Colored intersection types: a bridge between higher-order model-checking and linear logic

The model-checking problem for higher-order recursive programs, expressed as higher-order recursion schemes (HORS), and where properties are specified in monadic second-order logic (MSO) has received much attention since it was proven decidable by Ong ten years ago. Every HORS may be understood as a simply-typed λ-term G with fixpoint operators Y whose free variables a, b, c . . . ∈ Σ are of order at most one. Following the principles of a Church encoding, these variables provide the tree constructors of a ranked alphabet Σ, so that the normalization of the recursion scheme G produces a typically infinite value tree 〈G〉 over this ranked alphabet. In order to check whether a given MSO formula φ holds at the root of such a value tree 〈G〉, a convenient and traditional approach is to run an equivalent automaton Aφ over it. In the specific case of MSO logic, the corresponding notion of automaton is provided by alternating parity tree automata (APT), a kind of non-deterministic top-down tree automaton enriched with alternation and coloring. Every run of such an automaton may be understood as a syntactic proof-search of the validity of the formula φ over the value tree 〈G〉. A typical transition over a binary symbol a ∈ Σ is of the following shape: δ(q0, a) = (2, q2) ∨ ((1, q1) ∧ (1, q2) ∧ (2, q0)) When reading the symbol a in the state q0, the automaton Aφ can either (1) drop the left subtree of a, and explore the right subtree with state q2, or (2) explore twice the left subtree of a in parallel, once with state q1 and the other time with state q2, and explore the right subtree of a with state q0. Kobayashi observed that the transitions of an alternating tree automaton A can be reflected by giving to the symbol a the following refined intersection type: