Realizability, Covers, and Sheaves I. Application to the Simply-Typed Lambda-Calculus

We present a general method for proving properties of typed λ-terms. This method is obtained by introducing a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory, a cover algebra being a Grothendieck topology in the case of a preorder). For this, we introduce a new class of semantic structures equipped with preorders, called pre-applicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Kleene\u27s recursive realizability and a variant of Kreisel\u27s modified realizability both fit into this framework. Applying this theorem to the special case of the term model, yields a general theorem for proving properties of typed λ-terms, in particular, strong normalization and confluence. This approach clarifies the reducibility method by showing that the closure conditions on candidates of reducibility can be viewed as sheaf conditions. Part I of this paper applies the above approach to the simply-typed λ-calculus (with types →, ×, +, and ⊥). Part II of this paper deals with the second-order (polymorphic) λ-calculus (with types → and ∀)

Proving Properties of Typed Lambda-Terms Using Realizability, Covers, and Sheaves (Preliminary Version)

We present a general method for proving properties of typed λ-terms. This method is obtained by introducing a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory). For this, we introduce a new class of semantic structures equipped with preorders, called pre-applicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Kleene\u27s recursive realizability and a variant of Kreisel\u27s modified realizability both fit into this framework. Applying this theorem to the special case of the term model, yields a general theorem for proving properties of typed λ-terms, in particular, strong normalization and confluence. This approach clarifies the reducibility method by showing that the closure conditions on candidates of reducibility can be viewed as sheaf conditions. The above approach is applied to the simply-typed λ-calculus (with types →, x, +, and ⊥), and to the second-order (polymorphic λ-calculus (with types → and ∀2), for which it yields a new theorem

The intended interpretation of the intuitionistic first-order logical operators.

The present thesis is an investigation on an open problem in mathematical logic: the problem of devising an explanation of the meaning of the intuitionistic first-order logical operators, which is both mathematically rigorous and faithful to the interpretation intended by the intuitionistic mathematicians who invented and have been using them. This problem has been outstanding since the early thirties, when it was formulated and addressed for the first time. The thesis includes a historical, expository part, which focuses on the contributions of Kolmogorov, Heyting, Gentzen and Kreisel, and a long and detailed discussion of the various interpretations which have been proposed by these and other authors. Special attention is paid to the decidability of the proof relation and the introduction of Kreisel's extra-clauses, to the various notions of 'canonical proof' and to the attempt to reformulate the semantic definition in terms of proofs from premises. In this thesis I include a conclusive argument to the effect that if one wants to withdraw the extra-clauses then one cannot maintain the concept of 'proof as the basic concept of the definition; instead, I describe an alternative interpretation based on the concept of a construction 'performing' the operations indicated by a given sentence, and I show that it is not equivalent to the verificationist interpretation. I point out a redundancy in the internal-pseudo-inductive-structure of Kreisel's interpretation and I propose a way to resolve it. Finally, I develop the interpretation in terms of proofs from premises and show that a precise formulation of it must also make use of non-inductive clauses, not for the definition of the conditional but -surprisingly enough- for the definitions of disjunction and of the existential quantifier

A Type Theory which is complete for Kreisel's Modified Realizability

We define a type theory with a strong elimination rule for existential quantification. As in Martin-Löf's type theory, the “axiom of choice” is thus derivable. Proofs are also annotated by realizers which are simply typed lambda-terms. A new rule called “type extraction” which extracts the type of a realizer allows us to derive the so-called “independance of premisses” schema. Consequently, any formula which is realizable in HA^omega according to Kreisel's modified realizability, is derivable in this type theory