A Dialectica-Like Approach to Tree Automata

We propose a fibred monoidal closed category of alternating tree automata, based on a Dialectica-like approach to the fibred game model of [17]. Alternating tree automata are equivalent in expressive power to the Monadic Second-Order Logic on infinite trees (MSO), which subsumes most of the logics used in verification (see e.g. [14, 5]). Acceptance of an input tree t by an automaton A can be described by a two-player game, where the Proponent P (also called Automaton or ∃löıse) tries to force the execution of the automaton on a successful path, while its Opponent O (∀belard) tries to find a failing path. Then A accepts t iff P has a winning strategy in this game. Alternating automata are easily closed under complement, and together with the translation of alternating automata to non-deterministic ones (the Simulation Theorem [14]), this provides a convenient decomposition of the translation MSO formulas to automata (see e.g. [19, 5]), implying the decidability of MSO [16]. This work shows that to some extent this decomposition can be reflected in the decomposition of intuitionistic logic in linear logic [4]. The fibred symmetric monoidal closed structure allows to organize tree automata in a deduction system for a first-order multiplicative linear logic. Our model, which is based on game semantics, provides following [17] a realizability interpretation of this system: From a derivation of say A( B, we can compute a strategy σ witnessing the inclusion L(A) ⊆ L(B) (where L(A) is the set of trees accepted by A), in the sense that for any input tree t and any strategy τ witnessing that t ∈ L(A), the strategy t∗(σ) ◦ τ witnesses that t ∈ L(B) (here t∗ is the substitution functor induced by t). We use Gödel’s Dialectica interpretation (see e.g. [1, 13]) in two related ways. First, transitions of alternating tree automata can be seen (following e.g. [14]) as being valued in a free distributive lattice, hence as being given by expressions in a ∨∧-form. Then, Dialectica, seen as a constructive notion of prenex ∃∀formulas, provides the transition function of the internal linear implication of our notion of tree automata. Second, our notion of morphism (issued from [17]) is based on zig-zag strategies, which can be represented by a Dialectica-like category (see e.g. [3, 7, 6]) based on the topos of trees (see e.g. [2]). This allows to conveniently describe the dependencies of strategies on tree directions, and to get a very simple fibred structure thanks to a variation of simple fibrations based on comonoid indexing (see e.g. [8, 9]). When restricting to parity automata, the winning conditions of games of the form A1⊗· · ·⊗An ( B are given by disjunctions of parity conditions (called Rabin conditions), and it is known that if P wins such a game, then he has a positional winning strategy (see e.g. [12, 11, 10, 20]). In this context, we show that a powerset operation translating an alternating automaton to an equivalent non-deterministic one satisfies the deduction rules of the ‘!’ modality of linear logic. Unfortunately, positional strategies do not compose, but we still get a deduction system for intuitionistic linear logic, which in particular gives deduction for minimal intuitionistic predicate logic via the Girard translation. Using a suitable negative translation based on the ‘?’ modality, we can interpret proofs of minimal classical logic, and also get a weak form of completeness of our realizers w.r.t. language inclusion, in the sense that if L(A) ⊆ L(B), then from a regular O-strategy witnessing L(!A) ∩ L(!(B⊥)) = ∅ (provided by an algorithm solving regular games on finite graphs, see e.g. [15]), we can build a winning P-strategy on !A( (!(B⊥))⊥. Details can be found in [18].