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

Abstract

Temporal logic has become essential for various areas in computer science, most notably for the 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 the linear-time setting, one-pass and two-pass tableau methods have been developed. In the repository of the CTL-type branching-time setting, the well-known logics ECTL and ECTL+ were developed to explicitly deal with fairness. However, due to the syntactical restrictions, these logics can only express restricted versions of fairness. The logic CTL*, often considered as "the full branching-time logic" overcomes these restrictions on expressing fairness. However, this logic itself is extremely challenging for the application of verification techniques, and the 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 the formulation of dual sequent calculi that are often treated as more "natural", being more friendly for human understanding. Based on these two considerations, the 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 the formulation of a new range of fairness constraints with "until" operator. The presentation of the tableau construction is accompanied by an algorithm for constructing a systematic tableau, for any given input of admissible branching-time formulae. We prove the termination, soundness, and completeness of the method. As tree-shaped one-pass tableaux are well suited for the automation and are amenable to the implementation and the formulation of sequent calculi, our results also open a prospect of relevant developments of the automation and implementation of the tableau method for ECTL#, and of a dual sequent calculi.