W. Penczek, Branching time and partial order in temporal logics, chapter 4 in [2].  Home/Search   Document Details and Download   Summary   Related Articles   Check

....concurrent and reactive programs. Since then, a number of linear and branching time temporal logical systems have been put forward with this purpose, some of the most popular ones of the latter type being UB, CTL, CTL and variations of them (see (Emerson and Srinivasan, 1989; Emerson, 1990) (Penczek, 1995), and (Stirling, 1992) for surveys on these) Since CTL subsumes the others, it will be the one in the focus of this paper, though the results will accordingly apply to the others as well. Besides the computational interpretation, branching time logics have important philosophical and purely ....

....1994) 2.2. Computation Tree Logics The full branching time computation tree logic CTL was introduced in (Emerson and Halpern, 1983) Here we o er a very brief summary of the syntax and semantics of CTL , referring the reader to (Emerson and Srinivasan, 1989) Emerson, 1990) or (Penczek, 1995) for more detail. 2.2.1. Syntax The language of CTL is a propositional language with a set of atomic propositions (propositional variables) AP, temporal operators X (nexttime) and U (until) and path quanti ers, 8 meaning for all paths , and its dual 9, meaning for some path , which will ....

Penczek W., Branching Time and Partial Order in Temporal Logics, in: Time and Logic: a Computational Approach, Univ. College of London, 1995, 179-228.

....alternative is to consider a base institution that may lead to categories of objects with products reflecting full concurrency . Work on this direction is reported in [19] and [12] considering event structures [52, 53] as basic interpretation structures and their associated temporal logics [31, 32, 37]. Another possibility would be to consider the temporal logic proposed in [41] Please note that, whichever is the base logic (institution) being considered, for each signature Sigma, every interpretation structure m 2 jInt( Sigma)j is a model of the initial theory h Sigma; ffl i. Therefore ....

W. Penczek. Branching time and partial order in temporal logics. In L. Bolc and A. Szalas, editors, Time and Logic: A Computational Approach, pages 179--228. UCL Press, 1995.

....a formula; ffl If a is a formula then (9F a) is a formula; ffl If a is a formula then (9X a) is a formula. The basic abbreviations are: ffl (8G a) j abv ( 9F ( a) 31 ffl (8F a) j abv ( 9G ( a) ffl (8X a) j abv ( 9X ( a) The set of UB formulae is equivalent to the one defined in [Pen95]. The temporal operators have intuitive meanings linked to linear temporal logic: 9G a) there is a path where (G a) 9F a) there is a path where (F a) and (9X a) there is a path where (X a) The abbreviations have dual meanings: 8G a) for all paths (G a) 8F a) for all paths (F a) ....

....Prop. Definition Entailment : Subset UBform UBform Prop : Phi:Subset UBform] f:UBform] t: Tree Subset P) a:UBform) Phi a) Sat t a) Sat t f) End UB. 4. 4 Soundness verification of UB The defined semantics of UB logic is now used to prove the soundness of the axiomatization in [Pen95], including the axioms: UBAx1: 9X (a b) 9X a) 9X b) UBAx2: 9F a) a (9X (9F a) UBAx3: 9G a) a (9X (9G a) UBAx4: 9X (a ) a) and inference rules: 34 MP : From a and (a ) b) it is derived b. UBR2: From (a ) b) it is derived ( 9X a) 9X b) UBR3: From (a ) b) ....

[Article contains additional citation context not shown here]

W. Penczek. Branching time and partial order in temporal logics. In L. Bolc and A. Szalas, editors, Time & logic: a computational approach, pages 179-- 228. UCL, 1995.

....to use local clock functions, thus leaving the underlying partial orders to be implicit, rather than interpret formulas on infinite traces or on event structures. For a discussion of temporal logics on partial order models of concurrency like traces and event structures, see [LRT92] T94] and [P95]. The choice of top level parallelism in the frames is deliberate to keep the agent structure simple. It is important to study hierarchical composition of such systems, but we do not address that issue here. Choosing synchronization as the basic communication mechanism between agents is again ....

Penczek, W., "Branching time and partial order in temporal logics", in L. Bolc and A. Szalas (ed), Time and logic : a computational approach, UCL Press, 1995.

....about non deterministic, concurrent and reactive programs. Since then, a number of linear and branching time temporal logical systems have been put forward with this purpose, some of the most popular ones of the latter type being UB, CTL, CTL and variations of them (see [ES 89, Eme 90] Pen 95] and [Sti 92] for surveys on these) The logic CTL will be in the focus of this paper, though the results will accordingly apply to all others subsumed by it, as well. Besides the computational interpretation, branching time logics have important philosophical and purely logical aspects, in ....

....(binders) without universal modality have been axiomatized. 4 1. 2 Computation Tree Logics The full branching time computation tree logic CTL was introduced in [EmH 83] Here we offer a very brief summary of the syntax and semantics of CTL , referring the reader to [ES 89] Eme 90] or [Pen 95] for more detail. 1.2.1 Syntax The language of CTL is a propositional language with a set of atomic propositions (propositional variables) AP, temporal operators X (nexttime) and U (until) and path quantifiers, 8 meaning for all paths , and its dual 9, meaning for some path , which will be ....

