Infinitary Lambda Calculi and Böhm Models

Infinitely long rewrite sequences of possibly infinite terms are of interest for several reasons. Firstly, infinitary rewriting is a natural generalisation of finitary rewriting which extends it with the notion of computing towards a possibily infinite limit. Such limits naturally arise in the semantics of lazy functional languages, in which it is possible to write and compute with expressions which intuitively denote infinite data structures, such as a list of all the integers. If the limit of a reduction sequence still contains redexes, then it is natural to consider sequences whose length is longer than w in fact, sequences of any ordinal length. The question of the computational meaning of such sequences will be dealt with below. Secondly, computations with terms implemented as graphs allow the possibility of using cyclic graphs, which correspond in a natural way to infinite terms. Finite computations on cyclic graphs correspond to infinite computations on terms. Finally, the infinitary theory suggests new ways of dealing with some of the concepts that arise in the finitary theory, such as notions of undefinedness of terms. In this connection, Berarducci and Intrigila ([Bet, BI94]) have independently developed an infinitary lambda calculus and applied it to the study of consistency problems in the finitary lambda calculus. In [KKSdV-] we developed the basic theory of transfinite reduction for orthogonal term rewrite systems. In this paper we perform the same task for the