this paper have been presented at the 4th European Summer School in Logic, Language and Information, University of Essex, 1992; at the Tempus Summer School for Algebraic and Categorical Methods in Computer Science, Masaryk University, Brno, 1993; and the Summer School in Logic Methods in Concurrency, Aarhus University, 1993. I would like to thank the organisers and the participants of these summer schools, and of the Banff higher order workshop. I would also like to thank Julian Bradfield for use of his Tex tree constructor for building derivation trees and Carron Kirkwood, Faron Moller, Perdita Stevens and David Walker for comments on earlier drafts.

The varieties RA of relation algebras and DA of dynamic algebras are similar with regard to definitional capacity, admitting essentially the same equational definitions of converse and star. They differ with regard to completeness and decidability. The RA definitions that are incomplete with respect to representable relation algebras, when expressed in their DA form are complete with respect to representable dynamic algebras. Moreover, whereas the theory of RA is undecidable, that of DA is decidable in exponential time. These results follow from representability of the free intensional dynamic algebras. Dept. of Computer Science, Stanford, CA 94305. This paper is based on a talk given at the conference Algebra and Computer Science, Ames, Iowa, June 2-4, 1988. It will appear in the proceedings of that conference, to be published by SpringerVerlag in the Lecture Notes in Computer Science series. This work was supported by the National Science Foundation under grant number CCR-8814921 ...

Digital Equipment Corporation The weakest liberal precondition and strongest postcondition predicate transformers are general-ized to the weakest invariant and strongest invariant. These new predicate transformers are useful for reasoning about concurrent programs containing operations in which the grain of atomicity is unspecified. They can also be used to replace behavioral arguments with more rigorous assertional ones.

We present a two-sorted algebra, called a Peirce algebra, of relations and sets interacting with each other. In a Peirce algebra, sets can combine with each other as in a Boolean algebra, relations can combine with each other as in a relation algebra, and in addition we have both a relationforming operator on sets (the Peirce product of Boolean modules) and a set-forming operator on relations (a cylindrification operation). Two applications of Peirce algebras are given. The first points out that Peirce algebras provide a natural algebraic framework for modelling certain programming constructs. The second shows that the so-called terminological logics arising in knowledge representation have evolved a semantics best described as a calculus of relations interacting with sets.

Quantum logic is static, describing automata having uncertain states but no state transitions and no Heisenberg uncertainty tradeoff. We cast Girard’s linear logic in the role of a dynamic quantum logic, regarded as an extension of quantum logic with time nonstandardly interpreted over a domain of linear automata and their dual linear schedules. In this extension the uncertainty tradeoff emerges via the “structure veil. ” When VLSI shrinks to where quantum effects are felt, their computer-aided design systems may benefit from such logics of computational behavior having a strong connection to quantum mechanics. 1

Dynamic algebras combine the classes of Boolean (B 0 0) and regular (R [ ; ) algebras into a single finitely axiomatized variety (B R 3) resembling an R-module with "scalar" multiplication 3. The basic result is that is reflexive transitive closure, contrary to the intuition that this concept should require quantifiers for its definition. Using this result we give several examples of dynamic algebras arising naturally in connection with additive functions, binary relations, state trajectories, languages, and flowcharts. The main result is that free dynamic algebras are residually finite (i.e. factor as a subdirect product of finite dynamic algebras), important because finite separable dynamic algebras are isomorphic to Kripke structures. Applications include a new completeness proof for the Segerberg axiomatization of propositional dynamic logic, and yet another notion of regular algebra. Key words: Dynamic algebra, logic, program verification, regular algebra. This paper or...

This paper makes several contributions towards a clarified view of schema-based program development. First, we propose that schemata can be understood, formalized, and used in a simple way: program development schemata are derived rules. We mean this in the standard sense of a derived rule of inference in logic. A schema like Figure i can be formulated as a rule stating that the conclusion follows from the premises defining F, G, and the applicability conditions. By deriving the rule in an axiomatic theory, we validate a semantic statement about it: the conclusion of the rule holds in every model where both the axioms of the theory and the premises of the rule are true. Hence, by selecting a language to work in we control which development schemata are formalizable, and by selecting a theory we determine which schemata are derivable

L’algèbre relationnelle abstraite est utilisée pour donner une définition sémantique d’un langage de programmation impératif simple. A cet effet, divers domaines sont spécifiés par des axiomes relationnels. Certaines spécifications définissent des relations sur les types de base du langage (Booléens et entiers non négatifs); leur présentation insiste sur l’importance de la notion de point. Les autres spécifications construisent les domaines dont les relations sont utilisées pour dénoter les fragments de programmes. Les fragments ainsi traités sont les expressions, les déclarations de variables, les instructions (affectation, séquence, condition et itération) et les procédures. Les relations qui dénotent un frag-ment dépendent seulement de ce fragment et non de son environnement (à l’exception des procédures), ce qui constitue une approche originale. Enfin, on montre comment prouver la correction d’un fragment, relativement à une spécification, en utilisant sa définition sémantique. Les spécifications, la sémantique et la dérivation de programmes sont donc traitées uniformément dans le cadre de l’algèbre relationnelle abstraite. i Abstract relational algebra is used to define the semantics of a simple imperative language. In order to carry out this task, various domains are specified by relational axioms. Some specifications define relations on the basic types of the language (Booleans and natural numbers); their presentation stresses the importance of the concept of point. Other spec-ifications construct the relational domains whose relations are used to denote programs. The programming constructs that are defined include expressions, variable declarations, assignment statements, while-program statements and procedures. A particularity of the semantic definitions is that the relations denoting a program fragment depend only on the fragment, and not on its environment (procedure calls excepted). Finally, it is shown how the semantics of a program fragment can be used to prove its correctness relative to a specification. The result is a uniform abstract relational setting for specification, semantics and program derivation. ii

Time can be understood as dual to information in extant models of both sequential and concurrent computation. The basis for this duality is phase space, coordinatized by time and information, whose axes are oriented respectively horizontally and vertically. We fit various basic phenomena of computation, and of behavior in general, to the phase space perspective. The extant two-dimensional logics of sequential behavior, the van Glabbeek map of branching time and true concurrency, event-state duality and schedule-automaton duality, and Chu spaces, all fit the phase space perspective well, in every case confirming our choice of orientation. 1 Introduction Our recent research has emphasized a basic duality between time and information in concurrent computation. In this paper we return to our earlier work on sequential computation and point out that a very similar duality is present there also. Our main goal here will be to compare concurrent and sequential computation in terms of this dua...

James Lipton Dept. of Mathematics Wesleyan University Emily Chapman Dept. of Mathematics Wesleyan University Abstract We study the use of relation calculi for compilation and execution of Horn Clause programs with an extended notion of input and output. We consider various other extensions to the Prolog core.