Characterizing determinacy in Kleene algebras

Abstract

Elements of Kleene algebras can be used, among others, as abstractions of the inputoutput semantics of nondeterministic programs or as models for the association of pointers with their target objects. In the first case, one seeks to distinguish the subclass of elements that correspond to deterministic programs. In the second case one is only interested in functional correspondences, since it does not make sense for a pointer to point to two different objects.We discuss several candidate notions of determinacy and clarify their relationship. Some characterizations that are equivalent in the case where the underlying Kleene algebra is an (abstract) relation algebra are not equivalent for general Kleene algebras.

Quantales and Temporal Logics

“…This is well defined, since in a Boolean left quantale · preserves arbitrary meets of tests in its left argument [4], and hence in left Boolean quantales domain always exists. By general properties of Galois connections, domain preserves arbitrary joins.…”

“…This semantic property can equivalently be characterised as follows (property (a) was already shown in [4]). …”

“…But in general quantales the special and the general case are not equivalent [4]. Moreover, again from [4], we know that in quantales such as WOR and STR an element n is left-distributive over meet iff it is prefix-free, i.e.…”

Extended transitive separation logic

“…We borrow it from [14]: Definition 5.3. An access element a is deterministic iff ∀ p : a| | | |a p ≤ p, where the dual diamond is defined by | |a p = (a · p) .…”

“…The subexpression a * left · ¬ a left occurring in a RLm is an algebraic representation of the loop while a left do a left . It has been shown in [19] that determinacy of a loop body is inherited by the corresponding while loop.…”