S. Abramsky and A. Jung. Domain Theory. Handbook of Logic in Computer Science. Volume III: Semantic Structures, pages 1--168. S. Abramsky, D. M. Gabbay and T. S. E. Maibaum, editors. Oxford University Press 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Volume 7, Released 1995, Published 2003 Inst. of Computer Science, Univ. of Bial ystok Reduction Relations Grzegorz Bancerek Institute of Mathematics Polish Academy of Sciences Summary. The goal of the article is to start the formalization of Knuth Bendix completion method (see [2] 10] or [1]; see also [11] 9] i.e. to formalize the concept of the completion of a reduction relation. The completion of a reduction relation R is a complete reduction relation equivalent to R such that convertible elements have the same normal forms. The theory formalized in the article includes concepts ....

S. Abramsky, D. M. Gabbay, and S. E. Maibaum, editors. Handbook of Logic in Computer Science, vol. 2: Computational structures. Clarendon Press, Oxford, 1992.

....formulas into interval logic ones, that is based on this semantic correspondence. We apply this translation to obtain our axioms for iteration from the corresponding axioms in PDL. The readers who are not familiar with PDL can skip this section. Basic definitions about PDL can be found in, e.g. [AGM92]. Let F = hhT ; i; hD; 0i; mi be an interval logic frame with time domain hT ; i, duration domain hD; 0i, and measure function m : I(T ) D, where I(T ) f[ 1 ; 2 ] j 1 ; 2 ; 2 T ; 1 2 g (cf. Dut95] for interval logic terminology) The set of time points T can be taken as the set ....

....interval logic model hF; Ii can be expressed using t by the following proposition: Proposition 1 Let 2 L PDL . Then T ; v; minT j= iff hF; Ii; min T ; maxT ] j= t( Proof: Direct check by induction on the construction . 2 PDL has the following axioms for iteration in its proof system ([AGM92]) ff] ff ] ff] ff The t translations of these axioms are equivalent to (I 1 ) ff ff ) ff (I 2 ) ff ) ff ) where b = for short. The validity of 1 for some given PDL relation term ff for all possible ....

S. Abramsky, D. Gabbay and T.S.E. Maibaum, eds. Handbook of Logic in Computer Science, Clarendon Press, Oxford, 1992.

....automata in this extension, using so called simple formulas. In this paper we present a complete system of axioms for iteration of simple formulas. We obtained our axioms by translating appropriately the schemata for iteration from the proof system of propositional dynamic logic ( Seg77] cf. e.g. [AGM92]) which is a well known formal system with iteration. We present this translation and the correspondence between the semantics of propositional dynamic logic and that of interval temporal logic that underlies it. The argument of completeness for the axioms for iteration in propositional dynamic ....

....formulas. We also discuss a way to generalise the result to broader classes of duration calculus and interval logic formulas. We derive our axiomatisation using a correspondence between the semantics of interval logic and the relational (Kripke ) semantics of propositional dynamic logic (cf. e.g. [AGM92]) which has iteration too. Propositional dynamic logic has been first given a complete proof system by K. Segerberg in 1977[Seg77] We translate the schemata that deal with iteration in a more recent version of the proof system[AGM92] and further subject them to some transformations so that we ....

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S. Abramsky, D. Gabbay and T.S.E. Maibaum, eds. Handbook of Logic in Computer Science, Clarendon Press, Oxford, 1992.

....issues of the specification environment Tbench which supports the modelling of complex information systems with omTroll and Troll. 2.1 Troll by example We illustrate the Troll language by an example. We are going to specify some aspects of the library of our computer science department. Our fragment of the library world consists of the librarian and the library itself which is composed of the documents, copies of documents and a file of registered library users. There may be several copies of every document in the library. Users may borrow these copies. The library has to trace the ....

