Graph lambda theories

Author:

BUCCIARELLI ANTONIO,SALIBRA ANTONINO

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.

Publisher

Cambridge University Press (CUP)

Subject

Computer Science Applications,Mathematics (miscellaneous)

Reference31 articles.

Cited by 9 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. RELATIONAL GRAPH MODELS AT WORK;LOG METH COMPUT SCI;2018

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3. Minimal lambda-theories by ultraproducts;Electronic Proceedings in Theoretical Computer Science;2013-03-28

4. Scott Is Always Simple;Mathematical Foundations of Computer Science 2012;2012

5. Observational Equivalence and Full Abstraction in the Symmetric Interaction Combinators;Logical Methods in Computer Science;2009-12-22

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