Penczek W., Branching Time and Partial Order in Temporal Logics, in: Time and Logic: a Computational Approach, Univ. College of London, 1995, 179-228.

....interleaving together with adequate assumptions of fairness [GPSS80, Fra86] on the execution of its components. The use of branching temporal logics like UB [BAMP81] CTL [CE81] or CTL [EH83] and even of linear temporal logic for such purposes has been deeply studied [Pnu77, MP92, Wol95, Pen95] In particular, most of the work on temporal logics for information systems and object orientation we rely on has been developed in this setting (e.g. Ser80, SFSE89, FSMS91, FM92, SCS94, SSC95, SSR96, DRCS97, SSC98a] However, certain subtleties of concurrent systems are lost under such ....

.... subtleties of concurrent systems are lost under such simplifications (see, for instance, Pra86] for a discussion on the subject) The need for a precise notion of causality as reflected by the time structures adopted for the logics led to the study of partial order temporal logics [PW84, KP87, Pen95] The basic difference is that the assumption of an omnipresent observer of the entire system being considered is dropped and replaced by a local causal perspective. It is the case of the logics for partially ordered computations introduced by Pinter and Wolper [PW84] of interleaving set ....

W. Penczek. Branching time and partial order in temporal logics. In L. Bolc and A. Szalas, editors, Time and Logic: A Computational Approach, pages 179--228. UCL Press, 1995.

....are theorems. f where f is a propositional tautology (7. 1) The following axioms deal with the temporal path component of BAL (they are adapted from Stirling s axiomatization of CTL [166] One other axiom from Stirling s axiomatization has been omitted, due to its complexity; see [128] for a discussion of the axiomatization. f y ) f y ) 7.2) f f f f (7.3) f (f y ) f f f y ) 7.4) f f f f (7.5) f f f) f f) 7.6) f U y y (7.7) f U y y (f f (f U y ) 7.8) f Af if f is atomic (7.9) Ef f if f ....

W. Penczek. Branching time and partial order in temporal logic. Technical Report UMCS--91--3--3, Department of Computer Science, Manchester University, Oxford Rd., Manchester M13 9PL, UK, 1991.

....interleaving together with adequate assumptions of fairness [GPSS80; Fra86] on the execution of its components. The use of branching temporal logics like UB [BAMP81] CTL [CE81] or CTL [EH83] and even of linear temporal logic for such purposes has been deeply studied [Pnu77; MP92; Wol95; Pen95] In particular, most of the work on temporal logics for information systems and object orientation we rely on has been developed in this setting (e.g. Ser80; SFSE89; FSMS91; FM92; SCS94; SSC95; SSR96] However, certain subtleties of concurrent systems are lost under such simplifications ....

.... subtleties of concurrent systems are lost under such simplifications (see, for instance, Pra86] for a discussion on the subject) The need for a precise notion of causality as reflected by the time structures adopted for the logics led to the study of partial order temporal logics [PW84; KP87; Pen95] The basic difference is that the assumption of an omnipresent observer of the entire system being considered is dropped and replaced by a local causal perspective. It is the case of the logics for partially ordered computations introduced by Pinter and Wolper [PW84] of interleaving set ....

[Article contains additional citation context not shown here]

W. Penczek. Branching time and partial order in temporal logics. In L. Bolc and A. Szalas, editors, Time and Logic: A Computational Approach, pages 179--228. UCL Press, 1995.

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W. Penczek, Branching time and partial order in temporal logics, chapter 4 in [2].

....our logic. 1 Introduction Temporal logic is an important tool for program verification. Depending on the notion of model, three kinds of temporal logic can be distinguished: temporal logic of linear time (LTL) 15, 10] temporal logic of branching time (BTL) 7] and partial order temporal logic [21]. Mazurkiewicz traces and trace systems [13] are partial order structures frequently used to give semantics to concurrent programs and interpreting propositional temporal logics ( 12, 6] ISTL [11] TrPTL [26] TSL [22] TLC [1] The first order versions of temporal logics are intended for ....

W. Penczek, Branching time and partial order in temporal logics, chapter 4 in [4].

....partial orders [25, 19, 20, 21, 22] allow a direct representation of properties involving causality, conflict, and concurrency. The first temporal logic on prime event structures has been put forward by Lodaya and Thiagarajan [9] Since then several new logics on event structures have been defined [20, 21, 13, 14, 23]. Most of these logics have been proved to be decidable and possessing complete axiomatizations. However, the model checking problem for event structure logics has never been addressed. The reason for this was the lack of two notions: automata on event structures and or finite representations of ....

....each w 2 Sigma , let last(w) a if w = w 0 a. For each trace 2 [ Sigma ] Max( flast(w) j [w] g. A trace is called prime, if jMax( j = 1, i.e. there is exactly one operation executed last in . 4 Logic on Event Structures We give a definition of Discrete Event Structure Logic [20, 23] without past modalities, for which a model checking algorithm is defined. The language contains modalities corresponding to the relations of (immediate) causality and (immediate) conflict. 4.1 Syntax and Semantics Let AP = fp 1 ; p 2 ; g be a countable set of atomic propositions. The ....

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W. Penczek, Branching time and partial order in temporal logics, chapter 4 in [2].

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