Taut Monads, Dynamic Logic and Determinism

Abstract

Studied are Kleisli categories of monads of sets which satisfy two properties motivated by functional properties of collections. Such categories have box and diamond operators which follow the laws of (loop-free) dynamic logic. A theorem of Kozen states that the category of sets and relations is complete for loop-free dynamic logic. It is shown that the Kleisli category of the filter monad is likewise complete. A morphism α is deterministic if <α>Q⊂[α]Q for all Q. Each “output value”αx is associated with a filter which is forced to be an ultrafilter when α is deterministic and αx is defined. Early work in the theory of domains abstracted from the partially ordered set of partial functions between two sets, ordered by extension. A different abstraction, suited to conditional constructs rather than recursive fixed point equations, is the notion of a locally Boolean poset, and this is used to compare restriction categories with deterministic Kleisli categories. The laws of dynamic logic in terms of [α]Q and <α>Q for a single α hold in any topological space with [α]Q the interior operator and <α>Q the closure operator.