On the Complexity of Computing Minimal Unsatisfiable LTL formulas

We show that (1) the Minimal False QCNF search-problem (MF-search) and the Minimal Unsatisfiable LTL formula search problem (MU-search) are FPSPACE complete because of the very expressive power of QBF/LTL, (2) we extend the PSPACE-hardness of the MF decision problem to the MU decision problem. As a consequence, we deduce a positive answer to the open question of PSPACE hardness of the inherent Vacuity Checking problem. We even show that the Inherent Non Vacuous formula search problem is also FPSPACE-complete.Comment: Minimal unsatisfiable cores For LTL causes inherent vacuity checking redundancy coverag

A Modal Logic for Subject-Oriented Spatial Reasoning

We present a modal logic for representing and reasoning about space seen from the subject\u27s perspective. The language of our logic comprises modal operators for the relations "in front", "behind", "to the left", and "to the right" of the subject, which introduce the intrinsic frame of reference; and operators for "behind an object", "between the subject and an object", "to the left of an object", and "to the right of an object", employing the relative frame of reference. The language allows us to express nominals, hybrid operators, and a restricted form of distance operators which, as we demonstrate by example, makes the logic interesting for potential applications. We prove that the satisfiability problem in the logic is decidable and in particular PSpace-complete

A clausal resolution method for branching-time logic ECTL+

We expand the applicability of the clausal resolution technique to the branching-time temporal logic ECTL_. ECTL_ is strictly more expressive than the basic computation tree logic CTL and its extension, ECTL, as it allows Boolean combinations of fairness and single temporal operators. We show that any ECTL_ formula can be translated to a normal form the structure of which was initially defined for CTL and then applied to ECTL. This enables us to apply to ECTL_ a resolution technique defined over the set of clauses. Our correctness argument also bridges the gap in the correctness proof for ECTL: we show that the transformation procedure for ECTL preserves unsatisfiability

Satisfiability Games for Branching-Time Logics

The satisfiability problem for branching-time temporal logics like CTL*, CTL and CTL+ has important applications in program specification and verification. Their computational complexities are known: CTL* and CTL+ are complete for doubly exponential time, CTL is complete for single exponential time. Some decision procedures for these logics are known; they use tree automata, tableaux or axiom systems. In this paper we present a uniform game-theoretic framework for the satisfiability problem of these branching-time temporal logics. We define satisfiability games for the full branching-time temporal logic CTL* using a high-level definition of winning condition that captures the essence of well-foundedness of least fixpoint unfoldings. These winning conditions form formal languages of \omega-words. We analyse which kinds of deterministic {\omega}-automata are needed in which case in order to recognise these languages. We then obtain a reduction to the problem of solving parity or B\"uchi games. The worst-case complexity of the obtained algorithms matches the known lower bounds for these logics. This approach provides a uniform, yet complexity-theoretically optimal treatment of satisfiability for branching-time temporal logics. It separates the use of temporal logic machinery from the use of automata thus preserving a syntactical relationship between the input formula and the object that represents satisfiability, i.e. a winning strategy in a parity or B\"uchi game. The games presented here work on a Fischer-Ladner closure of the input formula only. Last but not least, the games presented here come with an attempt at providing tool support for the satisfiability problem of complex branching-time logics like CTL* and CTL+

A clausal resolution for extended computation tree logic ECTL

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

Specification and verification of reconfiguration protocols in grid component systems

In this work we present an approach for the formal specification and verification of the reconfiguration protocols in Grid component systems. We consider Fractal, a modular and extensible component model. As a specification tool we invoke a specific temporal language, separated clausal normal form, which has been shown to be capable of expressing any ECTL+ expression thus, we are able to express the complex fairness properties of a component system. The structure of the normal enables us to directly apply the deductive verification technique, temporal resolution defined in the framework of branching-time temporal logic

The complexity of clausal fragments of LTL

We introduce and investigate a number of fragments of propositional temporal logic LTL over the flow of time (ℤ, <). The fragments are defined in terms of the available temporal operators and the structure of the clausal normal form of the temporal formulas. We determine the computational complexity of the satisfiability problem for each of the fragments, which ranges from NLogSpace to PTime, NP and PSpace