....and semantics Temporal logic and interpretation. Let Delta be a local class signature. For temporal specification, we use propositional linear time temporal logic Ptl( Delta) including past and future directed temporal operators [Ehr96] For simplicity, we concentrate on the future directed fragment Ptl Definition 9. The syntax of Ptl ( Delta) i.e. its set of well formed formulae Phi, is given by Phi : P ) j false j ( Phi ) Phi) j ( Phi U Phi) A proposition (p) is intended to mean that class proposition p holds. These propositions are flexible, i.e. we intend to give ....

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S. Abramski, D. Gabbay and T. Maibaum. Handbook of Logic in Computer Science, Vols. 1-5. Oxford Science Publications, Oxford 1992ff

....S P = S f S n i=1 x i j n 2 N and x i 2 P for i = 1; ng. It follows that, on ( f) stable pavings the unions can always be assumed to be directed and on ( f) stable pavings the intersections ltered. 2. 1 Partial orders and domain theory Our main references for domain theory are [1][22] 15] The non initiated reader can nd a comprehensive introduction to the subject in [31] 40] 10] A preorder is a pair (P; v) where P is a set and v is a binary relation on P satisfying: a v a (re exivity) a v b and b v c implies a v c (transitivity) for all a; b; c 2 P . A preorder (P; v) ....

....the Lawson topology on L is compact and Hausdor [15, Theorem III 1.10] 5.1 Representation as the spectrum of Q For a discussion of this representation when the space is Hausdor the reader is referred to Section 3 of [9] for the relationship with the Smyth powerdomain see Section 6.2. 3 in [1]. See also [57, Exercise 7.7.8 15] Proposition 5.3 Let (X; G) be a locally compact sober space. Then 1. Q; is a continuous semilattice with maximum element ; 2. The map s : X Q given by s(x) sat(fxg) x is an embedding of X onto Spec Q with the relative Scott topology induced ....

S. Abramsky and A. Jung, Domain theory, Handbook of Logic in Computer Science, Vol. 3 (S. Abramsky, D.M. Gabbay, and T.S.E. Maibaum, eds.), Oxford University Press, Oxford, 1994, pp. 1-168.

....However, there are many di erent opinions on this issue concerning mathematical and real worlds, and this is also a strong piece of evidence that computation is a philosophical notion. This almost entails that philosophy is a theoretical basis of computation and computer science. However, we can[1] (and, indeed, should) do our best in mathematics here. A somewhat similar practical problem between traditional logics and the real world[25] is the closed world assumption[50] which, despite its usefulness 25 for some predicate de nitions, does not match well an open world, such as the ....

....C 1 jC 2 j= hr p1 t ; x 1 ; y 1 ; z 1 ; ti) hr p2 t ; x 2 ; y 2 ; z 2 ; ti) c (s t i s 0 t i ) 14) Parallel computation presents behaviors in parallel. Notice that computation might interfere in each other. Our abstract model of computation is an extension of the while language[1]. Nonetheless, although that model has if, we observe that this statement can be de nable: that is, if p then q is de ned as x : 0; while x = 0 p do q ; x : 1; endwhile 35 and, for de ning the if then else, we only use two similar loops, and this change will simplify our job. We are ....

S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, editors. Handbook of Logic in Computer Science, volume 1. Oxford University Press Inc., 1992.

....on propositional level, while the added complications of the first order machinery are avoided. References for this section: There is ample philosophical and formal logical literature on time and reasoning about time. A recommended book is [Ben93] Also, see the surveys on temporal reasoning in [AGM92] 4 1.2 Preliminaries on modal logic Here we provide a very basic background on modal logic, just to get started. 1.2.1 Modes of truth and the variety of modal logics. There are various modes of truth and they determine various modal operators on propositions, and respectively various kinds of ....

S. Abramsky, D.M. Gabbay, and T.S.E. Maibaum, editors. Handbook of Logic in Computer Science, volume 2. Clarendon Press, Oxford, 1992.

