The sensible graph theories of lambda calculus

Sensible λ-theories are equational extensions of the untyped lambda calculus that equate all the unsolvable λ-terms and are closed under derivation. A longstanding open problem in lambda calculus is whether there exists a non-syntactic model whose equational theory is the least sensible λ-theory H (generated by equating all the unsolvable terms). A related question is whether, given a class of models, there exist a minimal and maximal sensible λ-theory represented by it. In This work we give a positive answer to this question for the semantics of lambda calculus given in terms of graph models. We conjecture that the least sensible graph theory, where "graph theory" means "λ-theory of a graph model", is equal to H, while in the main result of the paper we characterize the greatest sensible graph theory as the lambda;-theory B generated by equating λ-terms with the same Bohm tree. This result is a consequence of the fact that all the equations between solvable λ-terms, which have different Bohm trees, fail in every sensible graph model. Further results of the paper are: (i) the existence of a continuum of different sensible graph theories strictly included in B (this result positively answers question 2 in [7, Section 6.3]); (ii) the non-existence of a graph model whose equational theory is exactly the minimal lambda theory λβ (this result negatively answers Question 1 in [7, Section 6.2] for the restricted class of graph models).