A universal innocent game model for the Böhm tree lambda theory

Abstract

We present a game model of the untyped λ-calculus, with equational theory equal to the Böhm tree λ-theory β, which is universal (i.e. every element of the model is definable by some term). This answers a question of Di Gianantonio, Franco and Honsell. We build on our earlier work, which uses the methods of innocent game semantics to develop a universal model inducing the maximal consistent sensible theory H*.To our knowledge these are the first syntax-independent universal models of the untyped λ-calculus.