V.R. Pratt. The Stone gamut: A coordinatization of mathematics. In Logic in Computer Science, pages 444--454. IEEE Computer Society, June 1995.  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, The Stone Gamut: A Coordinatization of Mathematics, In: Proceedings of Logic in Computer Science Conference LICS'95, IEEE Computer Society, June 1995, pp. 444--454.

.... propositional theories, or history preserving process graphs [GP95] Chu spaces are of independent mathematical interest, forming a bicomplete self dual closed category first studied by Barr and Chu [Bar79] which has found applications elsewhere in proof theory, game theory, and Stone duality [Pra95] Chu spaces are the subject of Vaughan Pratt s position statement, accessible by http: boole.stanford.edu stmt.html. 3.3 Design and Verification Methodologies There are two important observations about the current state in the design and verification of concurrent systems: ffl There are many ....

V. R. Pratt. The Stone gamut: A coordinatization of mathematics. In Logic in Computer Science, pages 444--454. IEEE Computer Society, June 1995.

....logic with an involutory constructive negation : asserting the effective construction of a counterexample to . cf. Zaslavskii(1973) for another symmetrization. Duality and symmetry are prevalent in the study of linear logic and concurrency(de Paiva 1989, de Paiva and Hyland 1993, Gupta 1994, Pratt 1995). The duality in linear logic arises from the presence of a constructive involutory negation and the symmetric nature of proofs (Girard 1987) The calculus(Kozen 1983) is another natural dual logic where its negation acts as a dualizing operator, turning least fixed points to greatest fixed ....

Pratt, V. 1995. The Stone Gamut: A Coordinatization of Mathematics. In Logic in Computer Science. IEEE Computer Society, June.

....concurrency theory, capturing the duality of states and y This work was partly supported under CEC grant ERBCHBGCT930496 and under ONR grant N00014 92 J 1974. D. Pavlovi c 2 events (Gupta and Pratt 1993; Pratt 1993a; Pratt 1993b; Pratt 1994a; Pratt 1994b; Gupta 1994; Glabbeek and Plotkin 1995; Pratt 1995). They were shown to be remarkably rich and versatile, accomodating concrete faithful functors from arbitrary small concrete categories. However, no categorical universal property (Mac Lane 1971, ch. III) of the Chu construction has been established so far, no explanation in the style it is the ....

Pratt, V.R. (1995) The Stone Gamut: A coordinatization of mathematics. In: Proc. of the 10th Symp. on Logics in Comp. Sci., 444--454. IEEE Computer Society Press.

....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, The Stone Gamut: A Coordinatization of Mathematics, In: Proceedings of Logic in Computer Science Conference LICS'95, IEEE Computer Society, June 1995, pp. 444--454.

....programming language structuring principles. We exemplify the duality structure between the category of data and the category of queries using relational databases, but there are many other examples. Stone dualities have many applications in mathematics, 18] and also in computer science, [24, 2, 22]. The dualities here are formally of this type, but the correspondence we emphasize is between data and queries. Now the data is always, for us, a category of models of a general sketch so the difference is only one of presentation. However, this perspective enables one to concentrate on the ....

V. Pratt, The Stone Gamut: A Coordinatization of Mathematics, Proc. Tenth Annual IEEE Symp. Logic in Comput. Sci., IEEE Computer Society Press, 1995, pp. 444-454.

....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. The Stone gamut: a coordinatization of mathematics. In Logic in Computer Science. IEEE Computer Society, 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, "The Stone Gamut: A Coordinatization of Mathematics", In: Proceedings of Logic in Computer Science Conference LICS'95, IEEE Computer Society, June 1995, pp. 444--454.

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V.R. Pratt. The Stone gamut: A coordinatization of mathematics. In Logic in Computer Science, pages 444--454. IEEE Computer Society, June 1995.

.... as universal topology, a generalization and simplification of universal algebra in which the operations of linear logic constitute pure versions of their counterparts in more application specific categories, e.g. direct sum U Phi V , tensor product U Omega V , and dual U of vector spaces [9, 10]. All proofs between MLL terms, such as of p p and r r) have interpretations as dinatural transformations, showing the soundness of MLL for Chu spaces. Here we prove the converse for a fragment of MLL: every dinatural transformation between the interpretations of terms of this fragment ....

V.R. Pratt. The Stone gamut: A coordinatization of mathematics. In Logic in Computer Science, pages 444--454. IEEE Computer Society, June 1995.

....and for foundations of mathematics and semantics of concurrency. Chu spaces provide a linear counterpart to relational structures as universal mathematical objects, inasmuch as a great many concrete categories arising in mathematical practice embed fully and concretely in Chu(Set; K) for some K [Pra95b]. Moreover the rows and columns of Chu spaces model events and states of concurrent processes with the same even handedness as Petri nets accord their places and transitions, but with a richer process algebraic structure [GP93, Gup94, Pra95a, VGP95] Given these considerations, the exact match of ....

