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Published Articles >> Table of Contents >> Abstract
19th Annual IEEE Symposium on Logic in Computer Science (LICS'04)
pp. 276-285
The Sensible Graph Theories of Lambda Calculus
Antonio Bucciarelli, Université Paris 7, France
Antonino Salibra, Università Ca'Foscari di Venezia, Italia
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DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/LICS.2004.1319622
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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 non-syntactic 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).
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Citation:
Antonio Bucciarelli, Antonino Salibra,
"The Sensible Graph Theories of Lambda Calculus,"
lics,
pp. 276-285,
19th Annual IEEE Symposium on Logic in Computer Science (LICS'04),
2004
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