Temporal Logic of Programs by Fred Kröger

By Fred Kröger

Creation to the temporal good judgment of - specifically paral- lel - programs.Divided into 3 major elements: - Presenta- tion of the natural temporal good judgment: language, semantics, and facts conception; - illustration of courses and their right- ties in the language of temporal good judgment; - program of the logical equipment to the verification of software right- ties together with a brand new embedding of Hoare's common sense into the temporal framework.

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If no node of ~ contains formulas of the kind IDA then ~ is obviously complete. Otherwise, let So be the first node in ~ containing such a formula. Ali, ... of nodes of T(§"): if A'; contains no IDA then the sequence ends with A'; ; otherwise, let -, oA' EA'; be such that, in the construction of So, ... , A';, IDA' has not been used (in this way) more times than the other formulas of this kind in A';. 0, ~, ... Ali, and so on. If So, ... , A'; is finite then we take the path from §" to A'; in the same way and iterate the whole construction by choosing an arbitrary path in T(A';).

Then m=2 x I and ~;=~{'U{Al}' ... , ~'=~"u{Ad, az-, - az-" U {- , A} U {- , A} l ' ... ----.... ----.... By the induction hypothesis we have f-~{, v ... v~" and from that we get f-~v ... v by (prop). Now the sets ~l*' ... , ~n* are just those ~' for which ~' u ~ is consistent. Suppose these are ~;, ... , ~' u ~ is inconsistent for ffi:: ~ i>n. ,. , f-ff ~ ~l* v '" v f7,,* , from this. 0 ~ f-~ ~ The informal meaning of some completion ~* of a consistent set ~ is that it gathers information about which subformulas of the formulas of ~ should be true in some state in order to get all formulas of ~ true in that same state.

Iff Ki(A) = f. iff Ki(A)=f or Ki(B)=t. iff K i + 1 (A)=t. iff Kj(A)=t for every r~i. iff Kj(B) = f for every j> i or Kk(A)=t for the smallest k>iwithKk(B)=t. iff K;(A) = t for every temporal structure K' =(S, e', W) with e'(Y) = e(y) for every yother than x. lSI Observe that lines 2-6 are the same as in the propositional case. Line 7 is the obvious extension of the respective classical definition. The definitions are transferred to the other propositional operators as before. For the existential quantifier we get: Ki(3xA)=t iff KaA)=t for some temporal structure K' =(S, e', W) with f(Y)=e(y) for every yother than x.

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