To my parents Understanding procedure calls is crucial in computer science and everyday pro-gramming. Among the most common strategies for passing procedure argu-ments (‘evaluation strategies’) are ‘call-by-name’, ‘call-by-need’, and ‘call-by-value’, where the latter is the most commonly used. While reasoning about procedure calls is simple for call-by-name, problems arise for call-by-need and call-by-value, because it matters how often and in which order the arguments of a procedure are evaluated. We shall classify these problems and see that all of them occur for call-by-value, some occur for call-by-need, and none occur for call-by-name. In that sense, call-by-value is the ‘greatest common denominator ’ of the three evaluation strategies. Reasoning about call-by-value programs has been tackled by Eugenio Moggi’s ‘computational lambda-calculus’, which is based on a distinction between ‘values’

This paper investigates some basic algebraic properties of the categories of spans and cospans (up to isomorphic supports) over the category Set of (small) sets and functions, analyzing the monoidal structures induced over both spans and cospans by the cartesian product and disjoint union of sets. Our results nd analogous counterparts in (and are partly inspired by) the theory of relational algebras, thus our paper also shed some light on the relationship between (co)spans and the categories of (multi)relations and of equivalence relations. And, since (co)spans yields an intuitive presentation in terms of dynamical system with input and output interfaces, our results introduce an expressive, two-fold algebra that can serve as a specication formalism for rewriting systems and for composing software modules and open programs. Key words: Spans, multi-relations, monoidal categories, system specications. Introduction The use of spans [1,6] (and of the dual notion of cospans) have been...

We develop the semantic foundations of the specification language HasCasl, which combines algebraic specification and functional programming on the basis of Moggi’s partial λ-calculus. Generalizing Lambek’s classical equivalence between the simply typed λ-calculus and cartesian closed categories, we establish an equivalence between partial cartesian closed categories (pccc’s) and partial λ-theories. Building on these results, we define (set-theoretic) notions of intensional Henkin model and syntactic λ-algebra for Moggi’s partial λ-calculus. These models are shown to be equivalent to the originally described categorical models in pccc’s via the global element construction. The semantics of HasCasl is defined in terms of syntactic λ-algebras. Correlations between logics and classes of categories facilitate reasoning both on the logical and on the categorical side; as an application, we pinpoint unique choice as the distinctive feature of topos logic (in comparison to intuitionistic higher-order logic of partial functions, which by our results is the logic of pccc’s with equality). Finally, we give some applications of the model-theoretic equivalence result to the semantics of HasCasl and its relation to first-order Casl.

Multi-algebras allow to model nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. Starting from a functorial presentation of multi-algebras based on gs-monoidal theories, we argue that speci cations for multi-algebras should be based on the notion of term graphs instead of on standard terms. We consider the simplest case of (term graph) equational specification, showing that it enjoys an unrestricted form of substitutivity. We discuss the expressive power of equational specification for multialgebras, and we sketch possible extensions of the calculus.

Along the lines of classical categorical type theory for total functions, we establish equivalence results between certain classes of partial equational theories on the one hand and corresponding classes of categories on the other hand, staying close to standard categorical notions.

Abstract. As part of a larger project, we have built a declarative assembly language that enables us to specify multiple code paths to compute particular quantities, giving the instruction scheduler more flexibility in balancing execution resources for superscalar execution. Since the key design points for this language are to only describe data flow, have built-in facilities for redundancies, and still have code that looks like assembler, by virtue of consisting mainly of assembly instructions, we are basing the theoretical foundations on data-flow graph theory, and have to accommodate also relational aspects. Using functorial semantics into a Kleene category of “hyper-paths”, we formally capture the data-flow-with-choice aspects of this language and its implementation, providing also the framework for the necessary correctness proofs. 1

Multi-algebras allow to model nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a simple inequational deduction system, based on term graphs, for inferring inclusions of derived relations in a multi-algebra, and we show that term graph rewriting provides a sound and complete implementation of it.

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Graph Types" and "Updatable Graph Views".