V. Pratt. Modeling concurrency with geometry. In Eighteenth annual ACM Symposium on Principles Of Programming Languages, pages 311--322. ACM, IEEE Computer Society Press, Jan. 1991.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....cuts and reduced homology 96 7 Regular cut 97 8 The branching semi globular nerve 99 9 Comparison with the branching semi cubical nerve 101 10 Concluding discussion 103 11 Acknowledgments 104 1. Introduction An # categorical model for higher dimensional automata (HDA) was first proposed in [13], followed by [9] for a first homological approach using these ideas and cubical models Received by the editors 2001 07 27 and, in revised form, 2003 02 25. Transmitted by Ross Street. Published on 2003 02 28. 2000 Mathematics Subject Classification: 55U10, 18G35, 68Q85. Key words and phrases: ....

V. Pratt. Modeling concurrency with geometry. In ACM Press, editor, Proc. of the 18th ACM Symposium on Principles of Programming Languages, 1991.

.... which discretizes this topological framework is that of a precubical set, which is a collection of points (states) edges (transitions) squares, cubes and hypercubes (higher dimensional transitions representing the truly concurrent execution of some number of actions) This is introduced in [Pra91] as well as possible formalizations using n categories, and a notion of homotopy. These precubical sets are called Higher4 0000000000000000011111111111111111 000 000 000 000 000 111 111 111 111 111 000 000 000 000 000 000 111 111 111 111 111 111 T1 T2 Pa (0,0) 1,1) Va Vb Pb Pb Vb Pa Va Figure ....

.... Higher4 0000000000000000011111111111111111 000 000 000 000 000 111 111 111 111 111 000 000 000 000 000 000 111 111 111 111 111 111 T1 T2 Pa (0,0) 1,1) Va Vb Pb Pb Vb Pa Va Figure 3: The progress graph corresponding to P b:V b:P a:V a j P a:V a:P b:V b Dimensional Automata (HDA) following [Pra91] because it really makes sense to consider a hypercube as some form of transition (as in transition systems, used in semantics of programming languages) We show the precise relation between this model and the new topological model we introduce here ( globular CW complexes ) in Section 3.2, the ....

V. Pratt. Modeling concurrency with geometry. In Proc. of the 18th ACM Symposium on Principles of Programming Languages. ACM Press, 1991.

....Models for concurrency, semantics, category theory. 1 Introduction There is a great variety of models for concurrency, as witnessed in [29] for instance. Most of the relationships between these models are known, but the newer geometric models for concurrency, such as cubical sets (HDA in [23] or in [16] or local po spaces [9] have not been so well formally linked with older models, such as transition systems or transition systems with independence. In fact, cubical sets have a notion of generalized transition in their very definition. The idea of relating these in the style of G. ....

.... is then aI s b ) a)I 0 oe(s) b) when (a) 6= and (b) 6= 4 Cubical sets Cubical sets, which are classical objects in combinatorial algebraic topology, see for instance [26] have been used as an alternative truly concurrent model for concurrency, in particular since the seminal paper [23]. More recently they have been used (in particular the precubical ones) in [8] and [9] for deriving new and interesting deadlock detection algorithms. More algorithms have been designed since then, see for instance [24] 7] and [10] 4.1 Precubical sets Definition 4. A precubical set K is a ....

V. Pratt. Modeling concurrency with geometry. In Proc. of the 18th ACM Symposium on Principles of Programming Languages. ACM Press, 1991.

....[24] and [26] I actually only realized the relationships between the combinatorial and the topological approaches quite recently, and have been aware of this line of research only after J. Gunawardena published his very enlightening paper [32] There are some ideas about using n categories in [49]. It is only quite recently that these have given their full flavour, see [20] for a start, where many algebraic invariants are also introduced. The unification of these approaches has lead us to the concept of globular CWcomplex [22] which I will briefly describe in Section 8. The interested ....

....flavoured model for concurrency. 3. A Combinatorial Approach Let us try to get back closer to transition systems now. In fact, there is another geometric model for concurrency which seems to relate more to transition systems than progress graphs, introduced in the article by Vaughan Pratt [49], and which has inspired a lot the following work on the subject (for instance [25] It was essentially motivated by a defect in the duality between event structures and automata 5 , two well known mathematical models for concurrency. The models which have been introduced to fix this defect, ....

