Y. Venema. A modal logic for chopping intervals. Journal of Logic and Computation, (1):453--476, 1991. 99  Home/Search   Document Not in Database   Summary   Related Articles   Check

....over a time interval w if P holds over w and there exists a direct constituent of w over which P densely holds. Most interval temporal logics, such as, for instance, Moszkowski s Interval Temporal Logic (ITL) 100] Halpern and Shoham s Modal Logic of Time Intervals (HS) 63] Venema s CDT Logic [119], and Chaochen and Hansen s Neighborhood Logic (NL) 16] have been shown to be undecidable. Decidable fragments of these logics have been obtained by imposing severe restrictions on their expressive power. As an example, Moszkowski [100] proves the decidability of the fragment of Propositional ....

Y. Venema. A Modal Logic for Chopping Intervals. Journal of Logic and Computation, 1(4):453--476, 1991.

....of a Duration Calculus with in nite intervals [7] and a neighbourhood logic [5] 2.1 Neighbourhood Logics In [5] Zhou and Hansen introduced a Neighbourhood Logic which is able to specify the liveness and fairness with a powerful proof system. This logics is similar to Venema s CDT logic ([51]) but more powerful. Namely it can reason about state durations. It extends the basic Duration Calculus with the neighbourhood modalities 3 l and 3 r . For a formula , the formula 3 l is satis ed by an interval [a; b] i there exists an interval which is a left neighbourhood of a, i.e. m; a] ....

Yde Venema. A modal logic for chopping intervals. Journal of Logic Computation, 1(4):453{ 476, 1991.

....takes place. Informally, S represents the sum of the lengths of those parts of an interval within which a modeled system satisfies some boolean condition. In particular, 1 will always evaluate to the length of this interval. Propositional systems with chop have been studied earlier by Venema [Ven91]. This chop operator, the temporal variables and the state variables of duration calculus, and the power of first order language have proved fit to express a wide range of properties of real time hybrid systems and bring the study of the correctness of their design to a rigorous formal level. Many ....

Y. Venema. A Modal Logic for Chopping Intervals. Journal of Logic and Computation, 1(4):453-476, 1991.

.... Interval Logic Thomas Marthedal Rasmussen 1 Various interval temporal logics have been proposed for specifying and reasoning about realtime systems [5, 4, 7, 8]. These logics interpret formulas over intervals of time and introduce di erent modalities to express the relationship between intervals. A typical modality is the chop modality [5, 7, 1] here denoted by ) A formula holds on an interval if there are two consecutive subintervals where ....

....1 Various interval temporal logics have been proposed for specifying and reasoning about realtime systems [5, 4, 7, 8] These logics interpret formulas over intervals of time and introduce di erent modalities to express the relationship between intervals. A typical modality is the chop modality [5, 7, 1] (here denoted by ) A formula holds on an interval if there are two consecutive subintervals where and holds, respectively: The chopping point can only lie inside the current interval, hence it is not possible to reason about properties outside the current interval. ....

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Yde Venema. A Modal Logic for Chopping Intervals. Journal of Logic and Computation, 1(4):453-476, 1991.

....must hold on all subintervals or property must hold on some interval eventually . One of the rst uses of such a formalism was the work of [5, 10] where timing aspects of hardware components were modeled. There has been a lot of work since, on many di erent aspects of interval logics, e.g. [9, 6, 21, 11, 14]. In the ProCoS project [1] in the end of the 1980 s and the beginning of the 1990 s it was realized that a convenient formalism for specifying and reasoning on accumulated durations of Boolean valued functions over time periods were required for expressing certain properties of real time ....

Y. Venema. A Modal Logic for Chopping Intervals. Journal of Logic and Computation, 1(4):453-476, 1991.

....to de ne future intervals in SIL for the speci cation of liveness properties. To characterize the expressive power of SIL we relate SIL to arrow logic and relational algebra. Keywords: interval logic, temporal intervals, arrow logic, real time systems, liveness. 1 Introduction Interval logics [4, 11, 17, 19, 8, 5, 22, 20, 12, 2, 13, 14, 6, 21] are logics of temporal intervals: One can express properties such as if holds on this interval then must hold on all subintervals or must hold on some interval eventually . Interval logics have proven useful in the speci cation and veri cation of real time and safety critical systems. ....

....following shortly consider related work on interval logic. In [5] an interval logic with six unary interval modalities is developed, and it is shown that they can express all thirteen possible relations between intervals [1] In [19] a complete proof system for the interval logic of [5] is given. [20] considers an even more expressive interval logic with three binary modalities instead of six unary. A complete proof system for this logic is also given. Unfortunately, both the proof systems of [19, 20] are somewhat complicated due to the fact that they are kept in a propositional setting. In ....

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Y. Venema. A Modal Logic for Chopping Intervals. Journal of Logic and Computation, 1(4):453-476, 1991.

....Interval Temporal Logic C build on a countable collection of propositions by closing under the logical connectives : and ( chop ) The other Boolean connectives are de ned as usual. We identify a special proposition 0 which we will use to mark intervals [a; a] consisting of one point [Ven91]. To be more precise, the logic can be called C 0 [LR00] but in this paper we use the simpler notation. Given a proposition p, a valuation V assigns intervals on which it is true, subject to the condition that the special proposition 0 is made true exactly on point intervals of the form [a; a] ....

