can be interpreted as the flow of word
meanings within sentences. Pregroups simplify Lambek's previous type-logic, the
Lambek calculus, which has been extensively used to formalise and reason about
various linguistic phenomena. The apparent reliance of the DisCoCat on
pregroups has been seen as a shortcoming. This paper addresses this concern, by
pointing out that one may as well realise a functorial passage from the
original type-logic of Lambek, a monoidal bi-closed category, to vector spaces,
or to any other model of meaning organised within a monoidal bi-closed
category. The corresponding string diagram calculus, due to Baez and Stay, now
depicts the flow of word meanings.Comment: 29 pages, pending publication in Annals of Pure and Applied Logi
Quantifiers on languages and codensity monads
This paper contributes to the techniques of topo-algebraic recognition for
languages beyond the regular setting as they relate to logic on words. In
particular, we provide a general construction on recognisers corresponding to
adding one layer of various kinds of quantifiers and prove a corresponding
Reutenauer-type theorem. Our main tools are codensity monads and duality
theory. Our construction hinges on a measure-theoretic characterisation of the
profinite monad of the free S-semimodule monad for finite and commutative
semirings S, which generalises our earlier insight that the Vietoris monad on
Boolean spaces is the codensity monad of the finite powerset functor.Comment: 30 pages. Presentation improved and details of several proofs added.
The main results are unchange
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
A Graph Model for Imperative Computation
Scott's graph model is a lambda-algebra based on the observation that
continuous endofunctions on the lattice of sets of natural numbers can be
represented via their graphs. A graph is a relation mapping finite sets of
input values to output values.
We consider a similar model based on relations whose input values are finite
sequences rather than sets. This alteration means that we are taking into
account the order in which observations are made. This new notion of graph
gives rise to a model of affine lambda-calculus that admits an interpretation
of imperative constructs including variable assignment, dereferencing and
allocation.
Extending this untyped model, we construct a category that provides a model
of typed higher-order imperative computation with an affine type system. An
appropriate language of this kind is Reynolds's Syntactic Control of
Interference. Our model turns out to be fully abstract for this language. At a
concrete level, it is the same as Reddy's object spaces model, which was the
first "state-free" model of a higher-order imperative programming language and
an important precursor of games models. The graph model can therefore be seen
as a universal domain for Reddy's model
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