I. M. Hodkinson. On Gabbay's temporal fixed point operator. Theoretical Computer Science, 139:1--25, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....that our modal logic could be related to some temporal logic of discrete, linear time with finite past and infinite future, where the modal operator corresponds to the previous time , or yesterday , modality. Gabbay and Hodkinson discussed such a temporal logic and its fixed point operator [12, 16]; however, the relationship with is not obvious since their temporal logic is based on classical logic. Acknowledgments The author is greatly indebted to Professor Solomon Feferman for the opportunity to develop the main part of this research in a stimulating environment at Stanford ....

I. M. Hodkinson. On Gabbay's temporal fixed point operator. Theoretical Computer Science, 139:1--25, 1995.

.... way: roughly speaking, if the free occurrences of the propositional symbol p are in the scope of past temporal operators in , then we can form the formula p: Since at any time point the past is finite (we have natural numbers flow of time) this guarantees the existence of a fixed point; see [7] for more details. We are going to interpret temporal formulas over flows of time on the natural numbers. In more detail, let N = 0; 1; n; be the set of natural numbers and their usual ordering (according to magnitude) Let v be an evaluation of the propositional variables in P , ....

....that satisfies for every atom q distinct from p, w(q) v(q) n 2 w(p) iff (N; w) n fl . Theorem 1. The complexity of the decision problem for temporal logic of U and S over natural numbers flow of time is pspace, 12] This remains true even if we include the fixed point operator, [7]. 2.3 Complex Event Definitions First we give the translations of the basic operations on events into temporal logic: e tr j e for primitive event e; e ; e 0 ) tr j e 0 tr Pe tr ; e k e 0 ) tr j e tr e 0 tr ; e u e 0 ) tr j (e tr e 0 tr ) e tr Pe 0 tr ) ....

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I. Hodkinson. On Gabbay's temporal fixed point operator. Journal of Theoretical Computer Science, 139:1--25, 1995.

....as opposed to predicate based. For example, in [Wolper, 1983] it was shown that a parity property namely q may only occur after an even number of p s can not be expressed in PML. Solving this problem has lead to the development of fixedpoint temporal logics of various kinds: see, for example, [Hodkinson, 1992] and [Barringer et al. 1986] Worse problems arise with the specification of first in first out (FIFO) buffers which have to eventually send out the messages which they take in in the right order even if messages can be repeated. This is shown to be beyond PML in [Sistla et al. 1982] In fact, ....

....have to implement them. We just have to construct a model so that they can be defined if one wants to (as indeed those responsible for proving correctness may have to do) However, in this case, we note that if one is adverse to using the stronger QML language then one could follow the idea in [Hodkinson, 1992] and simply rewrite the specification without the predicate quantifications. The resulting FML specification would certainly be enough for our purposes but would involve the implementer in a little extra work to actually make sure the predicates id and od were built. Thus, technically, it is not ....

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I Hodkinson. On Gabbay's temporal fixed point operator. Technical report, Imperial College, 1992. J. Theoretical Computer Science, submitted.

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