Graph Lambda Theories † a N T O N I O B U C C I a R E L L I and a N T O N I N O S a L I B R A

A longstanding open problem in lambda calculus is whether there exist continuous models of the untyped lambda calculus whose theory is exactly the λβ or the least sensible λ-theory H (generated by equating all the unsolvable terms). A related question is whether, given a class of lambda models, there are a minimal λ-theory and a minimal sensible λ-theory represented by it. In this paper, we give a positive answer to this question for the class of graph modelsà la Plotkin-Scott-Engeler. In particular, we build two graph models whose theories are respectively the set of equations satisfied in any graph model and in any sensible graph model. We conjecture that the least sensible graph theory, where " graph theory " means " λ-theory of a graph model " , is equal to H, while in one of the main results of the paper we show the non-existence of a graph model whose equational theory is exactly the λβ theory. Another related question is whether, given a class of lambda models, there is a maximal sensible λ-theory represented by it. In the main result of the paper we characterize the greatest sensible graph theory as the λ-theory B generated by equating λ-terms with the same B ¨ ohm tree. This result is a consequence of the main technical theorem of the paper: all the equations between solvable λ-terms, which have different Böhm trees, fail in every sensible graph model. A further result of the paper is the existence of a continuum of different sensible graph theories strictly included in B.