V. Gupta and V.R. Pratt. Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pages 62--71, November 1993.  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. Gupta and V. R. Pratt, Gates Accept Concurrent Behavior, Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., 1993, pp. 62--71. 8

....an event structure. In [10] we proposed to drop the closure conditions, thereby obtaining a more general model of concurrency, capturing both disjunctive causality and resolvable conflict. The resulting configuration structures are, up to isomorphism, the extensional Chu spaces of Gupta Pratt [13], but equipped with a slightly different computational interpretation. Through suitable translations we showed that these configuration structures are equally expressive as general Petri nets without self loops. Such nets are called pure. To this end we defined a 1 occurrence net to be a Petri net ....

V. Gupta & V.R. Pratt (1993): Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pp. 62--71. More material on Chu spaces can be found at http://boole.stanford.edu/chuguide.html.

....as a model of computation for process algebras, constraint and logic programming, term and graph rewriting, and mobile and coordination systems. Please see the position statement of Ugo Montanari at http: www.di.unipi.it ugo ACM text. html for a closer look at causal computing. A Chu space [GP93] is an event state symmetric generalization of the notion of event structure consisting of a set of events, a set of states, and a binary relation of occurrence between them. The interpretation of process algebra over event structures extends straightforwardly to Chu spaces, while the language of ....

V. Gupta and V. R. Pratt. Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pages 62--71, November 1993.

....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. Gupta & V.R. Pratt (1993): Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pp. 62--71.

....to 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 ....

Gupta, V. and Pratt, V.R. (1993) Gates accept concurrent behavior. In Proc. 34th Ann. Symp.

....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. Gupta and V. R. Pratt, Gates Accept Concurrent Behavior, Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., 1993, pp. 62--71.

....UNITY in [35] This system is based on shared variables and assignment statements. Lamport de ned the Temporal Logic of Actions (TLA) in [72] It is based on shared variables and actions, events that change state. Chu spaces, an automaton based approach, are described by Pratt and Gupta in [48, 49]. Lynch et al. describe I O Automata in [84, 85] Milner describes CCS [87] a process algebra, and later developed the calculus [88, 89] a formalism based on the idea of naming. The ability to prove properties of the composition of smaller systems is considered in various works. Barringer, ....

Vineet Gupta and Vaughan Pratt. Gates accept concurrent behavior. In FOCS '93, pages 6271, Palo Alto, CA, USA, 35 November 1993. IEEE Computer Society, IEEE Computer Society Press.

....and interaction at a very high level and abstract manner. Many tools for system specification and verification which are based on these models have been developed [10] 35] 22] 33] 32] 29] 4] and others. Now attempts are being made to define common theories of concurrency [3] 26] [12] (see [9] for a general statement) Special emphasis has been given on action and interaction categories [25] 1] 2] 36] 37] Our approach is general: we have been developing simple unifying framework for interactive concurrent computation in order to be able model most schemes of ....

V. Gupta and V.R. Pratt. Gates accept concurrent behavior. In Thirty-Four Annual IEE Symosium on Foundation of Computer Science, pages 62--71, November 1993.

....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 ....

Gupta, V., and Pratt, V. Gates accept concurrent behavior. In FOCS '93 (Palo Alto, CA, USA, 3--5 Nov. 1993), pp. 62--71.

....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. Gupta & V.R. Pratt (1993): Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pp. 62--71.

....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. Gupta and V. R. Pratt, "Gates Accept Concurrent Behavior", Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., 1993, pp. 62--71.

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V. Gupta and V.R. Pratt. Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pages 62--71, November 1993.

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V. Gupta and V.R. Pratt. Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pages 62--71, November 1993.

....lattices, and categories. This is because the closed sets of a comonoid are the open sets of another comonoid on the same set of points. The participants in this coalgebra workshop would recognize it most readily as a comonoid (A, #, #) in chu, the monoidal category of (bi)extensional Chu spaces [1,4,3,8], where is such a Chu space and # : A# # : A are Chu morphisms satisfying the coassociativity and two counit equations. Compare this with the notion of monoid in a monoidal category , I) as a triple (A, #) where is an object of C and : are morphisms of C satisfying the ....