V.R. Pratt. The Stone gamut: A coordinatization of mathematics. In Logic in Computer Science, pages 444--454. IEEE Computer Society, June 1995.

....and for foundations of mathematics and semantics of concurrency. Chu spaces provide a linear counterpart to relational structures as universal mathematical objects, inasmuch as a great many concrete categories arising in mathematical practice emded fully and concretely in Chu(Set; K) for some K [Pra95b]. Moreover the rows and columns of Chu spaces model events and states of concurrent processes with the same even handedness that Petri nets grant to their places and transitions, but with a richer process algebraic structure [Pra95a, VGP95] Given these considerations, the exact match of the ....

V.R. Pratt. The Stone gamut: A coordinatization of mathematics. In Logic in Computer Science, pages 444--454. IEEE Computer Society, June 1995.

....model theory. And the selfduality of finite dimensional vector spaces, along with its infinite dimensional extensions to topological vector spaces and Hilbert spaces, is the basis for linear algebra and its many applications. Observation (ii) constitutes the present author s present interest [23,24]. The core theorems of this observation are that all small concrete categories, as well as all categories of algebraic or relational structures and their homomorphisms, are concretely representable as categories of biextensional Chu Mackey spaces [15,3] and their continuous maps. A Chu Mackey ....

.... method of constructing such categories is the Chu Mackey construction, described in the appendix of [3] In the case of ordinary Chu spaces, the Chu construction applied to the category Set, these have the additional advantages of accommodating essentially all of transformational mathematics [24], and of extending event structures in a natural way [10,9,27] These circumstances combine to make involutary complementation very appealing. We assume it henceforth. Pratt 2.3 Aggregation Aggregation forms large collections from small. As an operation aggregation is definable at all arities, ....

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V.R. Pratt. The Stone gamut: A coordinatization of mathematics. In Logic in Computer Science, pages 444--454. IEEE Computer Society, June 1995.

....in which the adjointness condition defining continuity is relaxed from an equality to an inequality. Our own interest in Chu spaces originated in their application to the representation of generalized event structures [GP93] but we have since found them also of interest as universal objects [Pra95,Pra96], broadening the denotational semantics of linear logic to a much larger, in fact universal, class of mathematical objects than previously associated with linear logic. 2 Representation A Chu space resembles a formal language, in that it may be understood intuitively as a set of words over an ....

....operations, for which a sufficient basis is implication and the constant 0. More generally, with Sigma still 2, the maximal structure with which we may equip Sigma X is that of a complete atomic Boolean algebra. If we ignore questions of concreteness, this remains true for larger Sigma [Pra95]. 5 Abstract Structure In this section we shall identify the states of any extensional Chu space (A; r; X) with the dual words representing them. Thus instead of an extensional Chu space being (A; r; X) with r : X Sigma A being injective, we have just (A; X) with r(a; x) being defined ....

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V.R. Pratt. The Stone gamut: A coordinatization of mathematics. In Logic in Computer Science, pages 444--454. IEEE Computer Society, June 1995.

....mapping of points; thus Chu spaces transform by looking before they leap. With regard to utility, besides the motivating application to concurrent computation, Chu spaces find application in mathematics, where 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 ....

....introduced, so no structure (in the form of disallowed states) is lost. If the target has structure absent from the source, the structure must first be added by deleting columns or the row mapping will not be possible. This gives Chu transforms the character of structure preserving homomorphisms [Pra95b], with continuous functions then falling out as an obvious special case when analyzed as above. All relational structures are realizable as Chu spaces [Pra93, Pra95b] as are topological spaces [LS91] These representations can be combined to represent topological relational structures such as ....

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V.R. Pratt. The Stone gamut: A coordinatization of mathematics. In Logic in Computer Science, pages 444--454. IEEE Computer Society, June 1995.

....meet the conditions for being a topological space with X consisting of its open sets, Chu transforms are exactly continuous functions. More generally, Chu transforms can be understood as exactly the homomorphisms preserving whatever structure is implicit in the selection of X as a subset of K A [Pra95, x4]: the smaller X is, the more structure there is to preserve. Definition 2 A quantale [Mul86, Ros90] is a complete join semilattice Q (and hence a complete lattice) with an associative binary operation p Delta q that distributes over arbitrary sups on both sides (p Delta W i q i = W i (p ....

....that hold pointwise for the rows of A. Theorem 8 Given extensional Chu spaces (A; j; X) and (A 0 ; j 0 ; X 0 ) and a function f : A A 0 , f is a Chu transform if and only if it is a homomorphism of the induced partial quantales on A and A 0 . This is a special case of Theorem 2 of [Pra95] identifying continuity with homomorphism, namely for those properties expressible in the equational language of quantales. These are only a small subset of the set 2 K A of all possible properties of a Chu space A, all of which are covered by the general theorem. What is shocking about this ....

