33 research outputs found
Infinite hypergraphs I. Basic properties
AbstractBasic properties of the category of infinite directed hyperedge-labelled hypergraphs are studied. An algebraic structure is given which enables us to describe such hypergraphs by means of infinite expressions. It is then shown that two expressions define the same hypergraphs if and only if they are congruent with respect to some rewriting system. These results will be used in the second part of this paper to solve systems of recursive equations on hypergraphs and characterize their solutions
A sheaf-theoretic approach to pattern matching and related problems
AbstractWe present a general theory of pattern matching by adopting an extensional, geometric view of patterns. Representing the geometry of the pattern via a Grothendieck topology, the extension of the matching relation for a constant target and varying pattern forms a sheaf. We derive a generalized version of the Knuth-Morris-Pratt string-matching algorithm by gradually converting this extensional description into an intensional description, i.e., an algorithm. The generality of this approach is illustrated by briefly considering other applications: Earley's algorithm for parsing, Waltz filtering for scene analysis, matching modulo commutativity, and the n-queens problem
Effective solutions of recursive domain equations
Solving recursive domain equations is one of the main concerns in the denotational semantics of programming languages, and in the algebraic specification of data types. Because we are to solve them for the specification of computable objects, effective solutions of them should be needed. Though general methods for obtaining solutions are well known, effectiveness of the solutions has not been explicitly investigated.* The main objective of this dissertation is to provide a categorical method for obtaining effective solutions of recursive domain equations. Thence we will provide effective models of denotational semantics and algebraic data types. The importance of considering the effectiveness of solutions is two-fold. First we can guarantee that for every denotational specification of a programming language and algebraic data type specification, implementation exists. Second, we have an instance of a computability theory where higher type computability and even infinite type computability can be discussed very smoothly.
*While this dissertation has been written, Plotkin and Smyth obtained an alternative to our method which worked only for effectively given categories with universal objects
Retracing some paths in categorical semantics: From process-propositions-as-types to categorified reals and computers
The logical parallelism of propositional connectives and type constructors
extends beyond the static realm of predicates, to the dynamic realm of
processes. Understanding the logical parallelism of process propositions and
dynamic types was one of the central problems of the semantics of computation,
albeit not always clear or explicit. It sprung into clarity through the early
work of Samson Abramsky, where the central ideas of denotational semantics and
process calculus were brought together and analyzed by categorical tools, e.g.
in the structure of interaction categories. While some logical structures borne
of dynamics of computation immediately started to emerge, others had to wait,
be it because the underlying logical principles (mainly those arising from
coinduction) were not yet sufficiently well-understood, or simply because the
research community was more interested in other semantical tasks. Looking back,
it seems that the process logic uncovered by those early semantical efforts
might still be starting to emerge and that the vast field of results that have
been obtained in the meantime might be a valley on a tip of an iceberg.
In the present paper, I try to provide a logical overview of the gamut of
interaction categories and to distinguish those that model computation from
those that capture processes in general. The main coinductive constructions
turn out to be of this latter kind, as illustrated towards the end of the paper
by a compact category of all real numbers as processes, computable and
uncomputable, with polarized bisimulations as morphisms. The addition of the
reals arises as the biproduct, real vector spaces are the enriched
bicompletions, and linear algebra arises from the enriched kan extensions. At
the final step, I sketch a structure that characterizes the computable fragment
of categorical semantics.Comment: 63 pages, 40 figures; cut two words from the title, tried to improve
(without lengthening) Sec.8; rewrote a proof in the Appendi
\'Etale structures and the Joyal-Tierney representation theorem in countable model theory
An \'etale structure over a topological space is a continuous family of
structures (in some first-order language) indexed over . We give an
exposition of this fundamental concept from sheaf theory and its relevance to
countable model theory and invariant descriptive set theory. We show that many
classical aspects of spaces of countable models can be naturally framed and
generalized in the context of \'etale structures, including the Lopez-Escobar
theorem on invariant Borel sets, an omitting types theorem, and various
characterizations of Scott rank. We also present and prove the countable
version of the Joyal-Tierney representation theorem, which states that the
isomorphism groupoid of an \'etale structure determines its theory up to
bi-interpretability; and we explain how special cases of this theorem recover
several recent results in the literature on groupoids of models and functors
between them.Comment: 41 page
Bicontexts and structural induction
This thesis introduces and explores the notion of bicontext, an order-enriched category equipped with a unary endofunctor of order two called reverse. The purpose is threefold. First, the important categories that arise in Scott-Strachey denotational semantics have this additional structure, where by the constructions used to solve "data-type equations" are both limits and colimits simultaneously. Second, it yields a pleasant "set-theoretic" treatment of algebraic data-types in terms of bicontexts of (1, 1) relations rather than pairs of continuous functions. The theory provides a general way of relating bicontexts which serves to connect these particular ones. Third, the least solutions of data-type equations often have an associated principle of structural induction. Properties in such solutions become arrows in the appropriate bicontext, making the defining functor directly applicable to them. In this way the structural induction can be derived systematically from the functor