V. R. Pratt, Rational Mechanics and Natural Mathematics, Proceedings of TAPSOFT'95, Aarhus, Denmark, 1995, Springer Lecture Notes in Computer Science, 1995, Vol. 915, pp. 108--122.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....measurement. As a result, it is not possible to uniquely reconstruct an object from measurement results. In other words, each measurement is a function r(x; y) of two variables: an object x and a (not completely known) measuring device y. Such a function describes a so called Chu space (see, e.g. [1, 2, 7, 8, 20, 21, 22, 23, 24, 25, 26]) 1.3 Precise Definition of a Chu Space To be more precise, to define a Chu space, we must fix a set K (of possible values) Then, a K Chu space is defined as a triple (X; r; Y ) where X and Y are sets, and r : X Theta Y K is a function which maps every 1 pair (x; y) of elements x 2 X and ....

....Y ) is called a morphism of Chu spaces if it satisfies the property (2) for all x 2 X and for all z 2 Y 0 . 1.7 Applications to Parallelism and to Information Flow The notion of Chu spaces was actively used by V. Pratt (Stanford) for describing parallel problem solving algorithms (see, e.g. [7, 8, 20, 21, 22, 23, 24, 25, 26]) and by J. Barwise (Indiana) to describe information flow in general (see, e.g. 3] 2 Fuzzy as a Natural Particular Case of Chu Spaces Before we describe how Chu spaces can be used to justify fuzzy heuristics, let us show that fuzzy methodology can indeed be reformulated in Chu space ....

V. R. Pratt, Rational Mechanics and Natural Mathematics, Proceedings of TAPSOFT'95, Aarhus, Denmark, 1995, Springer Lecture Notes in Computer Science, 1995, Vol. 915, pp. 108--122.

....the constructive intend of linear logic in the sense of the Curry Howard isomorphism is concurrency as found by Samson Abramsky. We start now in making these ideas formal by developing this in the frame of Chu spaces. 5. 1 Chu spaces Chu spaces are the mathematical modeling of the idea of duality [Pra94b, Pra95a, Pra95b, Gup94, Gor95]. It can be seen as the abstraction of the stone duality for classical logic. The stone duality 4 observes that boolean algebras and its valuations can be modeled by a set system. Either boolean formulas are the elements and valuations the sets or dually the other way. A Chu space over K is a ....

Vaughan Pratt. Rational mechanics and natural mathematics. In TAPSOFT' 95, pages 108--122, Aarhus, Denmark, 1995. Springer-Verlag. LNCS 915.

....As a result, it is not possible to uniquely reconstruct an object from measurement results. In other words, each measurement is a function r(x; y) of two variables: an object x and a (not completely known) measuring device y. Such a function describes a so called Chu space (see, e.g. [1, 2, 7, 8, 16, 17, 18, 19, 20, 21, 22]) 1.3. Precise definition of a Chu space To be more precise, to define a Chu space, we must fix a set K (of possible values) Then, a K Chu space is defined as a triple (X; r; Y ) where X and Y are sets, and r : X Theta Y K is a function which maps every pair (x; y) of elements x 2 X and y ....

.... Y ) is called a morphism of Chu spaces if it satisfies the property (2) for all x 2 X and for all z 2 Y 0 . 1.7. Applications to parallelism and to information flow The notion of Chu spaces was actively used by V. Pratt (Stanford) for describing parallel problemsolving algorithms (see, e.g. [7, 8, 16, 17, 18, 19, 20, 21, 22]) and by J. Barwise (Indiana) to describe information flow in general (see, e.g. 3] 2. Chu spaces as a uniform justification for fuzzy techniques 2.1. Fuzzy is a particular case of Chu spaces Fuzzy knowledge can be naturally described as a Chu space (X; r; Y ) where X is the set of all ....

V. R. Pratt, Rational Mechanics and Natural Mathematics, Proceedings of TAPSOFT'95, Aarhus, Denmark, 1995, Springer Lecture Notes in Computer Science, 1995, Vol. 915, pp. 108--122.