V.R. Pratt. The Stone gamut: A coordinatization of mathematics. In Logic in Computer Science, pages 444--454. IEEE Computer Society, June 1995.

.... At that level of abstraction duality and interaction are themselves duals, as witnessed by the interchange of vertices and edges when dualizing a linear order [Pra92] Furthermore duality and interaction interact fruitfully as witnessed for concurrency theory by [GP93] and mathematics by [Pra95], both applications of the Chu construction, the mathematical quintessence of dual interaction. Dual interaction rests on itself. The category Set is where categories and sets must get al..ong. We shall give a new axiomatization of Set that stresses duality and simplicity throughout. The objects of ....

.... was made by the tensor product, which is a very simple yet remarkably effective deduction engine The blending process tends to average the knowledge to concept balance of its operands, which can range from coherent (knowing a lot about a little) to discrete (knowing little about a lot) see [Pra95] for more detailed numerical aspects of this measure. Perp negates this balance; thus if A is off center (the origin) in one direction, A will be just as off balance in the other. A Omega A then produces a larger but more balanced object. Depending on which way round one looks at the ....

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V.R. Pratt. The Stone gamut: A coordinatization of mathematics. In Logic in Computer Science, pages 444--454. IEEE Computer Society, June 1995.

....same functions, no more or less. From this viewpoint of Chu spaces as universal objects, the operations of linear logic constitute pure versions of their counterparts in more application specific categories, e.g. direct sum U Phi V , tensor product U Omega V , and dual U of vector spaces [14, 15]. They are pure in the sense that their definition makes no concessions to special features of any category of mathematics such as vector spaces or complete semilattices, whose tensor products are defined to take into account that the result should remain within the category at hand. While this ....

V.R. Pratt. The Stone gamut: A coordinatization of mathematics. In Logic in Computer Science, pages 444--454. IEEE Computer Society, June 1995.

....homogeneous universe of Chu spaces appears to span the entire range of structures, having sets at the discrete end, complete atomic Boolean algebras at the coherent end, and finite dimensional vector spaces, complete semilattices, etc. in the middle, constituting what have called the Stone gamut [Pra95]. As we have previously shown [Pra93, Pra95] archivally documented here, this representation of relational and topological structures is formalized as a full, faithful, and concrete functor from the category of k ary relational structures and their homomorphisms to the category of Chu spaces ....

....to span the entire range of structures, having sets at the discrete end, complete atomic Boolean algebras at the coherent end, and finite dimensional vector spaces, complete semilattices, etc. in the middle, constituting what have called the Stone gamut [Pra95] As we have previously shown [Pra93, Pra95], archivally documented here, this representation of relational and topological structures is formalized as a full, faithful, and concrete functor from the category of k ary relational structures and their homomorphisms to the category of Chu spaces over an alphabet of cardinality 2 k , or 2 ....

[Article contains additional citation context not shown here]

V.R. Pratt. The Stone gamut: A coordinatization of mathematics. In Logic in Computer Science, pages 444--454. IEEE Computer Society, June 1995.

....of linear logic. We first encountered Chu spaces ourselves while searching for a suitably general and natural model of true concurrency [GP93] We have shown that Chu realizes many large concrete categories of mathematics, in particular that of all relational structures and their homomorphisms [Pra93, Pra95], which in turn realizes Grp, Vct k , and most other familiar concrete mathematical categories. It also realizes the usual categories that can be formed from these by adding topology, such as topological groups, topological Boolean algebras, topological vector spaces, etc. 2 Result We first ....

....of a set A, the underlying set or carrier, and a set 1 X of functions g : A K, the states. A Chu morphism between normal Chu spaces (A; X) A 0 ; X 0 ) is a function f : A A 0 such that for every g : A 0 K in X 0 , gf 2 X . More details about Chu spaces may be found elsewhere [Pra95]. The essential idea of our representation is to represent the morphisms of C as their underlying functions with their codomains expanded or astricted to a single common codomain K. Composing with an inclusion on the right is called a restriction, on the left an astriction. That is, let B B 0 ....

V.R. Pratt. The stone gamut: A coordinatization of mathematics. In Logic in Computer Science. IEEE Computer Society, June 1995.

No context found.

V. R. Pratt, The Stone Gamut: A Coordinatization of Mathematics, In: Proceedings of Logic in Computer Science Conference LICS'95, IEEE Computer Society, June 1995, pp. 444--454.

No context found.

V.R. Pratt (1995), The Stone Gamut: A Coordinatization of Mathematics. In Proceedings of the conference Logic in Computer Science IEEE Computer Society.

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V.R. Pratt (1995), The Stone Gamut: A Coordinatization of Mathematics. In Proceedings of the conference Logic in Computer Science IEEE Computer Society.

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