Decidability in Intuitionistic Type Theory is Functionally Decidable

Decidability in Intuitionistic Type Theory is Functionally Decidable

In this paper we show that the usual intuitionistic characterization of the decidability of the propositional function B(x) prop [x : A], i.e. to require that the predicate (∀x ∈ A)B(x)∨¬B(x) is provable, is equivalent, when working within the framework of Martin-Löf’s Intuitionistic Type Theory, to require that there exists a decision function φ : A → Boole such that (∀x ∈ A) (φ(x) =Boole true...

Intuitionistic Type Theory

Intuitionistic Type Theory (also Constructive Type Theory or Martin-Löf Type Theory) is a formal logical system and philosophical foundation for constructive mathematics (link). It is a fullscale system which aims to play a similar role for constructive mathematics as Zermelo-Fraenkel Set Theory (link) does for classical mathematics. It is based on the propositions-as-types principle and clarif...

A Decidable Intuitionistic Temporal Logic

We introduce the logic ITLe, an intuitionistic temporal logic based on structures (W,4, S), where 4 is used to interpret intuitionistic implication and S is a 4-monotone function used to interpret temporal modalities. Our main result is that the satisfiability and validity problems for ITLe are decidable. We prove this by showing that the logic enjoys the strong finite model property. In contra...

Constructing Recursion Operators in Intuitionistic Type Theory

Martin-Löf’s Intuitionistic Theory of Types is becoming popular for formal reasoning about computer programs. To handle recursion schemes other than primitive recursion, a theory of well-founded relations is presented. Using primitive recursion over higher types, induction and recursion are formally derived for a large class of well-founded relations. Included are < on natural numbers, and rela...

Decidable Kripke Models of Intuitionistic Theories