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Abstract. We introduce a generic extension of the popular branchingtime logic CTL which refines the temporal until and release operators with formal languages. For instance, a language may determine the moments along a path that an until property may be fulfilled. We consider several classes of languages leading to logics with different expressive power and complexity, whose importance is motivated by their use in model checking, synthesis, abstract interpretation, etc. We show that even with context-free languages on the until operator the logic still allows for polynomial time model-checking despite the significant increase in expressive power. This makes the logic a promising candidate for applications in verification. In addition, we analyse the complexity of satisfiability and compare the expressive power of these logics to CTL * and extensions of PDL.

Abstract. 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 ω-words. We analyse which kinds of deterministic ω-automata are needed in which case in order to recognise these languages. We then obtain a reduction to the problem of solving parity or Büchi 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 Büchi 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 + . ACM CCS: [Theory of computiation]:Logic-Modal and temporal logics; Computational complexity and cryptography-Complexity theory and logic .

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