.... chop modality, Halpern and Shoham consider the logic HS with the three unary modalities B , and A . Observe that D is de nable in HS as B . Venema s paper [Ven90] shows that chop is not de nable in HS. The logic having the chop modality as well as the three unary modalities is called CDT [Ven91]. CDT , A C HS B , A H H H H H HY BE B , H H H H H HY DD D , D H H H H H HY D H H H H H HY We will also be interested in two sublogics BE and DD of HS. BE only has the modalities B ....

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Y. Venema. A modal logic for chopping intervals, J. Logic Comput. 1,4 (1991) 453-476.

....been introduced [10, 19] for which model checking is decidable [3, 17] In the above logics, formulas are interpreted over states which represent instantaneous situations; time points are the basic entities. Other formalisms adopt a different semantics and interpret formulas over intervals of time [20, 12, 27]. Among such interval modal logics, ITL [20] and more specifically the duration calculus [8, 24] have been proposed for reasoning about real time systems. These two formalisms are first order modal logics which incorporate a binary modal operator (denoted by ; interpreted as the operation of ....

.... which has the same expressive power as full regular expressions [25] A decision procedure and a complete proof system for such a propositional logic are given in [25] Other complete deductive systems for propositional modal logics which include the operator chop can also be found in [23] and [27]. 4 INTRODUCTION In the first order case, different deductive systems exist for both ITL [21] and the duration calculus [14, 26] but little is known about their power. Close links between the two logics have been established in [14] a complete proof system for a dense timed ITL would yield a ....

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Y. Venema. A modal logic for chopping intervals. Journal of Logic and Computation, 1(4):453--476, 1991.

....specification in section 3. 2. 4 Related work The development of requirements for a model of a dynamic system presented above is similar in spirit although not in the chosen formalism to work by Leveson [40, 34] and Parnas [64] The Duration Calculus builds on Moszkowski s interval temporal logic [53, 54, 75]. Time may also be handled explicitly as in TLA [38, 39] or within a conventional temporal logic [36, 65, 30] In a proof assistant, model checking [20, 7] might be very useful. The refinement approach outlined above is inspired by the hierarchical state machines of Harel [23] Designs can also be ....

Y. Venema. A modal logic for chopping intervals. J. Logic of Computation, 1(4):453--476, 1991.

....using paper and pencil; but for proofs of formulas involving the notion neighbourhood, it is impossible to get support from theorem provers for ITL logics. In order to improve the expressiveness of ITL, people have introduced into ITL infinite intervals [18, 6] and expanding modalities [23, 10, 19, 21]. However, all those modalities are a little complicated for different reasons. For example, 23] establishes a complete propositional calculus for three binary interval modalities. In addition to the chop (designated as C in [23] it introduces to modalities T and D, which are expanding in the ....

....to get support from theorem provers for ITL logics. In order to improve the expressiveness of ITL, people have introduced into ITL infinite intervals [18, 6] and expanding modalities [23, 10, 19, 21] However, all those modalities are a little complicated for different reasons. For example, [23] establishes a complete propositional calculus for three binary interval modalities. In addition to the chop (designated as C in [23] it introduces to modalities T and D, which are expanding in the sense that the truth value of formulas OET and OED on an interval [b; e] depends on intervals ....

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Y. Venema. A modal logic for chopping intervals. J. Logic Computat., 1(4):453--476, 1991.

....place. Informally, R S represents the sum of the lengths of those parts of an interval within which a modeled system satisfies some boolean condition. In particular, R 1 will always evaluate to the length of this interval. Propositional systems with chop have been studied earlier by Venema [Ven91]. This chop operator, the temporal variables and the state variables of duration calculus, and the power of first order language have proved fit to express a wide range of properties of real time hybrid systems and bring the study of the correctness of their design to a rigorous formal level. Many ....

Y. Venema. A Modal Logic for Chopping Intervals. Journal of Logic and Computation, 1(4):453-476, 1991.

....of the old relational model in the sense that it satisfies Axioms (S 1 S 4 ) but we shall see that it has some additional structure. An ITL formula is built from atomic propositions, Boolean connectives, and a binary modal operator for concatenating intervals, usually referred to as chop [12, 16]. A minimal grammer is : p j j [ j ; where p ranges over the set of propositions. The truth of a formula depends on an observation interval and a possible world . A possible world is represented as a function f : Time Sigma, where Time is a given totally ordered set and Sigma ....

Y. Venema. A modal logic for chopping intervals. J. Logic Computat., 1(4):453--476, 1991.

....of the old relational model in the sense that it satisfies Axioms (S 1 S 4 ) but we shall see that it has some additional structure. An ITL formula is built from atomic propositions, Boolean connectives, and a binary modal operator for concatenating intervals, usually referred to as chop [12, 16]. A minimal grammer is # : p # # # # #; # , where p ranges over the set of propositions. The truth of a formula depends on an observation interval and a possible world . A possible world is represented as a function f : Time # #, where Time is a given totally ordered set and # ....