....of cardinality up to three find valuable employment in the process interpretation of Chu spaces, for respectively untimed automata making discrete transitions, higher dimensional automata making continuous transitions, and causal automata making circumspective transitions. For references, see, [59, 60, 61]. On the decidability of a distributed decision task Sergio Rajsbaum, UNAM, Mexico rajsbaum concha.matem.unam.mx A task is a distributed coordination problem in which each process starts with a private input value taken from some finite set, communicates with the other processes by applying ....

V. R. Pratt. Rational mechanics and natural mathematics. In TAPSOFT'95, pages 108-- 122. Springer LNCS 915, 1995.

....theorem, simple semigroups are in 1 1 correspondence with functions Y Theta X H . Such functions form the basis of a new approach to foundations of concurrency and foundations of computer science in general which is promoted by V. R. Pratt from Stanford under the name of Chu spaces (see, e. g, [5, 6, 7, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]) Thus, a general extension of t norms naturally leads us to Chu spaces. Auxiliary Results and Their Relationship With the Existence and Borderline Character of Classical Truth Values. According to [1] Theorem 1.8, and [11] Theorem 1.4.2, if a compact topological semigroup S is not a group ....

V. R. Pratt, "Rational Mechanics and Natural Mathematics", Proceedings of TAPSOFT'95, Aarhus, Denmark, 1995, Springer Lecture Notes in Computer Science, 1995, Vol. 915, pp. 108--122.

....something of a jigsaw puzzle. One is then naturally led to ask whether the pieces of this puzzle can be arranged in some recognizable order. In this paper we review the outcome to date of our recent investigations of Chu spaces as a candidate unifying framework for these aspects of concurrency [Pra92, Pra93, Gup93, GP93, Gup94, Pra94b, Pra95a]. Their simplicity is deceptive, and two years of experience with them have convinced us that they are more than adequately equipped for this role. Chu spaces are simple, useful, and well organized. With regard to simplicity, a Chu space is merely a matrix that transforms by deleting and copying ....

.... they organize relational structures, topology, and duality into a unified framework [Pra95b] physics, where they provide a process interpretation of wavefunctions [Pra94a] and philosophy, where they offer a solution to Descartes problem of the mechanism by which the mind interacts with the body [Pra95a]. Common to these applications is the construction of everything in the domain in question in terms of the interaction of appropriate polar opposites in the domain: in mathematics as the interaction of sets and antisets, in physics as the interaction of particles and waves, in philosophy as the ....

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V.R. Pratt. Rational mechanics and natural mathematics. In TAPSOFT'95, volume 915 of Lecture Notes in Computer Science, pages 108--122, Aarhus, Denmark, 1995. Springer-Verlag.

....computation in June of 1992 when we found with our student V. Gupta [Gup93, GP93, Gup94] that they captured exactly the notion of partial distributive lattice that we had been trying to pin down [Pra92d] as an extension of our earlier notion of event space [Pra92c, Pra92a, Pra92b] More recently [Pra95] we have arrived at the view of =j and its converse j= as the dual relations of causality and consciousness. The entry a=jx expresses the causal link from physical event a to mental state x whereby the actual time of a (relative to x) stamps itself on x. The converse link xj=a points backwards ....

.... is ambiguously interpretable as physical time and mental information (of that time) This simple ambiguity forms the basis for one direction of Descartes causal interaction of body and mind, a mathematically more attractive seat of causal interaction than Descartes proposal of the pineal gland [Pra95]. Each Chu space has its own clock, whose possible values are its events, and its own history, whose possible values are its states. Events constitute the points or instants of the space s idea of time, which is structured by the data in its rows. States form the possible records of its view of ....

V.R. Pratt. Rational mechanics and natural mathematics. In TAPSOFT'95, LNCS 915, pages 108--122, Aarhus, Denmark, 1995. Springer-Verlag.

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V. R. Pratt, Rational Mechanics and Natural Mathematics, Proceedings of TAPSOFT'95, Aarhus, Denmark, 1995, Springer Lecture Notes in Computer Science, 1995, Vol. 915, pp. 108--122.

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