The Sensible Graph Theories of Lambda Calculus

Antonio Bucciarelli (1) , Antonino Salibra (2)

1 PPS - Preuves, Programmes et Systèmes
2 Dipartimento di Informatica

Sensible lambda theories are equational extensions of the untyped lambda-calculus that equate all the unsolvable lambda-terms and are closed under derivation. The least sensible lambda theory is the lambda theory $\cH$ (generated by equating all the unsolvable terms), while the greatest sensible lambda theory is the lambda theory $\cH^*$ (generated by equating terms with the same Böhm tree up to possibly infinite $\eta$-equivalence). A longstanding open problem in lambda calculus is whether there exists a non-syntactic model of lambda calculus whose equational theory is the least sensible $\gl$-theory $\cH$. A related question is whether, given a class of models, there exist a minimal and maximal sensible lambda-theory represented by it. In this paper we give a positive answer to this question for the semantics of lambda calculus given in terms of graph models (graph semantics, for short). We conjecture that the minimal sensible graph theory (where ``graph theory" means ``lambda-theory of a graph model") is equal to $\cH$, while in the main result of the paper we characterize the maximal sensible graph theory as the lambda-theory $\cB$ (generated by equating $\gl$-terms with the same Böhm tree). In fact, we prove that all the equation $M = N$ between solvable lambda-terms $M$ and $N$, which have different Böhm trees, fail in every graph model. In a further result of the paper we prove the existence of a continuum of different graph models whose equational theories are sensible and strictly included in $\cB$

Domaines: Logique en informatique [cs.LO]

Antonio Bucciarelli, Antonino Salibra. The Sensible Graph Theories of Lambda Calculus. Logic in Computer Science, 2004, Turku, Finland. pp.276-285. ⟨hal-00149558⟩