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Pratt, V., Modeling concurrency with geometry, in: Proc. of the 18th ACM Symposium on Principles of Programming Languages (1991).

.... simplicial cut 7 4 The negative semi path category 11 5 The semi path category of the hypercube 14 6 The branching semi globular cut 20 7 Comparison with the branching semi cubical nerve 21 1 1 Introduction An categorical model for higher dimensional automata has been rst proposed in [Pra91], followed by [Gou95] for a rst homological approach using these ideas and cubical models of topological spaces as in [BH81b] Since [Gau00b, Gau01a] it is clear that non contracting categories are a good framework to introduce new algebraic tools (in particular new homology theories) to ....

V. Pratt. Modeling concurrency with geometry. In ACM Press, editor, Proc. of the 18th ACM Symposium on Principles of Programming Languages, 1991.

.... categorical approach, where strict globular categories 1 Even if the limited required number of pages for this paper too entails to make some shortcuts. 2 are supposed to encode the algebraic structure of the possible compositions of execution paths and homotopies between them, initiated by [Pra91] and continued in [Gau00] where connections with homological ideas of [Gou95] were made. 2) The topological approach which consists, loosely speaking, to locally endow a topological space with a closed partial ordering which is supposed to represent the time : this is the notion of local po space ....

V. Pratt. Modeling concurrency with geometry. In ACM Press, editor, Proc. of the 18th ACM Symposium on Principles of Programming Languages, 1991.

.... which discretizes this topological framework is that of a precubical set, which is a collection of points (states) edges (transitions) squares, cubes and hypercubes (higher dimensional transitions representing the truly concurrent execution of some number of actions) This is introduced in [Pra91] as well as possible formalizations using n categories, and a notion of homotopy. These precubical sets are called HigherDimensional Automata (HDA) following [Pra91] because it really makes sense to consider a hypercube as some form of transition (as in transition systems, used in semantics of 4 ....

....(higher dimensional transitions representing the truly concurrent execution of some number of actions) This is introduced in [Pra91] as well as possible formalizations using n categories, and a notion of homotopy. These precubical sets are called HigherDimensional Automata (HDA) following [Pra91] because it really makes sense to consider a hypercube as some form of transition (as in transition systems, used in semantics of 4 0000000000000000011111111111111111 000 000 000 000 000 111 111 111 111 111 000 000 000 000 000 000 111 111 111 111 111 111 T1 T2 Pa (0,0) 1,1) Va Vb Pb Pb Vb Pa Va ....

V. Pratt. Modeling concurrency with geometry. In Proc. of the 18th ACM Symposium on Principles of Programming Languages. ACM Press, 1991.

....homology of some # categories and we give some invariance results for the reduced branching homology. We only treat the branching side. The merging side, that is the case of merging areas of execution paths is similar and can be easily deduced from the branching side. 1. Introduction After [22, 14], one knows that it is possible to model higher dimensional automata (HDA) using precubical sets (Definition 2.1) In such a model, a n cube corresponds to a n transition, that is the concurrent execution of n 1 transitions. This theoretical idea would be implemented later. Indeed a CaML program ....

V. Pratt. Modeling concurrency with geometry. In ACM Press, editor, Proc. of the 18th ACM Symposium on Principles of Programming Languages, 1991.

....and corner homologies of a cubical set . 46 11 Acknowledgments 50 12 The categories and functors of this work 50 2 1 Introduction The use of geometric notions to describe the behaviour of concurrent machines is certainly not new since progress graphs [Dij68] HDA [Pra91][vG91] and simplicial models of [HS94] HR94] The purpose of this article is to provide a new setting for the homotopy of execution paths in concurrent automata, in order to improve the homological approach of [Gou95] We can point out that some other approaches of this question already exist. ....

