S. Abramsky. Sequentiality vs. concurrency in games and logic. Mathematical Structures in Computer Science, 13:531--565, 2003.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....papers has been generalized by Rutten and Turi [192] Ibidem it is shown how TSSs in tyft tyxt format induce a denotational semantics, and the essential properties of semantic domains that make their definitions possible are investigated in a categorical perspective. Abramsky and Vickers [3] consider various notions of process observations in a uniform algebraic framework provided by the theory of quantales (see, e.g. 188] The methods developed in [3] yield, in a uniform fashion, observational logics and denotational models for each notion of process observation they consider. ....

....of semantic domains that make their definitions possible are investigated in a categorical perspective. Abramsky and Vickers [3] consider various notions of process observations in a uniform algebraic framework provided by the theory of quantales (see, e.g. 188] The methods developed in [3] yield, in a uniform fashion, observational logics and denotational models for each notion of process observation they consider. Their work is, however, semantic in nature, and ignores the algebraic structure of process expressions. In the area of the semantics of functional programs, ....

S. Abramsky and S. Vickers, Quantales, observational logic and process semantics, Mathematical Structures in Computer Science, 3 (1993), pp. 161--227.

....papers has been generalized by Rutten and Turi [201] Ibidem it is shown how TSSs in tyft tyxt format induce a denotational semantics, and the es sential properties of semantic domains that make their definitions possible are investigated in a categorical perspective. Abramsky and Vickers [3] consider various notions of process observa tions in a uniform algebraic framework provided by the theory of quantales (see, e.g. 197] The methods developed in [3] yield, in a uniform fashion, observational logics and denotational models for each notion of process observation they consider. ....

....of semantic domains that make their definitions possible are investigated in a categorical perspective. Abramsky and Vickers [3] consider various notions of process observa tions in a uniform algebraic framework provided by the theory of quantales (see, e.g. 197] The methods developed in [3] yield, in a uniform fashion, observational logics and denotational models for each notion of process observation they consider. Their work is, however, semantic in nature, and ignores the algebraic structure of process expressions. In the area of the semantics of functional programs, ....

S. ABRAMSKY AND S. VICKERS, Quantales, observational logic and process semantics, Mathematical Structures in Computer Science, 3 (1993), pp. 161-227.

....property for a composite system S k T to a model checking problem for S. The results that we have developed show that a timed version of ready simulation is testable, in the sense of this paper. This conclusion seems to be in agreement with the analysis of behavioural relations carried out in [AV93] within the framework of quantales. Whether our results can be justified by means of a general theory la Abramsky and Vickers is an interesting topic for further theoretical research. It would also be interesting to investigate the connections between our investigations and the seminal study ....

ABRAMSKY, S. and VICKERS, S. Quantales, Observational Logic and Process Semantics. Mathematical Structures in Computer Science, vol. 3(2):161--227, 1993.

....property for a composite system S k T to a model checking problem for S. The results that we have developed show that a timed version of ready simulation is testable, in the sense of this paper. This conclusion seems to be in agreement with the analysis of behavioural relations carried out in [AV93] within the framework of quantales. Whether our results can be justified by means of a general theory la Abramsky and Vickers is an interesting topic for further theoretical research. It would also be interesting to investigate the connections between our investigations and the seminal study ....

ABRAMSKY, S. and VICKERS, S. Quantales, Observational Logic and Process Semantics. Mathematical Structures in Computer Science, vol. 3(2):161--227, 1993.

....as in branching time temporal logics, Hennessy Milner logic or temporal logic based on partial order semantics [15, 33, 4, 30, 31] but in a way compatible with multiplicative conjunction. There have, to date, been a number of suggestive uses of substructural logics in concurrency (e.g. [11, 3, 2, 22, 8, 19, 9, 24, 25]) While there has not yet been a de nitive or completely convincing account, the idea of using multiplicative conjunction to describe spatial distribution of resources or processes remains appealing. In this connection it is worth stressing, in conclusion, that there appears to be no compelling ....

