Pratt, V.R. (1993). The Second Calculus of Binary Relations. Proc. 18th International Symposium on Mathematical Foundations of Computer Science, Gdansk, Poland, Springer-Verlag, p. 142-155.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of this proposal and the need to embed Intuitionistic Logic into Linear Logic. We can deal with this embedding simply because is a definable operator in ILL #2 . For evidence on the naturality of the proposal to split into two operators we mention that one finds in V. Pratt s Chu spaces [15] two maximal solutions for a storage operator and Girard s operator emerges as the set theoretic intersection of the two (V. Pratt [16] De Paiva s [14] construction of the operator also goes through a preliminary construction of two auxiliary comonads S and T related by a natural ....

V. Pratt, "The Second Calculus of Binary Relations", MFCS'93, Dda'nsk, Poland (1993) 142-155.

....as naturally as the closed monoidal category, and the challenge is to find a natural and useful construction for it. In this respect, we only note that the ongoing works on Chu construction by Pratt and Gupta, and game semantics by Abramsky and Jagadeesan seem to point to an interesting direction [1, 23]. 1.3.2 Interpretation of linear logic The meaning of a logical system is manifested most evidently in its entailment relation. We describe our intuitive understanding of the logical entailment in linear logic in terms of the useful analogy, consumption. This is in contrast to accumulation, which ....

V. R. Pratt. "The second calculus of binary relations." MFCS'93, Gda'nsk, Poland, 1993. BIBLIOGRAPHY 145

....C configurations. An event e in E represents an occurrence of an action the system may perform; a configuration X in C represents a state of the system, namely the state in which the events in X have been performed. Set systems are, up to isomorphism, the extensional Chu spaces of Gupta and Pratt [7, 8, 16, 14]. It was in their work that the idea of considering such structures as a model of concurrency arose. They pointed out that they generalise the event structures of Winskel and others [12, 19, 20, 2, 9] This is because the behaviour of the latter is described in terms of their configurations: the ....

....event isomorphic. 2.2 The isomorphisms We study configuration and history isomorphism in more detail. A configuration structure is standard if all its configurations are reachable, and all its events occur in some configuration. Standard configuration structures are coincidence free [19] and T 0 [16]. A configuration structure is T 0 (respectively, coincidencefree) iff for any two distinct events (occurring in a configuration) there is a (sub)configuration containing one but not the other) By deleting all non reachable configurations from a labelled configuration structure and all events ....

V.R. Pratt (1993): The second calculus of binary relations. In Proceedings of MFCS'93, Springer-Verlag, Gda'nsk, Poland, pp. 142--155.

....the category of sets, have been proposed by Pratt and his collaborators as a foundational structure for 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 ....

....of sets, have been proposed by Pratt and his collaborators as a foundational structure for 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) ....

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Pratt, V.R. (1993) The second calculus of binary relations, In Proc. Conference on Math. Foundations of Comp. Sci. IEEE Computer Society Press.

....can be embellished with the other operators from linear logic, and eks s then form a model for it, but this model is also not complete. However it does give us a logic to reason with eks s, and is useful in constructing a verification language for concurrency. This connection is explored in[Pra93]. ....

V.R. Pratt. The second calculus of binary relations. In Proceedings of MFCS'93, Gda'nsk, Poland, 1993. Springer-Verlag.

....of this proposal and the need to embed Intuitionistic Logic into Linear Logic. We can deal with this embedding simply because is a definable operator in ILL #2 . For evidence on the naturality of the proposal to split into two operators we mention that one finds in V. Pratt s Chu spaces [15] two maximal solutions for a storage operator and Girard s operator emerges as the set theoretic intersection of the two (V. Pratt [16] De Paiva s [14] construction of the operator also goes through a preliminary construction of two auxiliary comonads S and T related by a natural ....

V. Pratt, "The Second Calculus of Binary Relations", MFCS'93, Dda'nsk, Poland (1993) 142-155.

....behaves as sequential consistency in the absence of data races. The implementations that we develop are similar in spirit, but based on a different set of primitives closer to the machine architecture. Pratt and Gupta have developed a theory of concurrency based on the algebra of Chu spaces [13, 14, 23]. In this theory, a duality exists between schedules (sequences of events) and automata (state based structures) This is similar in spirit to our construction of relatively complete implementations, since the possible behaviors of an automaton are captured by the schedule which is its dual, and ....

Pratt, V. The second calculus of binary relations. In MFCS '93 (Gdansk, Poland, 30 Aug.--3 Sept. 1993), A. M. Borzyszkowski and S. Sokolowski, Eds., SpringerVerlag, pp. 142--155.

....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 Second Calculus of Binary Relations", Proceedings of MFCS'93, Gda'nsk, Poland, 1993, pp. 142--155.

