Optimal Tableaux Method for Constructive Satisfiability Testing and Model Synthesis in the Alternating-time Temporal Logic ATL+

We develop a sound, complete and practically implementable tableaux-based decision method for constructive satisfiability testing and model synthesis in the fragment ATL+ of the full Alternating time temporal logic ATL*. The method extends in an essential way a previously developed tableaux-based decision method for ATL and works in 2EXPTIME, which is the optimal worst case complexity of the satisfiability problem for ATL+ . We also discuss how suitable parametrizations and syntactic restrictions on the class of input ATL+ formulae can reduce the complexity of the satisfiability problem.Comment: 45 page

Tableau-based decision procedures for logics of strategic ability in multi-agent systems

We develop an incremental tableau-based decision procedures for the Alternating-time temporal logic ATL and some of its variants. While running within the theoretically established complexity upper bound, we claim that our tableau is practically more efficient in the average case than other decision procedures for ATL known so far. Besides, the ease of its adaptation to variants of ATL demonstrates the flexibility of the proposed procedure.Comment: To appear in ACM Transactions on Computational Logic. 48 page

Tableau-based decision procedure for the multi-agent epistemic logic with operators of common and distributed knowledge

We develop an incremental-tableau-based decision procedure for the multi-agent epistemic logic MAEL(CD) (aka S5_n (CD)), whose language contains operators of individual knowledge for a finite set Ag of agents, as well as operators of distributed and common knowledge among all agents in Ag. Our tableau procedure works in (deterministic) exponential time, thus establishing an upper bound for MAEL(CD)-satisfiability that matches the (implicit) lower-bound known from earlier results, which implies ExpTime-completeness of MAEL(CD)-satisfiability. Therefore, our procedure provides a complexity-optimal algorithm for checking MAEL(CD)-satisfiability, which, however, in most cases is much more efficient. We prove soundness and completeness of the procedure, and illustrate it with an example.Comment: To appear in the Proceedings of the 6th IEEE Conference on Software Engineering and Formal Methods (SEFM 2008

Tableau-based decision procedure for the multi-agent epistemic logic with all coalitional operators for common and distributed knowledge

We develop a conceptually clear, intuitive, and feasible decision procedure for testing satisfiability in the full multi-agent epistemic logic CMAEL(CD) with operators for common and distributed knowledge for all coalitions of agents mentioned in the language. To that end, we introduce Hintikka structures for CMAEL(CD) and prove that satisfiability in such structures is equivalent to satisfiability in standard models. Using that result, we design an incremental tableau-building procedure that eventually constructs a satisfying Hintikka structure for every satisfiable input set of formulae of CMAEL(CD) and closes for every unsatisfiable input set of formulae.Comment: Substantially extended and corrected version of arXiv:0902.2125. To appear in: Logic Journal of the IGPL, special issue on Formal Aspects of Multi-Agent System

Completeness of Flat Coalgebraic Fixpoint Logics

Modal fixpoint logics traditionally play a central role in computer science, in particular in artificial intelligence and concurrency. The mu-calculus and its relatives are among the most expressive logics of this type. However, popular fixpoint logics tend to trade expressivity for simplicity and readability, and in fact often live within the single variable fragment of the mu-calculus. The family of such flat fixpoint logics includes, e.g., LTL, CTL, and the logic of common knowledge. Extending this notion to the generic semantic framework of coalgebraic logic enables covering a wide range of logics beyond the standard mu-calculus including, e.g., flat fragments of the graded mu-calculus and the alternating-time mu-calculus (such as alternating-time temporal logic ATL), as well as probabilistic and monotone fixpoint logics. We give a generic proof of completeness of the Kozen-Park axiomatization for such flat coalgebraic fixpoint logics.Comment: Short version appeared in Proc. 21st International Conference on Concurrency Theory, CONCUR 2010, Vol. 6269 of Lecture Notes in Computer Science, Springer, 2010, pp. 524-53

Using automata to characterise fixed point temporal logics

This work examines propositional fixed point temporal and modal logics called mu-calculi and their relationship to automata on infinite strings and trees. We use correspondences between formulae and automata to explore definability in mu-calculi and their fragments, to provide normal forms for formulae, and to prove completeness of axiomatisations. The study of such methods for describing infinitary languages is of fundamental importance to the areas of computer science dealing with non-terminating computations, in particular to the specification and verification of concurrent and reactive systems. To emphasise the close relationship between formulae of mu-calculi and alternating automata, we introduce a new first recurrence acceptance condition for automata, checking intuitively whether the first infinitely often occurring state in a run is accepting. Alternating first recurrence automata can be identified with mu-calculus formulae, and ordinary, non-alternating first recurrence automata with formulae in a particular normal form, the strongly aconjunctive form. Automata with more traditional Büchi and Rabin acceptance conditions can be easily unwound to first recurrence automata, i.e. to mu-calculus formulae. In the other direction, we describe a powerset operation for automata that corresponds to fixpoints, allowing us to translate formulae inductively to ordinary Büchi and Rabin-automata. These translations give easy proofs of the facts that Rabin-automata, the full mu-calculus, its strongly aconjunctive fragment and the monadic second-order calculus of n successors SnS are all equiexpressive, that Büchi-automata, the fixpoint alternation class Pi_2 and the strongly aconjunctive fragment of Pi_2 are similarly related, and that the weak SnS and the fixpoint-alternation-free fragment of mu-calculus also coincide. As corollaries we obtain Rabin's complementation lemma and the powerful decidability result of SnS. We then describe a direct tableau decision method for modal and linear-time mu-calculi, based on the notion of definition trees. The tableaux can be interpreted as first recurrence automata, so the construction can also be viewed as a transformation to the strongly aconjunctive normal form. Finally, we present solutions to two open axiomatisation problems, for the linear-time mu-calculus and its extension with path quantifiers. Both completeness proofs are based on transforming formulae to normal forms inspired by automata. In extending the completeness result of the linear-time mu-calculus to the version with path quantifiers, the essential problem is capturing the limit closure property of paths in an axiomatisation. To this purpose, we introduce a new \exists\nu-induction inference rule

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

Sublogics of a Branching Time Logic of Robustness

In this paper we study sublogics of RoCTL*, a recently proposed logic for specifying robustness. RoCTL* allows specifying robustness in terms of properties that are robust to a certain number of failures. RoCTL* is an extension of the branching time logic CTL* which in turn extends CTL by removing the requirement that temporal operators be paired with path quantifiers. In this paper we consider three sublogics of RoCTL*. We present a tableau for RoBCTL*, a bundled variant of RoCTL* that allows fairness constraints to be placed on allowable paths. We then examine two CTL-like restrictions of CTL*. Pair-RoCTL* requires a temporal operator to be paired with a path quantifier; we show that Pair-RoCTL* is as hard to reason about as the full CTL*. State-RoCTL* is restricted to State formulas, and we show that there is a linear truth preserving translation of State-RoCTL into CTL, allowing State-RoCTL to be reasoned about as efficiently as CTL