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 ...

In this paper we show the embedding of Hybrid Probabilistic Logic Programs into the rather general framework of Residuated Logic Programs, where the main results of (definite) logic programming are validly extrapolated, namely the extension of the immediate consequences operator of van Emden and Kowalski. The importance of this result is that for the first time a framework encompassing several quite distinct logic programming semantics is described, namely Generalized Annotated Logic Programs, Fuzzy Logic Programming, Hybrid Probabilistic Logic Programs, and Possibilistic Logic Programming. Moreover, the embedding provides a more general semantical structure paving the way for defining paraconsistent probabilistic reasoning with a logic programming semantics.

properties of Galois connections 29 4.1 Pre-orders : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 30 4.1.1 Calculating in pre-orders : : : : : : : : : : : : : : : : : : : : : : : : 30 4.1.2 Alternative definitions : : : : : : : : : : : : : : : : : : : : : : : : : 33 4.1.3 Uniqueness of adjoints in a pre-order : : : : : : : : : : : : : : : : : 34 4.2 Partial orders : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 35 4.2.1 More cancellation laws : : : : : : : : : : : : : : : : : : : : : : : : : 35 4.2.2 Existence of adjoints : : : : : : : : : : : : : : : : : : : : : : : : : : 36 4.2.3 The closure connection : : : : : : : : : : : : : : : : : : : : : : : : : 40 4.2.4 "Perfect" connections : : : : : : : : : : : : : : : : : : : : : : : : : : 41 4.3 Complete lattices : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 44 i 5 Application: The Domain Operator 47 5.1 Monotypes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :...

The aim of this paper is to build a formal model for fuzzy unification in multi-adjoint logic programs containing both a declarative and a procedural part, and prove its soundness and completeness. Our approach is based on a general framework for logic programming, which gives a formal model of fuzzy logic programming extended by fuzzy similarities and axioms of first-order logic with equality.

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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...

Chu spaces have found applications in computer science, mathematics, and physics. They enjoy a useful categorical duality analogous to that of lattice theory and projective geometry. As natural mathematics Chu spaces borrow ideas from the natural sciences, particularly physics, while as rational mechanics they cast Hamiltonian mechanics in terms of the interaction of body and mind. This paper addresses the chief stumbling block for Descartes ’ 17thcentury philosophy of mind-body dualism, how can the fundamentally dissimilar mental and physical planes causally interact with each other? We apply Cartesian logic to reject not only divine intervention, preordained synchronization, and the eventual mass retreat to monism, but also an assumption Descartes himself somehow neglected to reject, that causal interaction within these planes is an easier problem than between. We use Chu spaces and residuation to derive all causal interaction, both between and within the two planes, from a uniform and algebraically rich theory of between-plane interaction alone. Lifting the two-valued Boolean logic of binary relations to the complex-valued fuzzy logic of quantum mechanics transforms residuation into a natural generalization of the inner product operation of a Hilbert space and demonstrates that this account of causal interaction is of essentially the same form as the Heisenberg-Schrödinger quantum-mechanical solution to analogous problems of causal interaction in physics. 1 Cartesian Dualism The Chu construction [Bar79] strikes us as extraordinarily useful, more so with every passing month. Elsewhere we have described the application of Chu spaces to process algebra [GP93], metamathematics [Pra93, Pra94a], and physics [Pra94b]. Here we make a first attempt at applying them to philosophy. It might seem that traditional philosophical questions would be beyond the scope of TAPSOFT. Bear in mind however that Boolean logic as the basis for