Seminaire d'Algebre Paul Dubreil et Marie-Paule Malliavin by M.P. Malliavin

By M.P. Malliavin

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In 1997, Schneider presented the temporal logic LeftCTL * that allows arbitrary deep nesting of temporal operators on one argument of the binary temporal operators without necessarily using path quantifiers. LeftCTL * can be translated to the alternationfree p,-calculus, and in particular, to CTL. The size of the obtained CTL formula for a given LeftCTL* formula if> is thereby bound by O(Iif>12 21 4>1), so that a LeftCTL * model checking procedure with runtime O(IKIIif>12214>1) is obtained. Note that the blow-up shows that there are LeftCTL * formulas that are exponentially more succinct than corresponding CTL formulas, and are therefore more readable.

While first order logics only quantify over elements of the considered domain V, higher order logics additionally quantify over sets of such elements, and functions of such elements. These additional extensions enhance the expressiveness of the logics so that all facts of mathematics can be expressed. For example, Peano's axioms for the natural numbers can be easily expressed in higher order logic, and hence, the natural numbers can be formalized with this formalism. Clearly, this means by Godel's result that there is no complete calculus for this logic.

As temporal logicS are less expressive than wautomata, there are properties that can not be expressed by temporal logics, but with w-automata. As an example, Wolper [519] mentioned the property that a variable a should hold for at least all even points of time. As these properties can, however, be expressed as J,t-calculus formulas and also as wautomata, Wolper suggested to extend temporal logic by operators based on regular languages [523]. Wolper also proposed a deductive proof system for his logic, which was shown to be complete, after some corrections, by Banieqbal and Barringer in 1986 [28].

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