....that a recursive program can be captured as the fixed point of a higher order function which is defined, by the corresponding recursion, on the domain of all programs of the given type. There are a number of basic categories of domains according to various additional properties that they satisfy [79, 2]. Algebraic domains are characterized by a subset of so called finite or compact elements representing computational results which can be obtained in a finite number of steps. The finite elements form a basis of the domain; every element of the domain is the limit of the basis elements ....

....on these types; PCF can be considered as the theoretical model for functional programming languages. Domain theory has developed extensively in the past three decades and is now a major paradigm in the semantics of programming languages. For a basic introduction to its theory and applications, see [59, 74, 96, 64, 2, 114]. Algebraic domains have also been used to represent classical spaces in mathematics in an e#ective framework. Weihrauch and Schreiber [124] constructed an embedding of a Polish space (a topologically complete separable metrizable space) into an algebraic domain. Stoltenberg Hansen and Tucker have ....

[Article contains additional citation context not shown here]

S. Abramsky and A. Jung, Domain theory, Handbook of logic in computer science (S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, editors), vol. 3, Clarendon Press, 1994.

....of Logic in Computer Science , avers that Turing s analysis of what is involved in computation . seems so general that it is hard to imagine some other method which falls outside the scope of his description . so . anything which can be computed can be computed by a Turing machine (Abramsky, Gabbay and Maibaum 1992: 123) Of course, Turing neither proved nor asserted that a universal Turing machine can compute any function that any computer, with any architecture, can compute . Nor did he have a result entailing that a Turing machine can display any systematic pattern of responses to the environment ....

Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds) 1992. Handbook of Logic in Computer Science. Vol.1. Oxford: Clarendon Press.

....of what is involved in computation . seems so general that it is hard to imagine some other method which falls outside the scope of his description . so . anything which can be computed can be computed by a Turing machine. Phillips writing in the Handbook of Logic and Computer Science ; Abramsky, Gabbay and Maibaum 1992: 123. v) Any problem for which we can find an algorithm that can be programmed in some programming language, any language, running on some computer, any computer, even one that has not been built yet but can be built, and even one that will require unbounded amounts of time and memory space ....

Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds) 1992. Handbook of Logic in Computer Science. Vol.1. Oxford: Clarendon Press.

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S. Abramsky and A. Jung. Domain Theory. Handbook of Logic in Computer Science. Volume III: Semantic Structures, pages 1--168. S. Abramsky, D. M. Gabbay and T. S. E. Maibaum, editors. Oxford University Press 1994.

....of games and strategies Let us begin by revisiting Blass games, with the idea that game trees can be viewed as partial orders, in which x y means that the position y can be reached from the position x by playing some additional moves. This is a natural information ordering as in Domain theory [AJ94]; it is the pre x ordering on the plays or paths through the game tree. If we add limit points corresponding to the in nite branches in the game tree, we obtain a complete partial order D. Viewed in these terms, the construction of sums and products of games as in Blass games can be described ....

....the order in which decisions were taken) We can take C = V = N for a universal choice of game board, into which any other (subject to countability) can be embedded. The positions are then partial functions on natural numbers, which under inclusion form one of the basic examples of a domain [AJ94]. Moreover, as play progresses, and more moves are made, the position increases in the natural information ordering: f(c 1 ; v 1 ) c 2 ; v 2 )g (c 3 ;v 3 ) f(c 1 ; v 1 ) c 2 ; v 2 ) c 3 ; v 3 )g: Call this universal game board G . A strategy is a function f : G G : For the reasons ....

S. Abramsky and A. Jung. Domain Theory. Handbook of Logic in Computer Science. Volume III: Semantic Structures, pages 1-168. S. Abramsky, D. M. Gabbay and T. S. E. Maibaum, editors. Oxford University Press 1994.

