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SOME GEOMETRIC PERSPECTIVES IN CONCURRENCY THEORY
 HOMOLOGY, HOMOTOPY AND APPLICATIONS, VOL.5(2), 2003, PP.95–136
, 2003
"... Concurrency, i.e., the domain in computer science which deals with parallel (asynchronous) computations, has very strong links with algebraic topology; this is what we are developing in this paper, giving a survey of “geometric” models for concurrency. We show that the properties we want to prove on ..."
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Cited by 42 (3 self)
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Concurrency, i.e., the domain in computer science which deals with parallel (asynchronous) computations, has very strong links with algebraic topology; this is what we are developing in this paper, giving a survey of “geometric” models for concurrency. We show that the properties we want to prove on concurrent systems are stable under some form of deformation, which is almost homotopy. In fact, as the “direction ” of time matters, we have to allow deformation only as long as we do not reverse the direction of time. This calls for a new homotopy theory: “directed ” or dihomotopy. We develop some of the geometric intuition behind this theory and give some hints about the algebraic objects one can associate with it (in particular homology groups). For some historic as well as for some deeper reasons, the theory is at a stage where there is a nice blend between cubical, ωcategorical and topological techniques.
Chu spaces and their interpretation as concurrent objects
, 2005
"... A Chu space is a binary relation =  from a set A to an antiset X defined as a set which transforms via converse functions. Chu spaces admit a great many interpretations by virtue of realizing all small concrete categories and most large ones arising in mathematical and computational practice. Of pa ..."
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Cited by 21 (0 self)
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A Chu space is a binary relation =  from a set A to an antiset X defined as a set which transforms via converse functions. Chu spaces admit a great many interpretations by virtue of realizing all small concrete categories and most large ones arising in mathematical and computational practice. Of particular interest for computer science is their interpretation as computational processes, which takes A to be a schedule of events distributed in time, X to be an automaton of states forming an information system in the sense of Scott, and the pairs (a, x) in the =  relation to be the individual transcriptions of the making of history. The traditional homogeneous binary relations of transition on X and precedence on A are recovered as respectively the right and left residuals of the heterogeneous binary relation =  with itself. The natural algebra of Chu spaces is that of linear logic, made a process algebra by the process interpretation.
Time and Information in Sequential and Concurrent Computation
 In Proc. Theory and Practice of Parallel Programming
, 1994
"... 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 computat ..."
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Cited by 6 (1 self)
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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 twodimensional logics of sequential behavior, the van Glabbeek map of branching time and true concurrency, eventstate duality and scheduleautomaton 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...
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 5 (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
Chu I: cofree equivalences, dualities and *autonomous categories
, 1993
"... ing from the technique of dual pairs in functional analysis (Kelley, Nanmioka et al. 1963, ch. 5), they defined the objects of their category to be the triples hA; B; A\Omega B OE !?i, where A and B are arbitrary objects of an autonomous category V, and ? is a fixed object, chosen to become duali ..."
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Cited by 5 (1 self)
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ing from the technique of dual pairs in functional analysis (Kelley, Nanmioka et al. 1963, ch. 5), they defined the objects of their category to be the triples hA; B; A\Omega B OE !?i, where A and B are arbitrary objects of an autonomous category V, and ? is a fixed object, chosen to become dualizing. A morphism from hA; B; OEi to hC; D; fli was defined as a pair hu : A ! C; B / D : vi of Varrows, making the square A\Omega D A\Omega B C\Omega D ? u\Omega D<Fnan><Fnan> fflffl A\Omega v<Fnan><Fnan> // OE<Fnan><Fnan> fflffl fl<Fnan><Fnan> (1) Cofree equivalences, dualities and autonomous categories 3 commute. This is the setting in which the autonomous structure of a Chu category was originally discovered. The starting point of the present paper is the fact that the category described by Chu is isomorphic to the comma category V=? ? , induced by the homming functor ? ? : V op \Gamma! V : A 7\Gamma! A ? = A \Gammaffi? : (2) By definition, the objects of V=? ? (i.e. Id V =?...
Quantum Computing: A new Paradigm and it's Type Theory
 Lecture given at the Quantum Computing Seminar, Lehrstuhl Prof. Beth, Universität
, 1996
"... To use quantum mechanical behavior for computing has been proposed by Feynman. Shor gave an algorithm for the quantum computer which raised a big stream of research. This was because Shor's algorithm did reduce the yet assumed exponential complexity of the security relevant factorization proble ..."
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Cited by 3 (0 self)
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To use quantum mechanical behavior for computing has been proposed by Feynman. Shor gave an algorithm for the quantum computer which raised a big stream of research. This was because Shor's algorithm did reduce the yet assumed exponential complexity of the security relevant factorization problem, to a quadratic complexity if quantum computed. In the paper a short introduction to quantum mechanics can be found in the appendix. With this material the operation of the quantum computer, and the ideas of quantum logic will be explained. The focus will be the argument that a connection of quantum logic and linear logic is the right type theory for quantum computing. These ideas are inspired by Vaughan Pratt's view that the intuitionistic formulas argue about states (i.e physical quantum states) and linear formulas argue about state transformations (i.e computation steps). 1 Introduction A calculus for programs on quantum computers is strongly missed. Here we present the material t...
The Pragmatics of HDA for Concurrent Program Analysis
, 1997
"... In this paper we discuss the relevance of HigherDimensional Automata to program analysis. Most of the work on that model for trueconcurrency dealt with the nature of the model itself [Pra91b, vG91], the ways one could use it to give semantics to concurrent programming languages [GJ92, Gou93]. In [ ..."
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In this paper we discuss the relevance of HigherDimensional Automata to program analysis. Most of the work on that model for trueconcurrency dealt with the nature of the model itself [Pra91b, vG91], the ways one could use it to give semantics to concurrent programming languages [GJ92, Gou93]. In [CG93, Gou95b] a first step was made towards a practical use of the tools provided by the HDA semantics, whereas in [Cri95, Cri96] a prototype analyser was presented for a concurrent Pascal (and for CML). In this paper, we explain the pragmatics of HDA for concurrent program analysis on various levels: definition and presentation of semantics, expressivity, complexity of the computer representation and implementation issues (complexity of deadlock detection in this model, following [FR96], being given as a good test case). We also introduce a new abstract interpretation for finding and verifying schedules of concurrent systems, some ideas of which have been inspired by some work in concurrent...
Extending TNorms Beyond [0,1]: Relevant Results of Semigroup Theory
, 1999
"... Originally, fuzzy logic was proposed to describe human reasoning. Lately, it turned out that fuzzy logic is also a convenient approximation tool, and that moreover, sometimes a better approximation can be obtained if we use real values outside the interval [0, 1]; it is therefore necessary to descri ..."
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Originally, fuzzy logic was proposed to describe human reasoning. Lately, it turned out that fuzzy logic is also a convenient approximation tool, and that moreover, sometimes a better approximation can be obtained if we use real values outside the interval [0, 1]; it is therefore necessary to describe possible extension of tnorms and tconorms to such new values. It is reasonable to require that this extension be associative, i.e., that the set of truth value with the corresponding operation form a semigroup. Semigroups have been extensively studied in mathematics. In this short paper, we describe several results from semigroup theory which we believe to be relevant for the proposed extension of tnorms and tconorms.