Extending Fairness Expressibility of ECTL+: A Tree-Style One-Pass Tableau Approach

Abstract

Temporal has become essential for various areas in computer science, most notably for specification and verification of hardware and software systems. For specification purposes rich temporal languages are required that, in particular, can express fairness constraints. For linear-time logics which deal with fairness in linear-time setting, one-pass and two-pass tableau methods have been developed. In repository of CTL-type branching-time setting, well-known logics ECTL and ECTL^+ were developed to explicitly deal with fairness. However, due to syntactical restrictions, these logics can only express restricted versions of fairness. The CTL^*, often considered as the full branching-time logic overcomes these restrictions on expressing fairness. However, this itself, is extremely challenging for application of verification techniques, and tableau technique, in particular. For example, there is no one-pass tableau construction for this logic, while it is known that one-pass tableau has an additional benefit enabling formulation of dual sequent calculi that are often treated as more natural being more friendly for human understanding. Based on these two considerations, following problem arises - are there logics that have richer expressiveness than ECTL^+ yet simpler than CTL^* for which a one-pass tableau can be developed? In this paper we give a solution to this problem. We present a tree-style one-pass tableau for a sub-logic of CTL^* that we call ECTL^#, which is more expressive than ECTL^+ allowing formulation of a new range of fairness constraints with until operator. The presentation of tableau construction is accompanied by an algorithm for constructing a systematic tableau, for any given input of admissible branching-time formulae. We prove termination, soundness and completeness of method. As tree-shaped one-pass tableaux are well suited for automation and are amenable for implementation and for formulation of sequent calculi, our results also open a prospect of relevant developments of automation and implementation of tableau method for ECTL^#, and of a dual sequent calculi.