Pratt, V.R., Chu Spaces: Complementarity and Uncertainty in Rational Mechanics. Technical Report, Stanford University, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and situation predicates. The corresponding relation RS contains all pairs of worlds that are members of this set. The relation R serves as the basis for PSL ordering operators. This relation is simplest to describe formally when expressions are restricted to event predicates on fixed messages[23, 66, 55]. In this case expressions in V are just characteristic predicates of sets of event occurrences, and OE(e; w) is true if w contains the events of e. For example, suppose a satisfies e a = out(m1) for some message m1, and b satisfies e b = out(m1) in(m1) The relation a b states that e b ) e a ....

....in a simple way under the intended mapping to raw events: If the instances of two events are ordered, then so are the corresponding instances of event predicates. PSL operators, situations, and partitionings are more closely related to corresponding constructs in event structures and its variants[66, 23, 55], as adapted for use in interface based specification. Derivation of more complete ties to such frameworks remains the subject of further study. Temporal and Modal Logic. As discussed in Section 2.9, PSL IDL is an application of temporal and modal logic[19, 62] akin to other frameworks (e.g. ....

Pratt, V.R., Chu Spaces: Complementarity and Uncertainty in Rational Mechanics. Technical Report, Stanford University, 1994.

....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 ....

....causally independent. Hence our transition relation Gamma and the corresponding computational interpretation of configuration structures could be called asynchronous. It should be noted that other computational interpretations of configuration structures are possible. The one of Gupta Pratt [8, 7, 14] is obtained by dropping the last two requirements in Definition 1. By labelling the events, we may observe the transitions: Definition 2 A labelled configuration structure (over an alphabet Act) is a triple C = E; C; l) with (E; C) a set system and l : E Act. The components of a labelled ....

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V. Pratt (1994): Chu spaces: complementarity and uncertainty in rational mechanics. Technical report, TEMPUS Summer School, Budapest. Manuscript available as pub/bud.tex.Z by anonymous FTP from Boole.Stanford.EDU.

....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. Chu spaces: Complementarity and uncertainty in rational mechanics. Technical report, Budapest, 1994. Course notes, TEMPUS summer school, 35 pp.

....and situation predicates. The corresponding relation RS contains all pairs of worlds that are members of this set. The relation R serves as the basis for PSL ordering operators. This relation is simplest to describe formally when expressions are restricted to event predicates on fixed messages [24, 70, 58]. In this case, expressions in V are just characteristic predicates of sets of event occurrences, and OE(e,w) is true if w contains the events of e. For example, suppose a satisfies e a = out(m1) for some message m1, and b satisfies e b = out(m1) in(m1) The relation a b states that e b ) e ....

....in a simple way under the intended mapping to raw events: If the instances of two events are ordered, then so are the corresponding instances of event predicates. PSL operators, situations, and partitionings are more closely related to corresponding constructs in event structures and its variants [70, 24, 58], as adapted for use in interface based specification. 7 Appendix Here we describe relations between PSL IDL conventions and underlying PSL constructs. We first introduce an expanded notation for representing situations as named, parameterized predicates of the form Name(Parameters) ....

Pratt, V.R., Chu Spaces: Complementarity and Uncertainty in Rational Mechanics. Technical Report, Stanford University, 1994.

....of this model was previously proposed by Pinna Poign e [17] Their event automata are rooted finitary configuration structures together with a transition relation between the configurations. Our configuration structures are, up to isomorphism, the extensional Chu spaces of Gupta Pratt [11, 10, 20]. It was in their work that the idea arose of using the full generality of such structures in modelling concurrency. It should be noted however that the computational interpretation in [11, 10, 20] differs slightly from that in [16, 22, 6, 17] Formulae In Section 3 we consider set systems from ....

....Our configuration structures are, up to isomorphism, the extensional Chu spaces of Gupta Pratt [11, 10, 20] It was in their work that the idea arose of using the full generality of such structures in modelling concurrency. It should be noted however that the computational interpretation in [11, 10, 20] differs slightly from that in [16, 22, 6, 17] Formulae In Section 3 we consider set systems from the point of view of (infinitary) propositional logic: E is now thought of as the set of propositions and C as the set of models. Following Pratt [20] we observe a bijective correspondence between ....

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V.R. Pratt (1994): Chu spaces: complementarity and uncertainty in rational mechanics. Tech. report, TEMPUS Summer School, Budapest. Available at ftp://boole.stanford.edu/pub/DVI/bud.dvi.gz.

....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, "Chu Spaces: Complementarity and Uncertainty in Rational Mechanics", Course notes, TEMPUS summer school Budapest, 1994.

....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 ....

....action and transition. An event is the performance of an action. Thus whereas a given action may happen repeatedly, a given event can happen at most once. A labeled Chu space associates with each event the action of which it is an occurrence; we shall not say any more about labeled Chu spaces. See [Pra94, Gup94] for a considerably more detailed treatment of automata and schedules, including the use of partial distributive lattices or two toned Hasse diagrams to depict Chu space automata and schedules graphically. This topic is of considerable interest to us and we wish we had more space to go into it ....

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V. Pratt. Chu spaces: complementarity and uncertainty in rational mechanics. Technical report, TEMPUS Summer School, Budapest, July 1994. Manuscript available as pub/bud.tex.Z by anonymous FTP from Boole.Stanford.EDU.

.... 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 ....

V. Pratt. Chu spaces: complementarity and uncertainty in rational mechanics. Technical report, TEMPUS Summer School, Budapest, July 1994. Manuscript available as pub/bud.tex.Z by anonymous FTP from Boole.Stanford.EDU.

....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. Pratt. Chu spaces: complementarity and uncertainty in rational mechanics. Technical report, TEMPUS Summer School, Budapest, July 1994. Manuscript available as pub/bud.tex.Z by anonymous FTP from Boole.Stanford.EDU.

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