Innocent game models of untyped -calculus

We present a new denotational model for the untyped -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, Universit6 at Gesamthochschule Siegen, Shaker-Verlag, 1996), but the traditional distinction between “question” and “answer” moves is removed. We :rst construct models D and DREC as global sections of a re=exive object in the categories A and AREC of arenas and innocent and recursive innocent strategies, respectively. We show that these are sensible -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 de:nable elements of DREC are precisely what we call e$ectively almost-everywhere copycat (EAC) strategies. The category AEAC with these strategies as morphisms gives rise to a -model DEAC which we show has the same expressive power as D∞, i.e. the equational theory of DEAC is the maximal consistent sensible theory H∗. We show that the model DEAC is sensible, order-extensional and universal (i.e. every strategy is the denotation of some -term). To our knowledge this is the :rst syntax-free model of the untyped -calculus with the universality property. c © 2002 Elsevier Science B.V. All rights reserved.