Compositional higher-order model checking via ω-regular games over Böhm trees

We introduce type-checking games, which are ω-regular games over Böhm trees, determined by a type of the Kobayashi-Ong intersection type system. These games are a higher-type extension of parity games over trees, determined by an alternating parity tree automaton. However, in contrast to these games over trees, the "game boards" of our type-checking games are composable, using the composition of Böhm trees. Moreover the winner (and winning strategies) of a composite game is completely determined by the respective winners (and winning strategies) of the component games. To our knowledge, type-checking games give the first compositional analysis of higher-order model checking, or the model checking of trees generated by recursion schemes. We study a higher-type analogue of higher-order model checking, namely, the problem to decide the winner of a type-checking game over the Böhm tree generated by an arbitrary λY-term. We introduce a new type-assignment system and use it to prove that the problem is decidable. On the semantic side, we develop a novel (two-level) arena game model for type-checking games, which is a cartesian closed category equipped with parametric monad and comonad that themselves form a parametrised adjunction.