....of games and strategies Let us begin by revisiting Blass games, with the idea that game trees can be viewed as partial orders, in which x y means that the position y can be reached from the position x by playing some additional moves. This is a natural information ordering as in Domain theory [AJ94]; it is the pre x ordering on the plays or paths through the game tree. If we add limit points corresponding to the in nite branches in the game tree, we obtain a complete partial order D. Viewed in these terms, the construction of sums and products of games as in Blass games can be described ....

....the order in which decisions were taken) We can take C = V = N for a universal choice of game board, into which any other (subject to countability) can be embedded. The positions are then partial functions on natural numbers, which under inclusion form one of the basic examples of a domain [AJ94]. Moreover, as play progresses, and more moves are made, the position increases in the natural information ordering: f(c 1 ; v 1 ) c 2 ; v 2 )g (c 3 ;v 3 ) f(c 1 ; v 1 ) c 2 ; v 2 ) c 3 ; v 3 )g: Call this universal game board G . A strategy is a function f : G G : For the reasons ....

S. Abramsky and A. Jung. Domain Theory. Handbook of Logic in Computer Science. Volume III: Semantic Structures, pages 1-168. S. Abramsky, D. M. Gabbay and T. S. E. Maibaum, editors. Oxford University Press 1994.

.... (so that the total information obtainable as output from an in nite sequence of input elements with re ning information is the sum of all information obtained from each input element) There are a number of categories of domains in according to various additional properties that they satisfy e.g. [1, 18]. In this work, to construct computational models for real valued functions and functionals we will use continuous This research was supported in part by the RFBR (grants N 99 01 00485, N 00 0100810) and by the Siberian Division of RAS (a grant for young researchers, 2000) domains. ....

....17, 22, 25, 28] In the framework proposed in this paper the trajectories of continuous parts of hybrid systems (performance speci cations) can be represented by computable functionals. This is a subject of a further paper. 2 Basic Notions For background material on continuous domain we refer to [1, 31, 32, 14, 7]. To propose notions of computability of operators and real valued functionals, we, following the papers [31, 7] recall the de nitions of continuous domain for the reals (the interval domain) and computable functions and introduce functional domains. 2.1 Terminology Throughout the article, we ....

S. Abramsky, A. Jung, Domain theory, Handbook of Logic in Computer Science, v. 3, Clarendon Press, 1994.

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S. Abramsky, D. Gabbay, and T. Maibaum, editors. Handbook of Logic in Computer Science. Oxford Science Publications, 1992.

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S. Abramsky, D.M. Gabbay, and T.S.E. Maibaum, editors. Handbook of Logic in Computer Science, volume 2. Clarendon Press, Oxford, 1992.

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S. Abramsky, D. M. Gabbay, T. S. E. Maibaum (Eds.), Handbook of Logic in Computer Science, Vol. 2, Oxford University Press, 1992. 46

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S. Abramsky, Dov M. Gabbay, and T.S.E. Maibaum, editors. Handbook of Logic in Computer Science, Volume 2. Oxford University Press, 1992.

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S. Abramsky, D. Gabbay, and T. Maibaum, editors. Handbook of Logic in Computer Science. Oxford University Press, 1992.

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S. ABRAMSKY, D. GABBAY, and T. MAIBAUM, Handbook of Logic in Computer Science, two vols. (Oxford University Press, 1992).

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S. Abramsky, Dov M. Gabbay, and T. S. E Maibaum, editors. Handbook of Logic in Computer Science, volume 1. Oxford Science Publications, 1992.

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S. Abramsky, A. Jung, Domain theory, Handbook of Logic in Computer Science, v. 3, Clarendon Press, 1994.

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S. Abramsky, Dov. M. Gabbay and T.S.E. Maibaum, editors. Handbook of logic in computer science. Oxford University Press, 1992.

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S. Abramsky, D. Gabbay, and T. Maibaum, editors. Handbook of Logic in Computer Science. Oxford University Press, 1992.

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Gabbay, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science,volume 2, chapter 1, pages 1#117. Oxford University Press, Oxford, 1992.

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