S. Abramsky and S. Vickers. Quantales, observational logic, and process semantics. Mathematical Structures in Computer Science, 3:161-227, 1993. 17

....areas of computer science. Most of the applications utilizes the concept of resource which is di#cult to represent in traditional logics. In the case of the Petri net, tokens are presented as atomic formulas (i.e. resources) and reachability of Petri net is mapped to provability of linear logic [2]. There are many other frameworks and computation models encodable into linear logic, such as counter machines, Turing machines, and # calculus [31] As for an application to functional programming language, linear logic can be used for e#cient memory management [29] and for a type system [1] ....

....names are sorted. The following rule finds two sorted lists from the goals, and replaces them with a single goal holding the new sorted list by merging. sorted(L1) # sorted(L2) merge(L1,L2,L)# sorted(L) Figure 5.3 and Figure 5. 4 are the LL proof and the IO proof for sorting the list [3, 2, 1] into ascending order. 24 #; # m( 3] 2] 2, 3] #; # m( 2, 3] 1] 1, 2, 3] X = 1, 2, 3] #; s(X) # s( 1, 2, 3] #; s(X) # s( 2, 3] s( 1] #; s(X) # s( 3] s( 2] s( 1] s( #; s(X) # p( 3, 2, 1] #; # sort( 3, 2, 1] X) Figure 5.3: An LL proof for sorting ....

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S. Abramsky and S. Vickers. Quantales, observational logic, and process semantics. Mathematical Structures in Computer Science, 3:161--227, 1993.

.... eld, the aim being to assess the extent to which quantales are also capable of describing spaces whose points are instances of a particular kind of dynamical system found in computer science, namely in concurrency theory; the de nition of such dynamical systems uses quantales in a natural way [1, 29, 27], and our intention is to relate these rather di erent applications of quantales. More precisely, we will see that the examples in concurrency can be recast into a form similar to that of [16] whereby spectra are described in the category of unital quantales, and in such a way that systems ....

....processes, and additional behavioural equivalences have to be supplied [8, 9] whereby certain states are considered to be equivalent in the sense that they have the same observable behaviour . It is commonly assumed that such equivalences rely on notions of experi2 mental observation, and in [1] this was made explicit by taking the actions to be some of the generators of a unital quantale (i.e. a monoid in the category of sup lattices SL [11] see x2.1) which however may have other generators. In other words, the quantale is an algebra of nite run time observations; performing ....

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S. Abramsky and S. Vickers. Quantales, observational logic and process semantics. Mathematical Structures in Computer Science, 3(2):161-227, 1993.

....of the algebraic closure of Q form, not an in nite discrete (Galois) group, but a compact topological group. Similarly, the powerset of even a discrete set is not itself a discrete set, but a non Hausdor topological lattice. Vickers has taken the same motivation in a di erent direction [Vic99] The types in our logic are therefore to be spaces. The topological structure is an indissoluble part of what it is to be a space: it is not a set of points together with a topology, any more than chipboard (which is made of sawdust and glue) is wood. When we bring (not necessarily Martin L of) ....

Steven Vickers. Topical categories of domains. Mathematical Structures in Computer Science, 9:569-616, 1999.

....are defined, motivated, compared and axiomatized. Of course many more equivalences can be given than the ones presented here. The reason for selecting just these, is that they can be motivated rather nicely and or play a role in the literature on semantic equivalences. In Abramsky Vickers [2] the observations which underly many of the semantics in this paper are placed in a uniform algebraic framework, and some general completeness criteria are stated and proved. They also introduce acceptance semantics, which can be obtained from acceptance refusal semantics (Section 7) by dropping ....

S. Abramsky & S. Vickers (1993): Quantales, observational logic and process semantics. Mathematical Structures in Computer Science 3, pp. 161--227.

....as in branching time temporal logics, Hennessy Milner logic or temporal logic based on partial order semantics [14, 29, 4, 27, 28] but in a way compatible with multiplicative conjunction. There have, to date, been a number of suggestive uses of substructural logics in concurrency (e.g. [11, 3, 2, 20, 8, 17, 9, 21, 22]) While there has not yet been a definitive or completely convincing account, the idea of using multiplicative conjunction to describe spatial distribution of resources or processes remains appealing. In this connection it is worth stressing, in conclusion, that there appears to be no compelling ....

