Automata Theory Approach to Predicate Intuitionistic Logic

Abstract

Abstract. Predicate intuitionistic logic is a well established fragment of dependent types. According to the Curry-Howard isomorphism proof construction in the logic corresponds well to synthesis of a program the type of which is a given formula. We present a model of automata that can handle proof construction in full intuitionistic first-order logic. The automata are constructed in such a way that any successful run corresponds directly to a normal proof in the logic. This makes it possible to discuss formal languages of proofs or programs, the closure properties of the automata and their connections with the traditional logical connectives.

Small Model Property Reflects in Games and Automata

“…We reason only about automata that are translated from a formula as defined in Sect. 4 of [7]. In Sect.…”

“…We work in intuitionistic first-order logic with no function symbols or constants. The logic is the same as in previous works on games [6] and automata [7]. There is a set of predicates P and every predicate P ∈ P has a defined arity.…”

“…Here we show a limit on resources of Arcadian automata [7] checking derivability of a formula ϕ from an effectively axiomatized class X that has the finite model property. Theorem 2 shows a limit on the size of the set of eigenvariables of the automaton in terms of the numbers of variables and subformulas in ϕ and the size of the small model.…”

“…An instantaneous description is q, κ, V, w, w , S where q ∈ Q and κ ∈ A are the current state and node, V is a set of eigenvariables, w and w are interpretations of bindings and S is the store. For more details see [7].…”

“…Note that our logic has the subformula property ( [7]). This means that there is only a limited number of possible targets τ .…”