Intuitionistic model constructions and normalization proofs

Abstract

The traditional notions of strong and weak normalization refer to properties of a binary reduction relation. In this paper we explore an alternative approach to normalization, where we bypass the reduction relation and instead focus on the normalization function, that is, the function which maps a term into its normal form. We work in an intuitionistic metalanguage, and characterize a normalization function as an algorithm which picks a canonical representative from the equivalence class of convertible terms. Hence we also get a decision algorithm for convertibility. Such a normalization function can be constructed by building an appropriate model and a function \quote" which inverts the interpretation function. The normalization function is then obtained by composing the quote function with the interpretation function. We also discuss how to get a simple proof of the property that constructors are one-to-one, which usually is obtained as a corollary of Church-Rosser and normalization in the traditional sense. We illustrate this approach by showing how a glueing model (closely related to the glueing construction used in category theory) gives rise to a normalization algorithm for a combinatory formulation of G odel System T. We then show how the method extends in a straightforward way when we add cartesian products and disjoint unions (full intuitionistic propositional logic under a Curry-Howard interpretation) and trans nite inductive types such as the Brouwer ordinals.

Type-and-scope safe programs and their proofs

“…It is possible to alter the model definition described earlier so that it avoids unnecessary η-expansions. We proceed by enriching the traditional model with extra syntactical artefacts in a manner reminiscent of Coquand and Dybjer's (1997) approach to defining an NBE procedure for the SK combinator calculus. Their resorting to glueing terms to elements of the model was dictated by the sheer impossibily to write a sensible reification procedure but, in hindsight, it provides us with a powerful technique to build models internalizing alternative equational theories.…”

Towards a formal semantics for a structurally dynamic noncausal modelling language

“…Two terms that are equationally related are said to be convertible. The equational theory dictates the behaviour of our normalisation function; that is, a term should be convertible to its normal form, and convertible terms should have equal normal forms [10].…”

Memoization in Type-Directed Partial Evaluation

“…Normalization by evaluation has been variously studied in logic, proof theory, and category theory [2,3,8,9,10,13] and in partial evaluation [14,16]. Typedirected partial evaluation, which we present next, has been investigated both practically [5,15,17,18,29,31,38] and foundationally [24,25,47].…”