....out that some other approaches of this question already exist. With partially ordered topological spaces in [FGR98a] And with partial posets in [Sok99] Pratt already noticed that a good structure to deal with execution paths and homotopies between them is the structure of globular category [Pra91]. The aim of this paper is threefold. First the use of the concept of globular category to describe concurrent machines is justified on some well known examples and in a very informal way. Secondly we associate to every globular category three homology theories and two natural maps between ....

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V. Pratt. Modeling concurrency with geometry. In ACM Press, editor, Proc. of the 18th ACM Symposium on Principles of Programming Languages, 1991.

....on database consistency. Fajstrup and Raussen have proposed an efficient deadlock detecting algorithm based on geometrical considerations see [4] Probably the most deeply algebraicized of all is the contribution by Goubault and Jensen [6] where the geometric view of concurrency (cf. Pratt [14] and van Glabbeek [5] is provided with a complete construction of homology groups in arbitrary dimensions. 0.1. How algebra helps topologists 3 0.1 How algebra helps topologists Algebraic topology is about a translation of difficult topological problems to easier algebraic ones. One of them is ....

V. Pratt. Modeling concurrency with geometry. Proc. of the 18th Annual ACM Symp. on Principles of Programming Languages, pp. 311--322, January 1991. 6. Final remarks and prospects 39

....of homotopy properties of certain simple topological spaces, that two phase locking is safe an important theorem on database consistency. Probably the most deeply algebraicized of all is the contribution by Goubault and Jensen [3] where the geometric view of concurrency (cf. Pratt [11] and van Glabbeek [2] is provided with a complete construction of homology groups in arbitrary dimensions. Informally, the fundamental (or first homotopy) group gives an account of the nature of holes in a topological space. In the realm of processes, such holes may correspond to forbidden ....

....of paths as defined by Mazurkiewicz. After some (not too many) technical discussions, we both agreed that this direction should be pursued. No effort has yet been undertaken to compare my underlying model of concurrency with those used in the database community; for instance, with Pratt s [11], with Goubault and Jensen s [3] or with the one used by Herlihy and Rajsbaum [6] in their applications of simplicial complexes. Surely, this has to be done. So far I have only told you what has to be checked before the approach is brought in order and before it may be considered robust and ....

V. Pratt. Modeling concurrency with geometry. Proc. of the 18th Annual ACM Symp. on Principles of Programming Languages, pp. 311--322, January 1991. 49

....parts of the system representation. Hence these translations cannot be expressed as reflexions or coreflexions in a suitable categorical framework. 3 Higher dimensional automata The concurrent interpretation of process graphs allows one to think of squares and cubes as being filled in . Pratt [20] proposes a geometric model of concurrency, refining this approach by not necessarily filling in all squares and cubes, but explicitly filling in only those that one wants to represent concurrency. Alternative formalisations of this idea appear in Van Glabbeek [8] Goubault Jensen [11] and ....

V.R. Pratt (1991): Modeling concurrency with geometry. In Proc. 18th Ann. ACM Symposium on Principles of Programming Languages, pp. 311--322.

....problems coming from concurrency theory in computer science. 1 Introduction We have already argued in [13] for modeling higher dimensional automata (HDA) using strict globular categories. To our knowledge, the link between globular categories and concurrent automata was rst noticed in [21]. Papers [13] 11] and [12] show that this way of formalizing HDA is very well adapted to getting interesting new functors deeply related to the computerscienti c properties of the HDA. We would like to explain here the construction of these functors, some of their known properties and some ....

....new functors deeply related to the computerscienti c properties of the HDA. We would like to explain here the construction of these functors, some of their known properties and some perspectives. Many explanations are given in a very informal way. We refer to the bibliography for more details. In [21] and [14] HDA are formalized using cubical sets in the sense of [6] The link between cubical sets and categories will be described at the end of Section 3. There are two equivalent approaches of the notion of category : a cubical one and a globular one [3] The globular approach will be used ....

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Pratt, V., Modeling concurrency with geometry, in: A. Press, editor, Proc. of the 18th ACM Symposium on Principles of Programming Languages, 1991.