S. Abramsky and S. Vickers. Quantales, observational logic, and process semantics. Mathematical Structures in Computer Science, 3:161--227, 1993.

....as in branching time temporal logics, Hennessy Milner logic or temporal logic based on partial order semantics [15, 33, 4, 30, 31] but in a way compatible with multiplicative conjunction. There have, to date, been a number of suggestive uses of substructural logics in concurrency (e.g. [11, 3, 2, 22, 8, 19, 9, 24, 25]) While there has not yet been a definitive or completely convincing account, the idea of using multiplicative conjunction to describe spatial distribution of resources or processes remains appealing. In this connection it is worth stressing, in conclusion, that there appears to be no compelling ....

S. Abramsky and S. Vickers. Quantales, observational logic, and process semantics. Mathematical Structures in Computer Science, 3:161--227, 1993. 17

....geometry, etc. Connes, 1994] Also the concept of a quantale originates here. Instead of considering the well behaved set of ideals of a commutative C algebra which lead to a Hausdorff structure space, we get in the non commutative case a more complicated T 0 space [Vickers, 1989, Abramsky and Vickers, 1993]. 4. Discrete Models and Paths C Algebras, either in their concrete or their abstract form, still might not be very intuitive objects. Perhaps, a more operational interpretation might be needed. In the following we will introduce a C algebra approach for computer scientist, based on graphs ....

Samson Abramsky and Steven Vickers. Quantales, observational logic and process semantics. Mathematical Structures in Computer Science, 3:161--227, 1993.

....semantics and logics of programs. Based on the fundamental insight of Smyth [Smy83b] that a topological space may be seen as a data type with the open sets as observable predicates , and functions between topological spaces as computations , Abramsky [Abr87, Abr91a] Zhang [Zha91] and Vickers [Vic89, AV93] developed a propositional program logic from a denotational semantics. Abramsky [Abr87, Abr91a] uses Stone duality to relate two views of SFPdomains (a special kind of complete partial orders) one in terms of logic theories and one in terms of semantic models. Abramsky s starting point is that ....

Abramsky, S., Vickers, S.J.: Quantales, observational logic and process semantics. Mathematical Structures in Computer Science, 3:161--227, 1993.

....preorder, giving some examples and proving that the induced precongruence is refined by a simple notion of bisimulation. There is an extensive theoretical literature discussing behavioural equivalences of process calculi that are induced by some kind of tests or observations of processes, e.g. [HM80, Mil81, DH84, Hoa85, Abr87, AV93, Gla90, Gla93, San93]. The argument that the testing scenario we adopt is appropriate for Pict relies on some essential differences between Pict and any process calculus considered only in the abstract. Firstly, Pict is a programming language, with a fixed interpretation of nondeterminism (as a loose specification of ....

S. Abramsky and S. J. Vickers. Quantales, observational logic and process semantics. Mathematical Structures in Computer Science, 3:161--227, 1993.

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S. Abramsky. Sequentiality vs. concurrency in games and logic. Mathematical Structures in Computer Science, 13:531--565, 2003.

....as above to any pre order on transition systems which is compatible with strong bisimulation and a precongruence with respect to the four basic combinators and guarded recursion. For a large selection of such pre orders including failures, acceptances, readies, and barbed versions of these, see [10]. 3.4 An Analysis of a Cyclic Scheduler As an illustration, we now use the ideas and techniques which have been presented so far to analyse a standard concurrency example the cyclic scheduler of [49] We begin by reviewing the specification and implementation of the scheduler, and then show ....

S. Abramsky and S. J. Vickers. Quantales, observational logic and process semantics. Mathematical Structures in Computer Science, 3:161--227, 1993.

