A clausal resolution method for extended computation tree logic ECTL

Abstract

A temporal clausal resolution method was originally developed for linear time temporal logic and further extended to the branching-time framework of Computation Tree Logic (CTL). In this paper, following our general idea to expand the applicability of this efficient method to more expressive formalisms useful in a variety of applications in computer science and AI requiring branching time logics, we define a clausal resolution technique for Extended Computation Tree Logic (ECTL). The branching-time temporal logic ECTL is strictly more expressive than CTL in allowing fairness operators. The key elements of the resolution method for ECTL, namely the clausal normal form, the concepts of step resolution and a temporal resolution, are introduced and justified with respect to this new framework. Although in developing these components we incorporate many of the techniques defined for CTL, we need novel mechanisms in order to capture fairness together with the limit closure property of the underlying tree models. We accompany our presentation of the relevant techniques by examples of the application of the temporal resolution method. Finally, we provide a correctness argument and consider future work discussing an extension of the method yet further, to the logic CTL * , the most powerful logic of this class.

A clausal resolution method for branching-time logic ECTL+

“…These results were essentially used in the formulation of the transformation rule for the ECTL fairness constraint A [2]. In this paper we will further extend their applicability in the transformation procedure for ECTL · .…”

“…The first stage of the transformation procedure outlined above, the procedure ½ , is taken from the translation of ECTL formulae to SNF ÌÄ [2]. Here we describe novel techniques to cope with Boolean combinations of temporal operators defined in addition to the rules of the procedure ¾ , introduced for ECTL [2].…”

“…Here we describe novel techniques to cope with Boolean combinations of temporal operators defined in addition to the rules of the procedure ¾ , introduced for ECTL [2]. We also recall some of those rules that will be used in our example given in Ü4.3.…”

“…The logic ECTL · further extends the expressiveness of ECTL by allowing Boolean combinations of elementary temporal operators and ECTL fairness constraints (but not permitting nesting of temporal operators or fairness constraints). In [2] a clausal resolution method has been developed for the logic ECTL. The introduction of the corresponding technique to cope with fairness constraints enabled the translation of an ECTL formula into the normal form, to which we apply a clausal resolution technique initially defined for the logic CTL.…”

Natural Deduction Calculus for Computation Tree Logic

“…Due to the dynamic and non-deterministic nature of the underlying complex systems, we believe that the appropriate formal framework is given in the so called branching-time setting. In [1] we have shown how the specification language of the normal form for branching-time logic [3] can be used in this framework to enable a resolution based deductive verification.…”