....22 References 24 0 Introduction Technically, this report belongs to topology. Its motivations are, however, derived from investigations of concurrent processes. For a general discussion of concurrent processes giving rise to topological spaces, the reader is referred for instance to Pratt [7] or [8]; or to Fajstrup, Goubault, Raussen [1] or to Goubault [6] In the most general terms, the mathematical structures modeling cooperation between processes are endowed with a partial order and with a topology. The partial order relates events to other events they may be causes of, which captures ....

V. Pratt. Modeling concurrency with geometry. Proc. of the 18th Annual ACM Symp. on Principles of Programming Languages, pp. 311--322, January 1991.

.... homology 46 12 Acknowledgment 50 1 Introduction Two homology theories, introduced in [Gau00] formalize respectively the branching areas of execution paths (the negative corner homology) and the merging areas of execution paths (the positive corner homology) in higher dimensional automata (HDA) [Pra91] [FGR98] These homology theories improve the idea of [Gou95] which consisted of considering the vertical and horizontal homologies of some bicomplex associated to a HDA. The main drawback of Goubault s homology theories was their dependence on the choice of the cubification of the HDA. It is one ....

V. Pratt. Modeling concurrency with geometry. In ACM Press, editor, Proc. of the 18th ACM Symposium on Principles of Programming Languages, 1991.

.... (CEA Technologies Avanc ees) DEIN SLA, CEA F91191 Gif sur Yvette Cedex Dept of Mathematics, Aalborg University DK 9220 Aalborg Abstract We study deadlocks using geometric methods based on generalized process graphs [Dij68] i.e. cubical complexes or Higher Dimensional Automata (HDA) Pra91, vG91, GJ92, Gun94] describing the semantics of the concurrent system of interest. A new algorithm is described and fully assessed, both theoretically and practically and compared with more well known traversing techniques. An implementation is available, applied to a toy language. This ....

.... some abstractions, or are compositional techniques [YY91] or based on dataflow analysis [DC94] or on integer programming techniques [ABC 91] but this in general only relies on necessary conditions for deadlocking behaviors) Based on some old ideas [Dij68] and some new semantic grounds [Pra91, vG91, Gun94, GJ92, Gou95a] see x2) we have developped an enhanced sort of reachability search (x2.3) This should mostly be compared to ordinary reachability analysis and not to virtual coarsening and symmetry techniques because these can also be used on top of ours. A first approach in the ....

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V. Pratt. Modeling concurrency with geometry. In Proc. of the 18th ACM Symposium on Principles of Programming Languages. ACM Press, 1991.

....tuples and substitution, thus leading to a calculus we call linda. 1 In the light of next section s formalism we can see this rule as allowing cartesian product of transitions M1 M 0 1 and M2 M 0 2 , which represents their synchronised execution. 4 Higher Dimensional Automata In [Pra91] and [Gla91] Pratt and Glabbeek advocate a model of concurrency based on geometry and in particular on the notion of a higher dimensional automaton. HDA are a generalisation of the usual non deterministic finite automata as described in e.g. HU79] The basic idea is to use the higher dimensions ....

V. Pratt. Modeling concurrency with geometry. In Proc. 18th ACM Symposium on Principles of Programming Languages. ACM Press, 1991.

....are completely independent, and there is no need to assume that they are performed either simultaneously or in a particular order. There is a diagonal transition labelled fa; bg (not part of the picture) that testifies of the independence of these events. In terms of higher dimensional automata [15] it indicates that the square is filled in . On the other hand, the absence of any diagonals in D indicates two distinct linearly ordered computations. In one c can only happen after a, and b in turn has to wait for c; the other has a causal ordering b c a. There is no way to view a and b ....

V.R. Pratt (1991): Modeling concurrency with geometry. In Proc. 18th Ann. ACM Symposium on Principles of Programming Languages, pp. 311--322.

....Next we turn our attention to shapes. There are many reasonable choices for universes of shapes including graphs and partial orders. A particularly interesting choice are higher dimensional automata (hda) introduced by Pratt in which he adds higher dimensional transitions to express concurrency [39]. This was expressed in terms of cubical sets by Goubault and Jensen [28,27] In these terms an hda determines a category via a construction analogous to that for a fundamental groupoid for a topological space. For the purposes of this paper, we will focus on the category of reflexive graphs, ....

