Is the Interesting Part of Process Logic uninteresting?: A Translation from PL to PDL

Abstract

With the (necessary) condition that atomic programs in process logic (PL) be binary, we present an algorithm for the translation of a PL formula p into a program '(p) of propositional dynamic logic (PDL) such that a finite path satisfies pitt it belongs to '(p). This reduction has two immediate corollaries: 1) validity in this PL can be tested by testing validity of formulas in PDL; 2) all state properties expressible in this PL are expressible in PDL. The translation, however, is of nonelementary time complexity.The significance of the result to the search for natural and powerful logics of programs is discussed.Key words, process logic, propositional dynamic logic, temporal logic 1. Introduction. The formalism of dynamic logic [Prl] has been successfully proposed as a unifying framework for the formal reasoning about programs. It generallogic, enjoying the advantages of both. Pratt's original process logic [Pr2], Parikh's SOAPL [PA] and Nishimura's language IN], were preliminary efforts in this direction. A recently proposed system which seems to have unified the basic concepts of both dynamic and temporal logic is the system of process logic (PL) presented in [HKP]. It borrows the program constructs and modal operators and from dynamic logic, and the temporal connectives t and sut from temporal logic and combines them into a single system.The declared purpose of this new system is to enable compositional reasoning about continuous behavior of programs. As such, one would expect it to be able to express properties on the propositional level inexpressible by either PDL or TL, the propositional versions of dynamic and temporal logic, respectively. Indeed, the PL expression [a]somep, for example, states that in every execution of the program there must be at least one state satisfying p. It can be shown [HI that this property cannot be expressed in PDL. If p is some property of a second program, say then it is not expressible in TL either.Having demonstrated this greater expressibility, PL certainly becomes an attractive system for study. It is shown in [HKP] that validity in (the propositional version of) PL is decidable by reduction to SnS, the second order theory of n successors JR]. This yields a nonelementary decision procedure in general, and it is still unknown whether an alternative elementary decision procedure exists. It has also been shown that various small extensions to the system lead to undecidability [CHMP], IS]. So much for

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