1. Syntax and semantics of DP L +

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1 Bulletin of the Section of Logic Volume 15/3 (1986), pp reedition 2005 [original edition, pp ] Zdzis law Habasiński DECIDABILITY IN PRATT S PROCESS LOGIC We present here the Dynamic Process Logic, DP L, which is a generalization of Pratt s process logics from [10]. DP L has been introduced in [4] and is designated for reasoning about events during regular programs computations. It should be regarded as a class of propositional logics with path formulae (interpreted over sequences of states) and state formulae interpreted as usually over states. A criterion of the decidability problem is given due to which any logic definable in the DP L-framework and meeting a natural regularity condition is decidable in time O (exp cn 3 ) where n is the length of the formula tested. For instance some logics strictly stronger than P L from [5] are still decidable in that time. DP L + is an extension of DP L allowing Boolean combinations of the elementary path formulae. Any DP L + -logic meeting the same regularity condition is decidable in time O(exp(n 3 exp ck)), where k is a number (usually appreciable) less than the length n of the given formula. 1. Syntax and semantics of DP L + Dynamic Process Logic generalizes the logics from [5,10]. The name Dynamic has been used in order to stress the connections with the Propositional Dynamic Logic [3]. Assume we have given two countable disjoint sets of atomic programs A 0, A 1,... and atomic formulae P 0, P 1,... and a finite set of path operators E 0,..., E z. Programs are defined exactly as in Propositional Dynamic Logic [3] (with tests over state formulae). Formulae of DP L + F1. Any atomic formula is a state formula.

2 Decidability in Pratt s Process Logic 89 F2. If a is a program and p is a path formula then < a > p is a state formula (called a diamond formula). F3. State formulae are closed under the usual Boolean connectives. F4. If E is a path operator of arity m and p 1,..., p m are state formulae then E(p 1,..., p m ) is a path formula (called an elementary one). F5 +. Path formulae are closed under the Boolean connectives. The set of DP L + -formulae consists of all the state formulae defined by the above rules. DP L-formulae are those defined by F1 - F4 only. As usual [a]p denotes < a > p. Semantics of DP L + Let S be a non-empty set. By S we denote the set of all non-empty finite sequences of elements from S. A structure for DP L + is any triple (S, =, T r) where S is a non-empty set of states, = is a satisfiability relation such that = (S {state formulae}) (S {path formulae}) and T r : {programs} Powerset(S). T r(a) is the set of traces of the program a. She above components should fulfill the following conditions: S1. s =< a > p iff there is s T r(a) : s = p and the first element of s is s. S2. T r(a) {(s) s S} =, for any atomic program A. S3. T r(a; b) = T r(a); T r(b) = {(s 0,..., s m ) (s 0,..., s k ) T r(a) and (s k,..., s m ) T r(b) for some k m}. S4. T r(a b) = T r(a) T r(b). S5. T r(a ) = {(s) s S} {T r(a n ) n 1}. S6. T r(p?) = {(s) s = p}. S7 - S9 describe the standard behaviour of the satisfiability relation on the Boolean connectives. 2. Examples In order to fix a DP L-logic we have to define path operators. Let us consider six two-ary operators: Until, while, before, pres, since and imp. We write simply p until q instead of until (p, q) etc. Let s = (s 0,..., s k ) for some k > 0.

3 90 Zdzis law Habasiński s = p until q iff i : s i = q and j i s j = p s = p while q iff i( j i s j = q) = s i = p s = p before q iff i : s i = p and j > i s j = q s = p pres q iff i : s i = p = j > i s j = q s = p since q iff i : s i = q and j > is j = p s = p imp q iff i : s i = p = j > i s j = q Using this formalism some other constructs may be defined: some p is simply true until p, all p corresponds to p while true and last p to false since p. Thus Pratt s construct during written in [5,10] as a p is expressible in DP L + as [a] some p and in DP L as < a >all p. Similarly preserves from the quoted papers is simply [a](p pres p) in DP L + i.e. < a > (p before p) in DP L. The formula < a > p from the Propositional Dynamic Logic is definable as < a >last p. ψ-formula from [5] is expressible in DP L + as [a](p imp q) or as < a > ( q since p) in DP L and is not definable in P L from [5]. Hence DP L over the six operators is an actual extension of P L from the Harel s paper. It follows from our criterion that it is decidable in one-exponential time. unchanged p operator which may be abstracted from [8] is easily defined as (p pres p) ( p pres p). 3. Decidability criterion Let us define for any formula p an alphabet Des(p) : D Des(p) iff D {state subformulae of p} and D is consistent i.e. q D q D. As usual finite automaton Aut accepting finite strings on the alphabet Des(p) defines the path formula p iff for any sequence (s 0,..., s k ) in any structure (S, =, T r) we have: (s 0,..., s k ) = p iff Aut accepts D 0,..., D k where D i = {q s i = q and q is a state subformula of p}. Let E be a path operator of arity m. E is regular iff there is a constant c : for any state formulae q 1,..., q m there is an usual finite-state automaton Aut on the alphabet Des(E(q 1,..., q m )) such that: Aut defines E(q 1,..., q m ) number of states in Aut is less than c Aut can be constructed in time proportional to the cardinality of its alphabet.

