Monoidal-Closed Categories of Tree Automata

We propose a realizability semantics for automata on infinite trees, based on categories of games built on usual simple games, and generalizing usual acceptance games of tree automata. Our approach can be summarized with the slogan “automata as objects, strategies as morphisms”. We show that the operations on tree automata used in the translations of MSOformulae to automata (underlying Rabin’s Theorem, that is the decidability of MSO on infinite trees) can be organized in a deduction system based on the multiplicative fragment of intuitionistic linear logic (ILL). Namely, we equip a variant of usual alternating tree automata (that we call uniform tree automata) with a fibred monoidal closed structure which in particular, via game determinacy handles a linear complementation of alternating automata, as well as deduction rules for existential and universal quantifications. This monoidal structure is actually Cartesian on non-deterministic automata. Moreover, an adaptation of a usual construction for the simulation of alternating automata by non-deterministic ones satisfies the deduction rules of the !(−) ILL-exponential modality. Our realizability semantics satisfies an expected property of witness extraction from proofs of existential statements. Moreover, it allows to combine realizers produced as interpretations of proofs with strategies witnessing (non-)emptiness of tree automata, possibly obtained using external algorithms.