## The sensible graph theories of lambda calculus (2004)

Venue: | IN: 19TH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE (LICS’04), IEEE COMPUTER |

Citations: | 12 - 8 self |

### BibTeX

@INPROCEEDINGS{Bucciarelli04thesensible,

author = {Antonio Bucciarelli and Antonino Salibra},

title = {The sensible graph theories of lambda calculus},

booktitle = {IN: 19TH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE (LICS’04), IEEE COMPUTER},

year = {2004},

publisher = {Society Publications}

}

### OpenURL

### Abstract

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 nonsyntactic 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 paper 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 λ-theory B generated by equating λ-terms with the same Böhm tree. This result is a consequence of the fact that all the equations between solvable λ-terms, which have different Böhm 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).