Innocent game models of untyped [formula omitted]-calculus

Abstract

We present a new denotational model for the untyped λ-calculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong (Inform. and Comput., to appear) and Nickau (Hereditarily Sequential Functionals: A Game-Theoretic Approach to Sequentiality, Shaker-Verlag, 1996. Dissertation, Universität Gesamthochschule Siegen, Shaker-Verlag, 1996), but the traditional distinction between “question” and “answer” moves is removed. We first construct models D and D REC as global sections of a reflexive object in the categories A and A REC of arenas and innocent and recursive innocent strategies, respectively. We show that these are sensible λη-algebras but are neither extensional nor universal. We then introduce a new representation of innocent strategies in an economical form. We show a strong connexion between the economical form of the denotation of a term in the game models and a variable-free form of the Nakajima tree of the term. Using this we show that the definable elements of D REC are precisely what we call effectively almost-everywhere copycat ( EAC) strategies. The category A EAC with these strategies as morphisms gives rise to a λη-model D EAC which we show has the same expressive power as D ∞ , i.e. the equational theory of D EAC is the maximal consistent sensible theory H ∗ . We show that the model D EAC is sensible, order-extensional and universal (i.e. every strategy is the denotation of some λ-term). To our knowledge this is the first syntax-free model of the untyped λ-calculus with the universality property.