Games and Weak-Head Reduction for Classical PCF

We present a game model for classical PCF, a nite version of PCF extended by a catch/throw mechanism. This model is build from E-dialogues, a kind of two-players game deened by Lorenzen. In the E-dialogues for classical PCF, the strategies of the rst player are isomorphic to the BB ohm trees of the language. We deene an interaction in E-dialogues and show that it models the weak-head reduction in classical PCF. The interaction is a variant of Coquand's debate and the weak-head reduction is a variant of the reduction in Krivine's Abstract Machine. We then extend E-dialogues to a kind of games similar to Hyland-Ong's games. Interaction in these games also models weak-head reduction. In the intuitionistic case (i.e. without the catch/throw mechanism), the extended E-dialogues are Hyland-Ong's games where the innocence condition on strategies is now a rule. Our model for classical PCF is diierent from Ong's model of Parigot's lambda-mu-calculus. His model works by adding new moves to the intu-itionistic case while ours works by relaxing the game rules.