V. Gupta and V.R. Pratt. Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pages 62--71, November 1993.

....today the underlying principle of our eventstate symmetric view of behavior, independently of whether sequential or concurrent. The basic framework for this view then was complete semilattices, modi ed to cater for con ict by replacing bottom by top. Within a month of writing [1] V. Gupta and I [2] had simpli ed and generalized this framework via Chu spaces [3] which has remained our current view for the past decade [http: chu.stanford.edu ] Yet earlier [4] we had applied categorical enrichment to a uni ed treatment of ordered time, real time, etc. but at that stage of thinking did ....

....algebra analogously. Ironically the original de nition of Chu spaces [3] was for the enriched case, with ordinary Chu spaces receiving only a passing mention. The rst detailed treatment of ordinary Chu spaces was by Lafont and Streicher, and they were subsequently adopted by Gupta and Pratt [2, 11] for the purpose of modeling behavior at a more fundamental level than possible with higher dimensional automata. 2 Event State Duality Computation is traditionally taught with a focus on states, a point of view that has permeated computer science so thoroughly that event oriented models are in ....

Gupta, V., Pratt, V.: Gates accept concurrent behavior. In: Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci. (1993) 62-71

....closed category, admitting functors perp, Gamma op Chu, and tensor, Omega Chu, which interpret the corresponding basic operations of MLL. Our interest in Chu spaces as a model of linear logic is two fold: as a model of concurrency, with linear logic serving as a process algebra [5, 4, 8], and 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 ....

V. Gupta and V.R. Pratt. Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pages 62--71, November 1993.

.... 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 the logicality semantics of Chu spaces with the proof structure of multiplicative logic simultaneously confers a degree of logical tractability on Chu spaces while broadening the applicability of linear logic. 2 Linear Logic, Chu Spaces Coherence ....

V. Gupta and V.R. Pratt. Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pages 62--71, November 1993.

.... 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, although for practical purposes it is customarily defined at ....

....fixed notion of morphism, they are as universal as arbitrary small categories. Thus as a universal source of structure with which to equip any type, they are ideally suited to type theory. On the other hand, Chu spaces over 2 are a natural generalization of event structures, as proposed in [10] and developed in more length in Gupta s thesis [9] and by van Glabbeek and Plotkin [27] To represent the event structure (A; #) simply take it to be the Chu space (A; r; X) where X consists of all order ideals of (A; that do not contain both a and b when a#b. The states of this Chu space ....

V. Gupta and V.R. Pratt. Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pages 62--71, November 1993.

....B, showing that this represents independent execution of A and B. A state in the composition is final iff both A and B are in their final states. The symbol for parallel composition used here is to conform to the usage in the process algebra community. However our preferred symbol for it is [GP93], since this is a coproduct in an appropriate category. Consequently we prefer 0 for the eks 1 defined below, since it is an initial object in that category. Sequential composition A; B represents the execution of B after completing the execution of A, i.e. after A has reached a final state. Once ....

V. Gupta and V.R. Pratt. Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., November 1993.

....Paiva [BGdP91] and Blass [Bla95] who have treated a lax continuous variant 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 ....

V. Gupta and V.R. Pratt. Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pages 62--71, November 1993.

....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 ....

V. Gupta and V.R. Pratt. Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pages 62--71, November 1993.

....for this particular organization, as well as the language of this metadiscussion, all these operations are definable in traditional set theoretic terms. Chu spaces first entered our investigations of the nature of concurrent 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 ....

V. Gupta and V.R. Pratt. Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pages 62-- 71, November 1993.

....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. Gupta and V.R. Pratt. Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pages 62--71, November 1993.

