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## The Sensible Graph Theories of Lambda Calculus

### Abstract

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

Topics: [INFO.INFO-LO] Computer Science [cs]/Logic in Computer Science [cs.LO]
Publisher: IEEE
Year: 2004
OAI identifier: oai:HAL:hal-00149558v1
Provided by: Hal-Diderot