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Gates accept concurrent behavior
 In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci
, 1993
"... We represent concurrent processes as Boolean propositions or gates, cast in the role of acceptors of concurrent behavior. This properly extends other mainstream representations of concurrent behavior such as event structures, yet is defined more simply. It admits an intrinsic notion of duality that ..."
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Cited by 32 (16 self)
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We represent concurrent processes as Boolean propositions or gates, cast in the role of acceptors of concurrent behavior. This properly extends other mainstream representations of concurrent behavior such as event structures, yet is defined more simply. It admits an intrinsic notion of duality that permits processes to be viewed as either schedules or automata. Its algebraic structure is essentially that of linear logic, with its morphisms being consequencepreserving renamings of propositions, and with its operations forming the core of a natural concurrent programming language. 1
The Stone gamut: A coordinatization of mathematics
 In Logic in Computer Science
, 1995
"... We give a uniform representation of the objects of mathematical practice as Chu spaces, forming a concrete selfdual bicomplete closed category and hence a constructive model of linear logic. This representation distributes mathematics over a twodimensional space we call the Stone gamut. The Stone ..."
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Cited by 30 (13 self)
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We give a uniform representation of the objects of mathematical practice as Chu spaces, forming a concrete selfdual bicomplete closed category and hence a constructive model of linear logic. This representation distributes mathematics over a twodimensional space we call the Stone gamut. The Stone gamut is coordinatized horizontally by coherence, ranging from −1 for sets to 1 for complete atomic Boolean algebras (CABA’s), and vertically by complexity of language. Complexity 0 contains only sets, CABA’s, and the inconsistent empty set. Complexity 1 admits noninteracting setCABA pairs. The entire Stone duality menagerie of partial distributive lattices enters at complexity 2. Groups, rings, fields, graphs, and categories have all entered by level 16, and every category of relational structures and their homomorphisms eventually appears. The key is the identification of continuous functions and homomorphisms, which puts StonePontrjagin duality on a uniform basis by merging algebra and topology into a simple common framework. 1 Mathematics from matrices We organize much of mathematics into a single category Chu of Chu spaces, or games as Lafont and Streicher have called them [LS91]. A Chu space is just a matrix that we shall denote =, but unlike the matrices of linear algebra, which serve as representations of linear transformations, Chu spaces serve as representations of the objects of mathematics, and their essence resides in how they transform. This organization permits a general twodimensional classification of mathematical objects that we call the Stone gamut 1, distributed horizontally by ∗This work was supported by ONR under grant number N0001492J1974. 1 “Spectrum, ” the obvious candidate for this appliction, already has a standard meaning in Stone duality, namely the representation of the dual space of a lattice by its prime ideals. “A
Linear logic for generalized quantum mechanics
 In Proc. Workshop on Physics and Computation (PhysComp'92
, 1993
"... 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 l ..."
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Cited by 16 (2 self)
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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 computeraided design systems may benefit from such logics of computational behavior having a strong connection to quantum mechanics. 1
Concurrent Kripke Structures
 In Proceedings of the North American Process Algebra Workshop, Cornell CSTR931369
, 1993
"... We consider a class of Kripke Structures in which the atomic propositions are events. This enables us to represent worlds as sets of events and the transition and satisfaction relations of Kripke structures as the subset and membership relations on sets. We use this class, called event Kripke struct ..."
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Cited by 10 (0 self)
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We consider a class of Kripke Structures in which the atomic propositions are events. This enables us to represent worlds as sets of events and the transition and satisfaction relations of Kripke structures as the subset and membership relations on sets. We use this class, called event Kripke structures, to model concurrency. The obvious semantics for these structures is a true concurrency semantics. We show how several aspects of concurrency can be easily defined, and in addition get distinctions between causality and enabling, and choice and nondeterminism. We define a duality for event Kripke structures, and show how this duality enables us to convert between imperative and declarative views of programs, by treating states and events on the same footing. We provide pictorial representations of both these views, each encoding all the information to convert to the other. We define a process algebra of event Kripke structures, showing how to combine them in the usual waysparallel co...
Chu spaces: Complementarity and Uncertainty in Rational Mechanics
, 1994
"... this paper will be realizations. The category of Boolean operations and their propertypreserving renamings is not selfdual since nonT 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 ..."
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Cited by 9 (0 self)
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this paper will be realizations. The category of Boolean operations and their propertypreserving renamings is not selfdual since nonT 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 selfdual. This is a very important fact: it means that to every full subcategory C of this selfdual category we may associate its dual as the image of C under the selfduality. This associates sets to complete atomic Boolean algebras, Boolean algebras to Stone spaces, distributive lattices to StonePriestley posets, semilattices to algebraic lattices, complete semilattices to themselves, and so on for many other familiar [Joh82] and not so familiar (selfduality of finitedimensional vector spaces over GF (2)) instances of Stone duality We now illustrate the general idea with some examples.
Chu Spaces: Automata with quantum aspects
 In Proc. Workshop on Physics and Computation (PhysComp’94
, 1994
"... Chu spaces are a recently developed model of concurrent computation extending automata theory to express branching time and true concurrency. They exhibit in a primitive form the quantum mechanical phenomena of complementarity and uncertainty. The complementarity arises as the duality of information ..."
