828 research outputs found
Algebraic recognizability of regular tree languages
We propose a new algebraic framework to discuss and classify recognizable
tree languages, and to characterize interesting classes of such languages. Our
algebraic tool, called preclones, encompasses the classical notion of syntactic
Sigma-algebra or minimal tree automaton, but adds new expressivity to it. The
main result in this paper is a variety theorem \`{a} la Eilenberg, but we also
discuss important examples of logically defined classes of recognizable tree
languages, whose characterization and decidability was established in recent
papers (by Benedikt and S\'{e}goufin, and by Bojanczyk and Walukiewicz) and can
be naturally formulated in terms of pseudovarieties of preclones. Finally, this
paper constitutes the foundation for another paper by the same authors, where
first-order definable tree languages receive an algebraic characterization
Variable binding, symmetric monoidal closed theories, and bigraphs
This paper investigates the use of symmetric monoidal closed (SMC) structure
for representing syntax with variable binding, in particular for languages with
linear aspects. In our setting, one first specifies an SMC theory T, which may
express binding operations, in a way reminiscent from higher-order abstract
syntax. This theory generates an SMC category S(T) whose morphisms are, in a
sense, terms in the desired syntax. We apply our approach to Jensen and
Milner's (abstract binding) bigraphs, which are linear w.r.t. processes. This
leads to an alternative category of bigraphs, which we compare to the original.Comment: An introduction to two more technical previous preprints. Accepted at
Concur '0
Functorial Data Migration
In this paper we present a simple database definition language: that of
categories and functors. A database schema is a small category and an instance
is a set-valued functor on it. We show that morphisms of schemas induce three
"data migration functors", which translate instances from one schema to the
other in canonical ways. These functors parameterize projections, unions, and
joins over all tables simultaneously and can be used in place of conjunctive
and disjunctive queries. We also show how to connect a database and a
functional programming language by introducing a functorial connection between
the schema and the category of types for that language. We begin the paper with
a multitude of examples to motivate the definitions, and near the end we
provide a dictionary whereby one can translate database concepts into
category-theoretic concepts and vice-versa.Comment: 30 page
Equality languages and fixed point languages
This paper considers equality languages and fixed-point languages of homomorphisms and deterministic gsm mappings. It provides some basic properties of these classes of languages. We introduce a new subclass of dgsm mappings, the so-called symmetric dgsm mappings. We prove that (unlike for arbitrary dgsm mappings) their fixed-point languages are regular but not effectively obtainable. This result has various consequences. In particular we strengthen a result from Ehrenfeucht, A., and Rozenberg, G. [(1978), Theor. Comp. Sci. 7, 169–184] by pointing out a class of homomorphisms which includes elementary homomorphisms but still has regular equality languages. Also we show that the result from Herman, G. T., and Walker, A. [(1976), Theor. Comp. Sci. 2, 115–130] that fixed-point languages of DIL mappings are regular, is not effective
A universe of processes and some of its guises
Our starting point is a particular `canvas' aimed to `draw' theories of
physics, which has symmetric monoidal categories as its mathematical backbone.
In this paper we consider the conceptual foundations for this canvas, and how
these can then be converted into mathematical structure. With very little
structural effort (i.e. in very abstract terms) and in a very short time span
the categorical quantum mechanics (CQM) research program has reproduced a
surprisingly large fragment of quantum theory. It also provides new insights
both in quantum foundations and in quantum information, and has even resulted
in automated reasoning software called `quantomatic' which exploits the
deductive power of CQM. In this paper we complement the available material by
not requiring prior knowledge of category theory, and by pointing at
connections to previous and current developments in the foundations of physics.
This research program is also in close synergy with developments elsewhere, for
example in representation theory, quantum algebra, knot theory, topological
quantum field theory and several other areas.Comment: Invited chapter in: "Deep Beauty: Understanding the Quantum World
through Mathematical Innovation", H. Halvorson, ed., Cambridge University
Press, forthcoming. (as usual, many pictures
Presenting Distributive Laws
Distributive laws of a monad T over a functor F are categorical tools for
specifying algebra-coalgebra interaction. They proved to be important for
solving systems of corecursive equations, for the specification of well-behaved
structural operational semantics and, more recently, also for enhancements of
the bisimulation proof method. If T is a free monad, then such distributive
laws correspond to simple natural transformations. However, when T is not free
it can be rather difficult to prove the defining axioms of a distributive law.
In this paper we describe how to obtain a distributive law for a monad with an
equational presentation from a distributive law for the underlying free monad.
We apply this result to show the equivalence between two different
representations of context-free languages
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