An Automated Theorem Prover for Intuitionistic Predicate Logic Based on Dialogue Games

1 Dialogue Logic In Dialogue Logic the validity of some given formula F is examined in two person, perfect information games. There are two players moving alternatively, both having complete information of the current situation of the game. The player who claims being able to justify F is called the proponent, his adversary the opponent. The initial state, the dialogue setting, is determined by a (possibly empty) set of hypotheses (brought into the game by the opponent) and a thesis F, the formula to be shown valid (brought into the game by the proponent). The consecutive moves of the dialogue game are attacks upon formulae set earlier or defences against previous attacks. Some moves include the setting of formulae that might be subject to subsequent attacks. The legal moves that a player can perform are deened by so-called particle rules and frame rules. For each logical connective a particle rule is given which speciies how attacks upon moves setting formulae that have as main connective and defences against such attacks have to be performed. The frame rules order the exchange of arguments, i.e. they impose restrictions on when attacks and defences may take place in the dialogue. The game has two possible outcomes: win and loss. A dialogue game is won by a player, if the other player cannot perform any action that is conform to the dialogue rules. The proponent is said to have a winning strategy for a formula F, if he is able to win any game with formula F as thesis (and a given set of hypotheses) by appropriate choices of his moves. A formula F is valid, if the proponent has a winning strategy for F. The winning strategy is not unique. In 2, 3], Felscher gives a proof that the notion of winning strategies in Dialogue Games (wrt. a well-deened set of dialogue rules) coincides with the notion of provability in Gentzen's calculus LJ for intuitionistic logic. We now give a formal description of Dialogue Logic. First, consider a rst order language L being deened in a standard way. We add the following to L: A set of special terms: ?, ?:left, ?:right, and ?:t, where t is a term. Two special symbols: P and O (symbolizing proponent and opponent). Two special variables: X and Y (as variables for P and O).