Propositional dynamic logics of programs: A survey

Abstract

ion, independent of the vagaries of syntax. Theoretical computer science provides many examples of dynamic algebras besides Kripke models; see [Pr5]. Dynamic algebras were introduced in [K1] and have also been studied by Pratt [Pr4,5,7], Reiterman and Trnkova [RTI], Nemeti [N], and the author [Kl-5] . A survey of these results follows. Definition ,of dynamic algebras PEDL has two sorts, propositions and programs. Accordingly, a dynamic algebra is a two-sorted algebraic structure (K,B,o), where: B is a Boolean algebra; K is a Kleene algebra or algebra of regular events [C] with operators ; , u, ~, 0, ~ (identity), and sometimes (reverse); and O is a "scalar multiplication" K x B-,B. There are several possible definitions of Kleene algebras (see ['C]). As the axioms we will use appear elsewhere in this volume [K2], we will restrict ourselves to some examples: the family of all binary relations on a set S, where u is set union, ; is relational composition, 0 is the null set, X is the identity relation, and * is reflexive transitive closure; the family of regular sets over {0,1}*, where ; is concatenation and ), is the set containing only the null string of {0,1}x; any Boolean algebra, where is 1, ; is A, and a*= 1 for all a; and the structure MIN consisting of the extended natural numbers tNO{oo}, where U gives the minimum of two numbers, ; is addition, the Kleene algebra constant g is the number 0, the Kleene algebra constant 0 is % and a* = 0 for all a. The last structure appears in the study of shortest path problems [AHU]. Other examples appear in [Pr5"t. The axioms for scalar multiplication are just the Segerberg axioms for PDL, with • the exception that Segerberg induction axiom (tnd) (X A [=:¢](X~Ca]X)) ~ Ca*IX is replaced by the stronger *-continuity condition