4 Decidability in Pratt s Process Logic 91 Note that each operator in the example is regular. Let us fix certain DP L + - formula p. A subformula of p is called maximal iff it is an elementary formula not in the scope of any path operator. Let DP L(k) denotes DP L + in which any path formula may contain at most k occurrences of the maximal subformulae. Theorem. Let us fix a finite set of regular path operators. For any state formula p 0 of DP L + the satisfiability problem is decidable in time proportional to exp(n 3 exp ck) where: n is the length of p 0, k = max{k 1,..., k N 1 }, k i is the number of occurrences of the maximal subformulae in p i (i = 1,..., N 1) and p 1,..., p N 1 are the diamond subformulae of p Concluding remarks In our opinion an introduction of whole class of regular operators in DP L is justified by the development of program verification methods. For example the operators below taken from this area are obviously regular. atnext from 7: s = atnext(p, q) iff ( i s i = q) or ( i : s i = p q and j < i s j = q) unless from [9]: s = unless(p, q) iff ( i s i = p) or ( i : s i = q and j < i s j = p) The complexity bound from the DP L + -criterion remains unchanged if we restrict the semantics of atomic programs in a way suggested in [12] to paths of length one or more generally shorter than a fixed number. However, our semantics cannot be simulated by the binary one, i.e. P DL-like semantics where programs are understood as binary relations. For consider an atomic program A with T r(a) = {(s 0, s 1, s 2 ), (t 0, s 1, t 2 )}. In this case we would mix the two paths getting unintentionally (s 0, s 1, t 2 ) as a trace of A. In other words DP L does not possess the crossing-path property. The arity of path operators does not make any difficulties in the decision procedure. We may consider DP L + -logics with zero-ary operators (non-local atomic formulae in parlance of [1]). As far as they are regular the resulting logic remains decidable by our criterion in contrast to the non-local P L from [6] which is known to be undecidable. One can prove (cf. [1]) that the set of paths {(s 0, s 1,...) s 2i = P } (P is a fixed atomic formula) cannot be defined in P L from [6]. Any regular DP L + -logic re-

5 92 Zdzis law Habasiński mains decidable when augmented by the operator even (s = even(p) iff every even state from s satisfies p). References The full version of this paper will appear in Proc. of the Conf. Mathematical Methods in Synthesis and Specification of Software Systems, Wendish- Rietz, April 85, Lecture Notes in Computer Sci [1] A. Chandra et al., Equations between regular terms and an application to Process Logic, Proc. ACM Symp. on Theory of Computing, 1981, pp [2] E. Emerson, J. Halpern, Decision procedures and expressiveness in the Temporal Logic of Branching Time, Proc. ACM Symp. on Theory of Computing, 1982, pp [3] M. Fisher, R. Lander, Propositional Dynamic Logic of regular programs, J. of Computer and System Sciences 18(2) (1979), pp [4] Z. Habasiński, Process Logics: two decidability results, Lecture Notes in Computer Sci. 176 (1984), pp [5] D. Harel, Two results on Process Logic, Information Processing Letters 8(4) (1979), pp [6] D. Harel et al., Process Logic: expressiveness, decidability, completeness, J. of Computer and System Sciences 25 (1982), pp [7] F. Kröger, On temporal program verification rules, RAIRO Informatique theorique Theoretical Informatics 19(5) (1985), pp [8] L. Lamport, Specyfying Concurrent Program Modules, ACM Transactions on Programming Languages and Systems 5(2). [9] Z. Manra, A. Pnueli, Proving precedence properties, Lecture Notes in Computer Science 154 (1983), pp [10] V. Pratt, Process Logic: preliminary report, Proc. of Principles of Programming Languages, 1979, pp [11] R. Street, Propositional Dynamic Logic of Looping and Converse, Proc. ACM Symp. on Theory of Computing, 1981.

6 Decidability in Pratt s Process Logic 93 [12] R. Sherman et al., In the interesting part of Process Logic uninteresting?, Rep. of the Veizmann Institute of Science, Rehovot, Israel, [13] P. Wolper, Synthesis of Communicating Processes from Temporal Logic Specifications, Rep. Dept. of Computer Science, Stanford Univ Computer Centre Technical University Poznań