....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 ....

....two such algebras as Chu transforms (h; h) where h is the unique right adjoint to h, A; f) being extensional by this construction. We have stated this for algebras, where it reads reasonably smoothly. It does however generalize to arbitrary relational structures. We have shown elsewhere [23,24] that this representation of such n ary relational structures and their homomorphisms by Chu spaces over 2 n and their Chu transforms is a full embedding of the category of the former in that of the latter. Furthermore the embedding is concrete, meaning that the representing Chu spaces have the ....

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V.R. Pratt. The second calculus of binary relations. In Proceedings of MFCS'93, volume 711 of Lecture Notes in Computer Science, pages 142--155, Gda'nsk, Poland, 1993. Springer-Verlag.

....these as gate based schedules that mention each event only once. This eliminates the usual dependence on labeling needed to distinguish these two notions from respectively late branching a(b c) and true concurrency akb. In addition, by tying this framework into the algebra of Chu spaces [Bar79, LS91, Bar91, Pra93] we implicitly bring to bear on gates as acceptors the by now considerable machinery of both Chuology and linear logic, with the expectation moreover of seeing many further developments in this very interesting and (we conjecture) extremely rich new field. 2 Processes as Gates 2.1 Dynamic ....

....and homomorphisms of PDLat s. The category Chu(V; k) of Chu spaces and their associated transforms, with dualizing object k, was first defined in Po Hsiang Chu s master s thesis, which appeared as an appendix to his advisor Michael Barr s monograph on autonomous categories [Bar79] Here and in [Pra93] we follow Lafont and Streicher [LS91] in focusing on Chu s construction for V = Set. In the present paper we further take k = f0; 1g. Numeric quantities are not the only mathematical objects on which one can perform arithmetic. One may also add and multiply vector spaces, groups, etc. Chu spaces ....

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V.R. Pratt. The second calculus of binary relations. In Proceedings of MFCS'93, Gda'nsk, Poland, 1993. Springer-Verlag.

....one measuring the generality of Chu spaces in terms of arbitrary relational structures and their homomorphisms, of which the foregoing are examples, the other in terms of arbitrary small categories, for which a forgetful functor may or may not be given. For all these situations we have shown [Pra93,Pra96], that the objects in question can be represented by Chu spaces in such a way that the continuous functions between the representing Chu spaces are exactly the homomorphisms of the corresponding represented objects. More precisely, in each case the category of interest embeds fully and concretely ....

....to arbitrary R by intersecting the X s that arise as above for each n tuple in R, and restricting r to A times that intersection. One can then show that a function from (A; R) to (B; S) is a homomorphism if and only if it is a continuous function between the respective representing Chu spaces [Pra93]. The construction generalizes to multiple relations by combining them into one super relation with arity the sum of its constituent arities. It further generalizes to multiple sorts by combining them into one sort with disjoint union and indicating the sorts with one unary relation per sort. ....

V.R. Pratt. The second calculus of binary relations. In Proceedings of MFCS'93, volume 711 of Lecture Notes in Computer Science, pages 142-- 155, Gda'nsk, Poland, 1993. Springer-Verlag.

....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 ....

....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 topological groups, topological vector spaces (the main autonomous category studied in [Bar79] ordered Stone spaces [Pri70] and so on. More recently (note in ....

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V.R. Pratt. The second calculus of binary relations. In Proceedings of MFCS'93, volume 711 of Lecture Notes in Computer Science, pages 142--155, Gda'nsk, Poland, 1993. Springer-Verlag.

....the partial distributive lattice categories (e.g. locales, semilattices, and algebraic lattices, but not abelian groups or nondistributive lattices) treated by Johnstone [Joh82] arise at K = 2. Discussions with M. Barr following an impromptu talk on Chu spaces at MFPS 93 led us to the theorem [Pra93] that every category C of (possibly multisorted) relational structures of total arity k and their homomorphisms was realizable in the category of Chu spaces over 2 k , in the sense that there exists a full and faithful functor F : C Chu 2 k (the notion of representation) which commutes with ....

....logic but because it has a maximal equational theory, dual to Set being the minimal (empty but for x = x) equational theory, both with regard to signature and equations. That is, it is as logical as it can be, at least with respect to K = 2. Our realization theorem for k ary relational structures [Pra93] as Chu spaces over K = 2 k shows that bringing in larger K lets us extend propositional logic to the rest of mathematics. Interestingly, any jKj 2 serves to represent the same (up to isomorphism) embedding of Set op in Set, since the cube K n behaves structurally like 2 n in that each ....

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V.R. Pratt. The second calculus of binary relations. In Proceedings of MFCS'93, pages 142-- 155, Gda'nsk, Poland, 1993. Springer-Verlag.

