Games for modal and temporal logics

Every logic comes with several decision problems. One of them is the model checking problem: does a given structure satisfy a given formula? Another is the satisfiability problem: for a given formula, is there a structure fulfilling it? For modal and temporal logics; tableaux, automata and games are commonly accepted as helpful techniques that solve these problems. The fact that these logics possess the tree model property makes tableau structures suitable for these tasks. On the other hand, starting with Büchi’s work, intimate connections between these logics and automata have been found. A formula can describe an automaton’s behaviour, and automata are constructed to accept exactly the word or tree models of a formula. In recent years the use of games has become more popular. There, an existential and a universal player play on a formula (and a structure) to decide whether the formula is satisfiable, resp. satisfied. The logical problem at hand is then characterised by the question of whether or not the existential player has a winning strategy for the game. These three methodologies are closely related. For example the non-emptiness test for an alternating automaton is nothing more than a 2-player game, while winning strategies for games are very similar to tableaux. Game-theoretic characterisations of logical problems give rise to an interactive semantics for the underlying logics. This is particularly useful in the specification and verification of concurrent systems where games can be used to generate counterexamples to failing properties in a very natural way. We start by defining simple model checking games for Propositional Dynamic Logic, PDL, in Chapter 4. These allow model checking for PDL in linear running time. In fact, they can be obtained from existing model checking games for the alternating free μ-calculus. However, we include them here because of their usefulness in proving correctness of the satisfiability games for PDL later on. Their winning strategies are history-free. Chapter 5 contains model checking games for branching time logics. Beginning with the Full Branching Time Logic CTL we introduce the notion of a focus game. Its key idea is to equip players with a tool that highlights a particular formula in