Showing papers presented at "International Conference on Mathematical Foundations of Programming Semantics in 1985"

Galois connections

Abstract: About 1830 E. Galois discovered and investigated a connection, for a given field extension K -* L, between the collection of all subfields of L containing K and the collection of all automorphisms of L leaving K pointwise fixed. The formal properties of this connection remain valid in more abstract settings. In 1940 G. Birkhoff [1] associated wi th any relation a connection, which he called a polarity. Generalizing this concept, O. Ore [8] introduced in 1944 Galois connexions between par t ia l ly ordered sets. These, as we l l as the polarities, have a contravariant form. Its covariant version was introduced in 1953 by J. Schmidt [11] under the name Galots connections of mixed type. Categorists observed that these connections are nothing else but adj0int s i tuations between par t ia l ly ordered sets, considered in the s tandard w a y as categories (see S. Mac Lane [7]). Unfor tuna te ly most properties of Galois connections, in fact a l l of the interesting ones, are no longer val id in the realm of adjoint situations. Hence, for Galois connections, adjoint functors form an inappropriate level of generality. The aim of this note is to provide suitable levels of generality.

Cartesian closed categories, quasitopoi and topological universes

Abstract: For a concrete, topological category K over a suitable base category, the interrelationship of the concepts in the title is investigated. K is cartesian closed iff regular sinks are finitely productive. K is a quasitopos iff regular sinks are universal. For categories over Set with constant maps, the latter are precisely the topological universes. These can also be described as categories of sieves for Grothendieck topologies.

A Fully Abstract Semantics and a Proof System for an ALGOL-Like Language with Sharing

Abstract: In this paper we discuss the semantics of a simple block-structured programming language which allows sharing or aliasing. Sharing, which arises naturally in procedural languages which permit certain forms of parameter passing, has typically been regarded as problematical for the semantic treatment of a language. Difficulties have been encountered in both denotational and axiomatic treatments of sharing in the literature. Nevertheless, we find that it is possible to define a clean and elegant formal semantics for sharing. The key to our success is the choice of semantic model; we show that conventional approaches based on locations are less than satisfactory for the purposes of reasoning about partial correctness, and that in a well defined sense locations are unnecessary.

Comparing Categories of Domains

Abstract: We discuss some of the reasons for the proliferation of categories of domains suggested for the mathematical foundations of the Scott-Strachey theory of programming semantics. Five general conditions are presented which such a category should satisfy and they are used to motivate a number of examples. An attempt is made to survey some of the methods whereby these examples may be compared and their relationships expressed. We also ask a few mathematical questions about the examples.

Free Constructions of Powerdomains

Abstract: A powerdomain is a CPO together with extra algebraic structure for handling nondeterministic values. In the first powerdomains, the algebraic structure was a continuous binary operation or, which met certain axioms. Plotkin [9] and Smyth [14] showed how such a structure could be added to certain kinds of CPOs in a free or universal manner. This paper extends the work of Plotkin and Smyth by giving free constructions of powerdomains for a more adaptable algebraic structure: semiring modules. Prior to the constructions, three detailed examples are given, showing how the semiring module structure can capture information about nondeterministic behavior. By putting the available information in an algebraic framework, the algebraic properties can supplement the usual order-theoretic properties in program proofs.

On the Syntax and Semantics of Concurrent Computing

Abstract: A mathematical model is presented as a common framework within which to discuss and compare different models of concurrent computation. Central to the model is the concept of flow net, which is used to describe concurrent computation, just as a conventional flowchart is used to describe serial computation. A flow-of-control algebra of flow nets is presented by defining a minimal set of operations for composing flow nets, together with an equational system they satisfy. These operations suggest a corresponding minimal syntax — the flow-of-control algebra of flow structures. In this algebra, which is continuous, a flow net is represented by its unfoldment — a finite system of recursion equations. Deadlock and equivalence are examples of properties of concurrent computation formulated in the presented syntax.

An FP Domain with Infinite Objects

Abstract: We describe a basis for extending Backus's FP languages to apply to infinite data objects, including streams, using data objects called prefixes, which can be viewed as approximations of either finite or infinite sequences. The resulting set of data objects, together with its limit points, admits not only an infinite number of entries in a sequence, but also infinitely nested expressions. Elements in this domain include arbitrarily good finite approximations to any of the infinite objects of the domain.