Clausal reasoning for branching-time logics

Computation Tree Logic (CTL) is a branching-time temporal logic whose underlying model of time is a choice of possibilities branching into the future. It has been used in a wide variety of areas in Computer Science and Artificial Intelligence, such as temporal databases, hardware verification, program reasoning, multi-agent systems, and concurrent and distributed systems. In this thesis, firstly we present a refined clausal resolution calculus R�,S CTL for CTL. The calculus requires a polynomial time computable transformation of an arbitrary CTL formula to an equisatisfiable clausal normal form formulated in an extension of CTL with indexed existential path quantifiers. The calculus itself consists of eight step resolution rules, two eventuality resolution rules and two rewrite rules, which can be used as the basis for an EXPTIME decision procedure for the satisfiability problem of CTL. We give a formal semantics for the clausal normal form, establish that the clausal normal form transformation preserves satisfiability, provide proofs for the soundness and completeness of the calculus R�,S CTL, and discuss the complexity of the decision procedure based on R�,S CTL. As R�,S CTL is based on the ideas underlying Bolotov’s clausal resolution calculus for CTL, we provide a comparison between our calculus R�,S CTL and Bolotov’s calculus for CTL in order to show that R�,S CTL improves Bolotov’s calculus in many areas. In particular, our calculus is designed to allow first-order resolution techniques to emulate resolution rules of R�,S CTL so that R�,S CTL can be implemented by reusing any first-order resolution theorem prover. Secondly, we introduce CTL-RP, our implementation of the calculus R�,S CTL. CTL-RP is the first implemented resolution-based theorem prover for CTL. The prover takes an arbitrary CTL formula as input and transforms it into a set of CTL formulae in clausal normal form. Furthermore, in order to use first-order techniques, formulae in clausal normal form are transformed into firstorder formulae, except for those formulae related to eventualities, i.e. formulae containing the eventuality operator 3. To implement step resolution and rewrite rules of the calculus R�,S CTL, we present an approach that uses first-order ordered resolution with selection to emulate the step resolution rules and related proofs. This approach enables us to make use of a first-order theorem prover, which implements the first-order ordered resolution with selection, in order to realise our calculus. Following this approach, CTL-RP utilises the first-order theorem prover SPASS to conduct resolution inferences for CTL and is implemented as a modification of SPASS. In particular, to implement the eventuality resolution rules, CTL-RP augments SPASS with an algorithm, called loop search algorithm for tackling eventualities in CTL. To study the performance of CTL-RP, we have compared CTL-RP with a tableau-based theorem prover for CTL. The experiments show good performance of CTL-RP. i ii ABSTRACT Thirdly, we apply the approach we used to develop R�,S CTL to the development of a clausal resolution calculus for a fragment of Alternating-time Temporal Logic (ATL). ATL is a generalisation and extension of branching-time temporal logic, in which the temporal operators are parameterised by sets of agents. Informally speaking, CTL formulae can be treated as ATL formulae with a single agent. Selective quantification over paths enables ATL to explicitly express coalition abilities, which naturally makes ATL a formalism for specification and verification of open systems and game-like multi-agent systems. In this thesis, we focus on the Next-time fragment of ATL (XATL), which is closely related to Coalition Logic. The satisfiability problem of XATL has lower complexity than ATL but there are still many applications in various strategic games and multi-agent systems that can be represented in and reasoned about in XATL. In this thesis, we present a resolution calculus RXATL for XATL to tackle its satisfiability problem. The calculus requires a polynomial time computable transformation of an arbitrary XATL formula to an equi-satisfiable clausal normal form. The calculus itself consists of a set of resolution rules and rewrite rules. We prove the soundness of the calculus and outline a completeness proof for the calculus RXATL. Also, we intend to extend our calculus RXATL to full ATL in the future.