V.R. Pratt. Modeling concurrency with geometry. In Proc. 18th Ann. ACM Symposium on Principles of Programming Languages, pages 311--322, January 1991.

....of homotopy properties of certain simple topological spaces, that two phase locking is safe an important theorem on database consistency. Probably the most deeply algebraicized of all is the contribution by Goubault and Jensen [3] where the geometric view of concurrency (cf. Pratt [11] and van Glabbeek [2] is provided with a complete construction of homology groups in arbitrary dimensions. 0.1 How algebra helps topologists Algebraic topology is about translation of difficult topological problems to easier algebraic ones. One of them is the classification problem: given two ....

V. Pratt. Modeling concurrency with geometry. Proc. of the 18th Annual ACM Symp. on Principles of Programming Languages, pp. 311--322, January 1991.

....[Gir87] The match up is uncannily accurate [DHPP99] given that linear logic was not intended at all as a process algebra but as a structuring of Gentzen style proof theory. There is another ostensibly altogether di#erent approach to the essence of concurrency, that of higher dimensional automata [Pra91,GJ92,FGR98,Gou00]. It is however possible to reconcile this approach with the Chu space approach by working with couples over 3, that is, a three letter alphabet for the basic event states of before, during, and after [Pra00] It is my belief that the threeway combination of duality based couples, geometry based ....

V.R. Pratt. Modeling concurrency with geometry. In Proc. 18th Ann. ACM Symposium on Principles of Programming Languages, pages 311--322, January 1991.

....the diverging of the two products But while [1] answered that puzzle with time information duality it did so only for ordered time and not for the generalized metrics of [4] leaving this as a loose end. In between [4] and [1] we introduced the notion of higher dimensional automaton (HDA) [6] as an algebraic topological form of automata theory supporting Papadimitriou s geometric view of concurrency control [7] in terms of higherdimensional state spaces, in which mutual exclusion takes the form of a hole. At that time we were unable to answer Boris Trakhtenbrot s question after our ....

Pratt, V.: Modeling concurrency with geometry. In: Proc. 18th Ann. ACM Symposium on Principles of Programming Languages. (1991) 311-322

....[Gir87] The match up is uncannily accurate [DHPP99] given that linear logic was not intended at all as a process algebra but as a structuring of Gentzen style proof theory. There is another ostensibly altogether di erent approach to the essence of concurrency, that of higher dimensional automata [Pra91,GJ92,FGR98,Gou00]. It is however possible to reconcile this approach with the Chu space approach by working with couples over 3, that is, a three letter alphabet for the basic event states of before, during, and after [Pra00] It is my belief that the threeway combination of duality based couples, geometry based ....

V.R. Pratt. Modeling concurrency with geometry. In Proc. 18th Ann. ACM Symposium on Principles of Programming Languages, pages 311-322, January 1991.

....1 Or Brouwer s included middle if we succumb to our Boolean instinct to cancel the double negative. 1 The automata theoretic way is to pass from conventional automata to higher dimensional automata. 2 . This is accomplished by introducing state transitions having dimension two or more [Pra91] as a natural representation of concurrent activity in the framework of automata theory. On the face of it, logic and geometry sound like fundamentally di erent modeling frameworks. As we shall see however they are simply di erent ways of thinking about a single common solution. 2 Event ....

....di erence whether we take :a to be a V as above or a W with semantic function 2 a. The latter leads to an alternative grammar in which :a is parsed as for a. 5 Conclusion This paper would have proceeded to develop the connections with couples (Chu spaces) over 3 and higher dimensional automata [Pra91] had we not just published a detailed account of this connection in a recent issue of this journal [Pra00] To avoid duplication of writing, printing, and reading we refer the reader to that paper as complementary to the ideas developed here. We should at least point out here however that 2 3 ....

V.R. Pratt. Modeling concurrency with geometry. In Proc. 18th Ann. ACM Symposium on Principles of Programming Languages, pages 311-322, January 1991.

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V. Pratt. Modeling concurrency with geometry. In Eighteenth annual ACM Symposium on Principles Of Programming Languages, pages 311--322. ACM, IEEE Computer Society Press, Jan. 1991.

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