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A Functorial Semantics for MultiAlgebras and Partial Algebras, With Applications to Syntax
, 2000
"... Multialgebras allow for the modeling of nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a functorial presentation of various categories of multialgebras and partial algebras, analogous to the classica ..."
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Cited by 14 (7 self)
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Multialgebras allow for the modeling of nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a functorial presentation of various categories of multialgebras and partial algebras, analogous to the classical presentation of algebras over a signature as cartesian functors from the algebraic theory over to Set. We introduce two dierent notions of theory over a signature, both having a structure weaker than cartesian, and we consider functors from them to Rel or Pfn, the categories of sets and relations or partial functions, respectively. Next we discuss how the functorial presentation provides guidelines for the choice of syntactical notions for a class of algebras, and as an application we argue that the natural generalization of usual terms are \conditioned terms" for partial algebras, and \term graphs" for multialgebras. Contents 1 Introduction 2 2 A short recap on multialgebras 4 3...
Functorial Semantics for Multialgebras
 Recent Trends in Algebraic Development Techniques, volume 1589 of LNCS
, 1998
"... . Multialgebras allow to model nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a functorial presentation of various categories of multialgebras and partial algebras, analogous to the classical pre ..."
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Cited by 6 (4 self)
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. Multialgebras allow to model nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a functorial presentation of various categories of multialgebras and partial algebras, analogous to the classical presentation of algebras over a signature \Sigma as cartesian functors from the algebraic theory of \Sigma , Th(\Sigma), to Set. The functors we introduce are based on variations of the notion of theory, having a structure weaker than cartesian, and their target is Rel, the category of sets and relations. We argue that this functorial presentation provides an original abstract syntax for partial and multialgebras. 1 Introduction Nondeterminism is a fundamental concept in Computer Science. It arises not only from the study of intrinsically nondeterministic computational models, like Turing machines and various kinds of automata, but also in the study of the behaviour of deterministic sys...
Modular Specifications: Constructions With Finite Colimits, Diagrams, Isomorphisms
, 1996
"... : The composition of modular specifications can be modeled, in a category theoretic framework, by colimits of diagrams. Pushouts in particular describe the combination of two specifications sharing a common part. This work extends this classic idea along three lines. First, we define a term language ..."
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: The composition of modular specifications can be modeled, in a category theoretic framework, by colimits of diagrams. Pushouts in particular describe the combination of two specifications sharing a common part. This work extends this classic idea along three lines. First, we define a term language to represent modular specifications built with colimit constructions over a category of base specifications. This language is formally characterized by a finitely cocomplete category. Then, we propose to associate with each term a diagram. This interpretation provides us with a more abstract representation of modular specifications because irrelevant steps of the construction are eliminated. We define a category of diagrams, which is a completion of the base category with finite colimits. We prove that the interpretation of terms as diagrams defines an equivalence between the corresponding categories, which shows the correctness of this interpretation. At last, we propose an algorithm to no...