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Graph lambda theories†

Published online by Cambridge University Press:  01 October 2008

ANTONIO BUCCIARELLI  and
ANTONINO SALIBRA
ANTONIO BUCCIARELLI
Affiliation:
Equipe PPS (case 7014), Université Paris 7, 2, place Jussieu, 72251 Paris Cedex 05, France Email: buccia@pps.jussieu.fr
ANTONINO SALIBRA
Affiliation:
Dipartimento di Informatica, Università Ca'Foscari di Venezia, Via Torino 155, 30172 Venezia, Italy Email: salibra@dsi.unive.it
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Abstract

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 (which is 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 models à la Plotkin, Scott and Engeler. In particular, we build two graph models whose theories are the set of equations satisfied in, respectively, any graph model and 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 characterise 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 main technical theorem of the paper, which says that all the equations between solvable λ-terms that 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.

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Type
Paper
Information
Mathematical Structures in Computer Science , Volume 18 , Issue 5 , October 2008 , pp. 975 - 1004
Copyright
Copyright © Cambridge University Press 2008

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