....between S and S satisfying certain conditions, then the so called C ideals of S form a frame with a certain universal property in the category of frames that can be conveniently described as a frame presentation by generators and relations. 5 A more refined analysis in Abramsky and Vickers [1] which we refer to as the suplattice coverage theorem shows that the suplattice of C ideals also has a universal property in the category of suplattices that can be conveniently described as a suplattice presentation. Hence it shows how to translate frame presentations into suplattice ....

....u W X (X covers u) Qua semilattice means that the injection of generators, a function from S to the frame, is to be a semilattice homomorphism. This can be achieved by adding extra relations to the presentation. Qua poset is similar. Proof The proof is given in Abramsky and Vickers [1] and what we must note here is that it is constructive. The first step to show that SupLat S(qua poset) j u W X (X covers u) exists. This can be done by explicit construction as Johnstone s [5] C Gamma Idl(S) or by the standard methods of universal algebra that first construct the free ....

Samson Abramsky and Steven J. Vickers. Quantales, observational logic and process semantics. Mathematical Structures in Computer Science, 3:161--227, 1993.

..... A map from D to PLE is just a suplattice homomorphism from Omega E to Omega D, and a crucial tool is a sharpening of Johnstone s [8] coverage theorem for frames that allows us to describe suplattice homomorphisms between frames. The sharpening is discussed in detail in Abramsky and Vickers [1] and we shall merely summarize it here. Theorem 4.3 (Johnstone s Coverage Theorem) Let S be a meet semilattice, and let C covers be a relation between S and S such that ffl if x 2 X C u then x u (i.e. if X C u then X # u) ffl if X C u and s 2 S then fx s : x 2 Xg C u s (Any ....

Samson Abramsky and Steven J. Vickers. Quantales, observational logic and process semantics. Mathematical Structures in Computer Science, 3:161--227, 1993.

....However, we have a deeper motivation: the failure to topologize is related to a failure to reason constructively, and we wish also to rectify that. As a general principle the relation is seen most clearly when the constructive discipline is the stringent geometric one, as used systematically in [19], giving a mathematics that is preserved by the inverse image functor of geometric morphisms. We want to use locales, for much of topology constructivizes more smoothly in localic form as has been seen both in toposes [7] and in type theory [16] However, constructing the set of points of a locale ....

S. Vickers. Topical categories of domains. Mathematical Structures in Computer Science, 9:569-616, 1999.

....for third completeness due to Abramsky and Vickers is reset constructively, and an example of application to failures semantics is given. Keywords: quantale, localic sup lattice, localic quantale module, localic tropological system, process semantics, third completeness. 1 Introduction In [2] and, subsequently, 12] di erent ideas of process semantics (in the computer science sense) are compared using the algebraic structures of quantales and modules over them. Research supported by the Anglo Portuguese Joint Research Programme Treaty of Windsor through grant B 29 99 Dynamic ....

....theory along an inverse image functor. It follows that a constructive treatment should properly pay attention to the topology. 1.1 Suplattice duality As a more concrete example, consider sup lattice duality. This was given an intuitionistically constructive treatment in [7] and used heavily in [2]. If M is a sup lattice a complete join semilattice then it also has all meets and so its opposite M op is also a sup lattice. Moreover, if f : M N is a sup lattice homomorphism preserving all joins then it has a right adjoint which preserves all meets, and hence corresponds to a sup lattice ....

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S. Abramsky and S. Vickers. Quantales, observational logic and process semantics. Mathematical Structures in Computer Science, 3(2):161-227, 1993.

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S. Abramsky and S. Vickers. Quantales, observational logic and process semantics. Mathematical Structures in Computer Science, 3:161--227, 1993.

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Abramsky, S. and Vickers, S. (1993) Quantales, observational logic, and process semantics, Mathematical Structures in Computer Science 3, 161-- 227.

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Abramsky, S. and Vickers, S. (1993) Quantales, observational logic and process semantics. Mathematical Structures in Computer Science, 3, 161-227.

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S. Abramsky and S. Vickers. Quantales, observational logic, and process semantics. Mathematical Structures in Computer Science, 3:161--227, 1993.

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