....such connections could prove useful in both computer science and physics. The full picture of Chu spaces that has emerged for us during the past two years since we began using them is beyond the scope (or at least available space) of this paper. We therefore refer the reader to other recent work [Pra94, Gup94, GP93, Pra93b], in that order. These papers are all available either as cited, or by anonymous ftp (start with pub ABSTRACTS from boole.stanford.edu) or via World Wide Web (WWW) via mosaic http: boole.stanford.edu . In computer science Chu spaces as an automatatheoretic abstraction of quantum mechanics ....

V.R. Pratt. The second calculus of binary relations. In Proceedings of MFCS'93, pages 142--155, Gda'nsk, Poland, 1993. SpringerVerlag.

....but as a field. Set does not cater for this notion. Standardly one implements algebraic structure with relational structure, sets of tuples over the carrier, and topological structure as sets of open sets. The Chu construction and its applications, not treated here but which may be found elsewhere [Bar79, LS91, Pra93], is, we have argued [Pra95] a workable universal framework for much if not most of concrete mathematics. This is the machinery that should be used in conjunction with this Set to create the rest of mathematics, as the chain of categories Chu(Set; K) where K is an arbitrary set, an ordinal in ....

V.R. Pratt. The second calculus of binary relations. In Proceedings of MFCS'93, volume 711 of Lecture Notes in Computer Science, pages 142--155, Gda'nsk, Poland, 1993. SpringerVerlag.

....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 ....

V.R. Pratt. The second calculus of binary relations. In Proceedings of MFCS'93, volume 711 of Lecture Notes in Computer Science, pages 142--155, Gda'nsk, Poland, 1993. Springer-Verlag.

.... row and column are interchanged everywhere, dealing immediately with the case of projection onto the time axis, yielding a schedule. That these objects are recognizable automata and schedules respectively (with loops and disjunctive confluences unfolded) can be appreciated from the examples in [Gup93, GP93, Pra93b, Gup94, Pra94a]. 6.4 Universality of Chu Spaces The categories Str of ary relational structures and their homomorphisms where is any ordinal are universal categories for mathematics to the extent that they realize many familiar categories: groups, lattices, and Boolean algebras when = 3, rings, fields, and ....

....of the category Chu(Set; 2) of Chu spaces over the twoelement set 2 = f0; 1g. Chu spaces have an equally evident duality obtained by transposition, which has the side effect of reversing the direction of the Chu transforms. As noticed by Barr [Bar91] and further developed by several authors [dP89, BG90, LS91, BGdP91, Pra93b], Chu spaces provide a straightforward interpretation of full linear logic; the fact that Set (as well as Pos and other even larger cartesian closed categories) is comonadic (cotripleable) in Chu gives a straightforward interpretation of Girard s bang operation A as the functor of that comonad. ....

V.R. Pratt. The second calculus of binary relations. In Proceedings of MFCS'93, pages 142--155, Gda'nsk, Poland, 1993. Springer-Verlag.

....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 ....

V.R. Pratt. The second calculus of binary relations. In Proceedings of MFCS'93, pages 142--155, Gda'nsk, Poland, 1993. Springer-Verlag.

....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 computer circuits was born of philosophy (and a little ....

....space over a field k, which requires k to be equipped with the four rationals; here K is simply a set with no additional structure. product in a cartesian closed category of partially ordered multisets (pomsets) but subsequently generalized it to the tensor product of any closed category [CCMP91, Pra93, GP93, Pra94a]. In all cases we took as our basic example the interaction of trains and stations described on the train station wall by the daily schedule. Whereas ordinary product must be capable of being projected consistently onto either component, tensor product requires only that each row or column of the ....

V.R. Pratt. The second calculus of binary relations. In Proceedings of MFCS'93, pages 142--155, Gda'nsk, Poland, 1993. Springer-Verlag.

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Pratt, V.R. (1993). The Second Calculus of Binary Relations. Proc. 18th International Symposium on Mathematical Foundations of Computer Science, Gdansk, Poland, Springer-Verlag, p. 142-155.

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Pratt, V.R. (1993). The Second Calculus of Binary Relations. Proc. 18th International Symposium on Mathematical Foundations of Computer Science, Gdansk, Poland, Springer-Verlag, p. 142-155.

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V.R. Pratt (1993b), The Second Calculus of Binary Relations. In Proceedings of MFCS'93, Gda'nsk, Poland, 142--155.

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V.R. Pratt (1993b), The Second Calculus of Binary Relations. In Proceedings of MFCS'93, Gda'nsk, Poland, 142--155.

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V.R. Pratt (1993), The Second Calculus of Binary Relations. In Proceedings of MFCS'93, Gda'nsk, Poland, 142--155.

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