....notions of enabling and conflict. What was not changed there was the 2 valued logic of configuration structures, the net effect then being to make their notion of general configuration structure equivalent to the extensional Chu spaces over 2 studied by Gupta and Pratt as a model of concurrency [Gup93, GP93, Gup94]. Our use of the above three element quantale will turn out to be equivalent to using Chu spaces over 3, unexpectedly with no mention made by the latter concerning that choice of quantale. This solves the basic problem in full, at least to the extent defined by its formulation above. For HDA s, ....

....ordinary 2 fuzzy sets. The question of what additional concepts besides enabling and conflict would be useful for this generalized notion of event structure is taken care of automatically by the straightforward yet structurally fruitful approach of permitting arbitrary configuration structures [GP93, VGP95]. 3 Definitions This section defines quantales and Chu spaces as the two main tools for this paper. Chu spaces are objects of type A Theta X K separating out the respective concerns of concurrency as a conjunctive or covariant set A of events, choice as a disjunctive or contravariant set X of ....

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V. Gupta and V.R. Pratt. Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pages 62--71, November 1993.

....duals, the edge is the interaction. 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 ....

V. Gupta and V.R. Pratt. Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pages 62--71, November 1993.

....of K, our results therefore apply equally well to Chu(Set; 2) and Chu(Set; K) for larger K. Our interest in Chu spaces as a model of linear logic is two fold. Our original and motivating application of Chu spaces has been as a model of concurrency, with linear logic serving as a process algebra [8, 7, 13]. More recently we have come to regard Chu spaces as universal topology analogous to universal algebra. A Chu space is to a topological space as an algebra (in the sense of one or more carriers and a family of operations) is to a group. The remarkable thing is that every algebra, indeed every ....

V. Gupta and V.R. Pratt. Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pages 62--71, November 1993.

....in the linear case no information was lost in the projection of a trajectory onto either axis. With suitable care in the definition of projection we may accomplish the same for biextensional Chu spaces, namely those having no repeated rows or columns, via a process we have previously described [GP93], which we sketch briefly here. We project onto the information axis to yield an automaton in two steps. First close the rows of the Chu space under arbitrary union and intersection (OR and AND of the rows as bit vectors) by adding new rows as needed. Now draw the poset of all resulting rows ....

....between any two distinct points of the chain lie two distinct points with no other point between them. We recover A as those elements of this lattice not the join of the lattice elements strictly below them, X as the set of black points of the lattice, and x j= a as the relation a x. The theorem [GP93] is that recovered space is isomorphic to the originally projected space. This construction works equally well when 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 ....

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V. Gupta and V.R. Pratt. Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pages 62--71, November 1993.

....we regard the former as more important and fundamental. As a concrete, complete, and self dual symmetric monoidal closed category, Chu is a natural model 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 ....

V. Gupta and V.R. Pratt. Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pages 62--71, November 1993.

....quantum mechanical 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 ....

....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. Gupta and V.R. Pratt. Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pages 62--71, November 1993.

..... other authors have used all sorts of weird combinations for writing it in L aT E X ) The work in chapter 6 has also benefited from discussions with Dr. R. J. van Glabbeek and Prof. Gordon D. Plotkin. Some of the results in the thesis have been published in [Gup93, GP93]. Chapter 2 Models of Concurrency Before we get into Chu space theory and applications we will briefly review some models of concurrency which contributed ideas to our model. These include Petri nets, traces, transition graphs and CCS, event structures, geometric automata and pomsets. All models ....

....back, so in a single run, it can be toggled exactly once, corresponding to the event taking place. A behavior is accepted if the output always stays 1, that is the process is always in an allowed state. So a gate is now an acceptor of a behavior, rather than a transducer as used traditionally [GP93]. Gates provide an alternative pictorial representation for Chu spaces, and are very useful in understanding the algebra of Chu spaces, as we will show in the next chapter. 5.4 Aspects of concurrency We will now show some properties that can be used to represent various features of concurrent ....

V. Gupta and V.R. Pratt. Gates accept concurrent behavior. In Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., pages 62--71, November 1993.

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V. Gupta and V. R. Pratt, Gates Accept Concurrent Behavior, Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci., 1993, pp. 62--71.

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