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Cited by 8 (3 self)
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Chu spaces are a recently developed model of concurrent computation extending automata theory to express branching time and true concurrency. They exhibit in a primitive form the quantum mechanical phenomena of complementarity and uncertainty. The complementarity arises as the duality of information and time, automata and schedules, and states and events. Uncertainty arises when we define a measurement to be a morphism and notice that increasing structure in the observed object reduces clarity of observation. For a Chu space this uncertainty can be calculated numerically in an attractively simple way directly from its form factor to yield the usual Heisenberg uncertainty relation. Chu spaces correspond to wavefunctions as vectors of Hilbert space, whose inner product operation is realized for Chu spaces as right residuation and whose quantum logic becomes Girard's linear logic. 1 Introduction 1.1 Prospects for Chu Spaces The automaton model of this paper, Chu spaces, is an outgrowth ...
A Dialectica Model of the Lambek Calculus
"... this paper. But it must be said from the start that the `answer' is only provided in semantical terms. The Proof Theory of the systems considered should be investigated in future work. Another warning is that the perspective of this note is basically from Category Theory as a branch of Mathematics, ..."
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Cited by 7 (1 self)
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this paper. But it must be said from the start that the `answer' is only provided in semantical terms. The Proof Theory of the systems considered should be investigated in future work. Another warning is that the perspective of this note is basically from Category Theory as a branch of Mathematics, so words like categories and functors are always meant in their mathematical, rather than linguistical or philosophical sense. We first recall Linear Logic and provide the transformations to show that the Lambek Calculus L really is the multiplicative fragment of (noncommutative) Intuitionistic Linear Logic. In the second section we describe the usual String Semantics for the Lambek Calculus L and generalise it, using a categorical perspective. In the third section we describe our Dialectica model for the Lambek Calculus. In the last section we discuss modalities and some `untidiness' of the CurryHoward correspondence for the fragments of Linear Logic in question. I would like to thank Jan van Eijck for inviting me to give the talk that became this note, thereby gently `forcing' me to think about the subject, as well as, for his generous hospitality. I also would like to thank Martin Hyland, Harold Schellinx, Dirk Roorda, Mark Hepple, Glyn Morrill and Michael Moortgat for several useful discussions. Many of the ideas in this paper have been shaped by these discussions, but of course the mistakes are all mine. Finally I want to thank Jim Lambek for `putting me right' about how completeness has nothing to do with the existence of two disjunctions, in the most friendly possible way. 1 From Linear Logic to the Lambek Calculus
Rational mechanics and natural mathematics
 In TAPSOFT'95
, 1995
"... 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 mec ..."
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Cited by 4 (1 self)
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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 mindbody 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 betweenplane interaction alone. Lifting the twovalued Boolean logic of binary relations to the complexvalued 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 HeisenbergSchrödinger quantummechanical 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
Categorical Multirelations, Linear Logic and Petri Nets
, 1991
"... This note presents a category of multirelations, which is, in a loose sense a generalisation of both our previous work (the categories GC, [dP'89]) and of Chu's construction ANC [Barr'79]. The main motivation for writing this note was the utilisation of the category GC by Brown and Gurr [BG90] to mo ..."
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Cited by 3 (1 self)
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This note presents a category of multirelations, which is, in a loose sense a generalisation of both our previous work (the categories GC, [dP'89]) and of Chu's construction ANC [Barr'79]. The main motivation for writing this note was the utilisation of the category GC by Brown and Gurr [BG90] to model Petri Nets. We wanted to extend their work to deal with multirelations, as Petri Nets are usually modelled using multirelations pre and post. That proved easy enough and people interested mainly in concurrency theory should refer to our joint work [BGdP'91]; this note deals with the mathematics underlying [BGdP'91]. The upshot of this work is that we build a model of Intuitionistic Linear Logic (without modalities) over any symmetric monoidal closed category C with a distinguished object (N; ; ffi; e \Gammaffi)  a closed poset. Moreover, if the category C is cartesian closed with free commutative monoids, we build a model of Intuitionistic Linear Logic with a nontrivial modality `!'...
Exponentials as Projections from Paraconsistent Logics
 First World Congress on Paraconsistency, Ghent Belgium
, 1997
"... An exponential is a unary operator, reminiscent of a modality. We suggest that exponentials be regarded as logic transformers, transforming paraconsistent logics to consistent logics. Consistent logics may appear as certain syntactic subsets of paraconsistent logics. We consider this embedding alge ..."
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An exponential is a unary operator, reminiscent of a modality. We suggest that exponentials be regarded as logic transformers, transforming paraconsistent logics to consistent logics. Consistent logics may appear as certain syntactic subsets of paraconsistent logics. We consider this embedding algebraically and see that an exponential is both a projection and an interior operation. We consider herein a logic constructible duality which is a conservative extension of the paraconsistent version of Nelson's constructible falsity. We give an onto representation theorem for the algebra of constructible duality, show that it provides a model for linear logic and in that model ! is an exponential (as defined by linear logicians) a projection operator and an interior operator. Another exponential ! is definable in constructible duality which has no linear logic counterpart. This is also a projection and an interior operator. Closure and interior operations can be thought of as approximations....