On the construction of stable models of untyped [formula omitted]-calculus

Abstract

Stable models of untyped λ-calculus have been introduced by Berry (Proceedings of ICALP, Lecture Notes in Computer Science, Vol. 62, Springer, Berlin, 1978, pp. 72–89) as a refinement of the continuous framework defined by Scott (unpublished manuscript, 1969, 53pp.). In this paper, we show that even in the stable case we have a fair amount of freedom during the construction of models. We introduce a uniform method to construct stable models that allows to obtain non standard models satisfying certain equational restraints. We apply this method in two particular examples: first, we construct a simple stable model that distinguishes the λ-terms Y and Θ . Second, we construct a family of 2 ℵ 0 stable models with pairwise distinct theories. We show that the latter models are sensible using some results of David (Proceedings of TLCA 1999, Lecture Notes in Computer Science, Vol. 1581, Springer, Berlin, 1999) and Kerth (J. Symbolic Logic 63 (1998) 1529–1548).