A semantics and a logic for Fuzzy Arden Syntax

Fuzzy programming languages, such as the Fuzzy Arden Syntax (FAS), are used to describe behaviours which evolve in a fuzzy way and thus cannot be characterized neither by a Boolean outcome nor by a probability distribution. This paper introduces a semantics for FAS, focusing on the weighted parallel interpretation of its conditional statement. The proposed construction is based on the notion of a fuzzy multirelation which associates with each state in a program a fuzzy set of weighted possible evolutions. The latter is parametric on a residuated lattice which models the underlying semantic 'truth space'. Finally, a family of dynamic logics, equally parametric on the residuated lattice, is introduced to reason about FAS programs.