Y. Venema. A modal logic for chopping intervals. J. Logic Computat., 1(4):453--476, 1991.

....specifications. More details on the semantics of the behavioural paradigm can be found in [5,6] Methodological aspects of the paradigm and worked examples can also be found in [8,9] As future work, we intend to study the relationship between DTL and interval based temporal logics, e.g. 3] and [10] (where we also find a completeness proof for an axiom system that involves operators based on ternary relations) as well as the realtime aspects that can be associated with durative actions [2] ....

Y.Venema, "A Modal Logic for Chopping Intervals", Journal of Logic and Computation 1 (4), pp. 453-476, 1991. -- 16 --

....of introducing infinite intervals for this purpose. However, it is no longer meaningful to evaluate duration measures over such intervals. Special care must be taken to distinguish between finite and infinite intervals, which causes the approach to lose some appeal. We have been inspired by Venema [24] to introduce modal operators which quantify over both super intervals and sub intervals. That is, properties are specified over an infinite number of finite intervals instead of over infinite intervals. This way, no special restrictions have to be placed on the evaluation of the duration ....

....and branching time temporal logic (CTL) Also a uniform model for DC, TL, and TLA has been proposed. These approaches are discussed in Section 7. 2 Interval Logic The following sections present the syntax and semantics of Interval Logic (IL) IL is a syntactic modification of dense time CDT [24]. 2.1 Syntax The syntax of a formula A in IL is: A : P j d e j :A j A 1 A 2 j A 1 ; A 2 j A 1 A 2 j A 1 . A 2 where P ranges over atomic propositions, d e denotes a point interval, and . the chopping operators, and : and the usual propositional connectives. Note, that we for ....

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Y. Venema. A modal logic for chopping intervals. Journal of Logic Computation, 1(4):453--476, 1991.

....have been introduced [8] for which model checking is decidable [2, 12] In the above logics, formulas are interpreted over states which represent instantaneous situations; time points are the basic entities. Other formalisms adopt a different semantics and interpret formulas over intervals of time [14, 10, 20]. Among such interval modal logics, ITL [14] and more specifically the duration calculus [6, 17] have been proposed for reasoning about real time systems. These two formalisms are first order logics which incorporate a binary modal operator (denoted by ; interpreted as the operation of chopping ....

....close links between the two logics have been established in [11] a complete proof system for a dense timed ITL would yield a complete deductive system for the duration calculus. In the propositional case, complete axiomatizations have been proposed for modal logics which contain the chop operator [18, 16, 20]. Some of these logics are known to be decidable [18] Except for restricted fragments, the duration calculus (and ITL) are not decidable [5] This paper presents completeness results for first order ITL, in a variant similar to the one used in [11] which contains no other modal operator than ....

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Y. Venema. A modal logic for chopping intervals. Journal of Logic and Computation, 1(4):453--476, 1991.

.... modal logic, cf. section 2, or start from an algebraic perspective by considering n ary operators on Boolean algebras, cf. J onsson Tarski [8] Examples of such (binary) operators can be found in categorial grammar (see Lambek [13] Roorda [15] in temporal logic of intervals (see Venema [18], Hansen Zhou [6] or in multidimensional modal logics such as arrow logic (see Marx et al..ii [14] A prime example in the theory of Boolean algebras with operators is the composition operator in relation algebras, see Hodkinson and Hirsch [7] If we are after a simulation of polyadic modal ....

Y. Venema. A modal logic for chopping intervals. Journal of Logic and Computation, 1:453-476, 1993. 48

.... modal logic, cf. section 2, or start from an algebraic perspective by considering n ary operators on Boolean algebras, cf. J onsson Tarski [8] Examples of such (binary) operators can be found in categorial grammar (see Lambek [13] Roorda [15] in temporal logic of intervals (see Venema [18], Hansen Zhou [6] or in multidimensional modal logics such as arrow logic (see Marx et al..ii [14] A prime example in the theory of Boolean algebras with operators is the composition operator in relation algebras, see Hodkinson and Hirsch [7] If we are after a simulation of polyadic modal ....

Y. Venema. A modal logic for chopping intervals. Journal of Logic and Computation, 1:453-476, 1993. 46

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Y. Venema. A modal logic for chopping intervals. Journal of Logic and Computation, (1):453--476, 1991. 99

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Y. Venema. A modal logic for chopping intervals. Journal of Logic and Computation, 1(4):453--476, 1991.

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Y. Venema. A modal logic for chopping intervals. Journal of Logic and Computation, 1:453--476, 1991.

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Y. Venema. A Modal Logic for Chopping Intervals. Journal of Logic and Computation, 1(4):453--476, 1991.

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Y. Venema. A modal logic for chopping intervals. J. Logic of Computation, 1(4):453--476, 1991.

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Venema Y., A Modal Logic for Chopping Intervals, Journal of Logic and Computation, Vol 1(4), pp. 453-476, 1991. References 25

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Y. Venema. A modal logic for chopping intervals. J. Logic of Computation, 1(4):453--476, 1991.

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