On the Construction of Multi-valued Concurrent Dynamic Logics

Abstract

Dynamic logic is a powerful framework for reasoning about imperative programs. An extension with a concurrent operator [18] was introduced to formalise programs running in parallel. In other direction, other authors proposed a systematic method for generating multi-valued propositional dynamic logics to reason about weighted programs [14]. This paper presents the first step of combining these two frameworks to introduce uncertainty in concurrent computations. In the developed framework, a weight is assigned to each branch of the parellel execution, resulting in a (possible) asymmetric parallelism, inherent to fuzzy programming paradigm [21,2]. By adopting such an approach, a family of logics is obtained, called multi-valued concurrent propositional dynamic logics (CGDL(A)), parametric on an action lattice A specifying a notion of "weight" assigned to program execution. Additionally, the validity of some axioms of CPDL is discussed in the new family of generated logics. IntroductionOver time, the different variations of dynamic logics developed went hand-inhand with the very notion of its object, the program. This resulted in a diverse myriad of dynamic logics tailored to specific programming paradigms. Examples include probabilistic [11], concurrent [18], quantum [1] and continuous [19] computations, and combinations thereof. An example of another non-trivial paradigm is the fuzzy one [21,2], where the execution of a program differs from both classical and probabilistic scenarios: a conditional statement may act as a concurrent execution with a weight associated to each branch. The formalisation of such behaviour encompasses two non-trivial computational settings: concurrency and uncertainty. An extensive research can be found in the literature on diverse formalisms to reason about programs running in parallel [9,10] and to deal with uncertainty [11,5,20,4]. However, even when these two components are combined into a single framework [16], the uncertainty models probabilistic nondeterminism. Thus we are still missing a proper semantics to describe the behaviour of the fuzzy paradigm.Recently, reference [14] initiated a research agenda on the systematic development of multi-valued propositional dynamic logics, parametric on an action ⋆