We view the Chu space interpretation of linear logic as an alternative interpretation of the language of the Peirce calculus of binary relations. Chu spaces amount to K-valued binary relations, which for K = 2 n we show generalize n-ary relational structures. We also exhibit a four-stage unique factorization system for Chu transforms that illuminates their operation. 1 Introduction In 1860 A. De Morgan [DM60] introduced a calculus of binary relations equivalent in expressive power to one whose formulas, written in today's notation, are inequalities a b between terms a; b; . . . built up from variables with the operations of composition a; b, converse a, and complement a \Gamma . In 1870 C.S. Peirce [Pei33] extended De Morgan's calculus with Boolean connectives a + b and ab, Boolean constants 0 and 1, and an identity 1 0 for composition. In 1895 E. Schroder devoted a book [Sch95] to the calculus, and further extended it with the operations of reflexive transitive closure, a ...

In Floyd-Hoare logic, programs are dynamic while assertions are static (hold at states). In action logic the two notions become one, with programs viewed as on-the-fly assertions whose truth is evaluated along intervals instead of at states. Action logic is an equational theory ACT conservatively extending the equational theory REG of regular expressions with operations preimplication a!b (had a then b) and postimplication b/a (b if-ever a). Unlike REG, ACT is finitely based, makes a reflexive transitive closure, and has an equivalent Hilbert system. The crucial axiom is that of pure induction, (a!a) = a!a. This work was supported by the National Science Foundation under grant number CCR-8814921. 1 Introduction Many logics of action have been proposed, most of them in the past two decades. Here we define action logic, ACT, a new yet simple juxtaposition of old ideas, and show off some of its attractive aspects. The language of action logic is that of equational regular expressio...

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 ...

this paper will be realizations. The category of Boolean operations and their property-preserving renamings is not self-dual since non-T 0 Chu spaces transpose to nonextensional ones. By the same reasoning the full subcategory consisting of T 0 operations, those with no properties a j b for distinct variables a; b, is self-dual. This is a very important fact: it means that to every full subcategory C of this self-dual category we may associate its dual as the image of C under the self-duality. This associates sets to complete atomic Boolean algebras, Boolean algebras to Stone spaces, distributive lattices to Stone-Priestley posets, semilattices to algebraic lattices, complete semilattices to themselves, and so on for many other familiar [Joh82] and not so familiar (self-duality of finite-dimensional vector spaces over GF (2)) instances of Stone duality We now illustrate the general idea with some examples.

A program derivation is said to be polytypic if some of its parameters are data types. Often these data types are container types, whose elements store data. Polytypic program derivations necessitate a general, non-inductive definition of `container (data) type'. Here we propose such a definition: a container type is a relator that has membership. It is shown how this definition implies various other properties that are shared by all container types. In particular, all container types have a unique strength, and all natural transformations between container types are strong. Capsule Review Progress in a scientific dicipline is readily equated with an increase in the volume of knowledge, but the true milestones are formed by the introduction of solid, precise and usable definitions. Here you will find the first generic (`polytypic') definition of the notion of `container type', a definition that is remarkably simple and suitable for formal generic proofs (as is amply illustrated in t...

This paper presents a brief introduction to relation algebras, including some examples motivated by work in computer science, namely, the ‘interval algebras’, relation algebras that arose from James Allen’s work on temporal reasoning, and by ‘compass algebras’, which are designed for similar reasoning about space. One kind of reasoning problem, called a ‘constraint satisfaction problem’, can be defined for arbitrary relation algebras. It will be shown here that the constraint satisfiability problem is NP-complete for almost all compass and interval algebras.

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...

We define the notion of two-dimensional logic and diagram the relative locations of a number of such. This work was supported by the National Science Foundation under grant number CCR-8814921 and a grant from Mitsubishi. 1 Background The theme of this note is logics with more or less independent disjunction and conjunction. At JELIA'90 I described one such logic, action logic, a single-sorted finitely based equational conservative extension of the equational logic of regular expressions, with the language part of the extension consisting of new operations preimplication A!B (had A then B) and postimplication B/A (B if-ever A) [Pra90a]. The organizers of the present conference requested that I talk again on action logic. Although I had nothing new to report on this subject it seemed to me that a walk around the neighborhood of action logic should be of some interest. Action logic being what I called a two-dimensional logic, a natural selection of neighbors would be the various two-dime...

Combining different knowledge representation languages is one of the main topics in Qualitative Spatial Reasoning (QSR). This allows the combined languages to compensate each other's representational de ciencies, and is seen as an asnwer to the emerging demand from real applications, such as Geographical Information Systems (GIS), robot navigation, or shape description, for the representation of more speci c knowledge than is allowed by each of the languages taken separately. Knowledge expressed in such a combined language decomposes then into parts, or components, each expressed in one of the combined languages. Reasoning internally within each component of such knowledge involves only the language the component is expressed in, which is not new. The challenging question is to come with methods for the interaction of the different components of such knowledge. With these considerations in mind, we propose a calculus, cCOA, combining, thus more expressive than each of, two calculi well-known in QSR: Frank's cardinal direction calculus, CDA,...

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