Hostname: page-component-77c89778f8-5wvtr Total loading time: 0 Render date: 2024-07-19T11:36:17.539Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramsky, S. (1991) Domain theory in logical form. Annals of Pure and Applied Logic 51 1–77.CrossRefGoogle Scholar
Alessi, F., Dezani, M. and Honsell, F. (2001) Filter models and easy terms. ICTCS'01. Springer-Verlag Lecture Notes in Computer Science 2202 17–37.CrossRefGoogle Scholar
Baeten, J. and Boerboom, B. (1979) Omega can be anything it should not be. Indag. Mathematicae 41 111–120.CrossRefGoogle Scholar
Barendregt, H. P. (1984) The lambda calculus: Its syntax and semantics, North-Holland.Google Scholar
Barendregt, H. P., Coppo, M. and Dezani-Ciancaglini, M. (1984) A filter lambda model and the completeness of type assignment. J. Symbolic Logic 48 931–940.CrossRefGoogle Scholar
Bastonero, O. and Gouy, X. (1999) Strong stability and the incompleteness of stable models of λ-calculus. Annals of Pure and Applied Logic 100 247–277.CrossRefGoogle Scholar
Berline, C. (2000) From computation to foundations via functions and application: The λ-calculus and its webbed models. Theoretical Computer Science 249 81–161.CrossRefGoogle Scholar
Berline, C. (2006) Graph models of λ-calculus at work, and variations. Mathematical Structures in Computer Science 16 185–221.CrossRefGoogle Scholar
Berline, C. and Salibra, A. (2006) Easiness in graph models. Theoretical Computer Science 354 4–23.CrossRefGoogle Scholar
Berline, C., Manzonetto, G. and Salibra, A. (2007) Lambda theories of effective lambda models. In: 16th EACSL Annual Conference on Computer Science and Logic (CSL'07). Springer-Verlag Lecture Notes in Computer Science 4646 298–312.Google Scholar
Berry, G. (1978) Stable models of typed lambda-calculi. In: Proc. 5th Int. Coll. on Automata, Languages and Programming. Springer-Verlag Lecture Notes in Computer Science 62.CrossRefGoogle Scholar
Bucciarelli, A. and Ehrhard, T. (1991) Sequentiality and strong stability. Sixth Annual IEEE Symposium on Logic in Computer Science, IEEE Computer Society Press 138–145.CrossRefGoogle Scholar
Bucciarelli, A. and Salibra, A. (2003) The minimal graph model of lambda calculus. 28th International Symposium on Mathematical Foundations of Computer Science. Springer-Verlag Lecture Notes in Computer Science 2747.CrossRefGoogle Scholar
Bucciarelli, A. and Salibra, A. (2004) The sensible graph theories of lambda calculus. 19th Annual IEEE Symposium on Logic in Computer Science (LICS 2004), IEEE Computer Society Press.Google Scholar
Church, A. (1933) A set of postulates for the foundation of logic. Annals of Math. 2 346–366.Google Scholar
Church, A. (1941) The calculi of lambda conversion, Princeton University Press.Google Scholar
Coppo, M. and Dezani, M. (1980) An extension of the basic functionality theory for the λ-calculus. Notre Dame J. Formal Logic 21 685–693.CrossRefGoogle Scholar
David, R. (2001) Computing with Böhm trees. Fundamenta Informaticae 45 53–77.Google Scholar
Di Gianantonio, P. and Honsell, F. (1993) An abstract notion of application. Typed lambda calculi and applications. Springer-Verlag Lecture Notes in Computer Science 664 124–138.CrossRefGoogle Scholar
Di Gianantonio, P., Honsell, F. and Plotkin, G. (1995) Uncountable limits and the lambda calculus. Nordic J. Comput. 2 126–145.Google Scholar
Gouy, X. (1995) Etude des théories équationnelles et des propriétés algébriques des modéles stables du λ-calcul, Thèse, Université de Paris 7.Google Scholar
Honsell, F. and Ronchi della Rocca, S. (1992) An approximation theorem for topological lambda models and the topological incompleteness of lambda calculus. Journal of Computer and System Sciences 45 49–75.CrossRefGoogle Scholar
Kerth, R. (1995) Isomorphisme et équivalence équationnelle entre modèles du λ-calcul, Thèse, Université de Paris 7.Google Scholar
Kerth, R. (1998) Isomorphism and equational equivalence of continuous lambda models. Studia Logica 61 403–415.CrossRefGoogle Scholar
Kerth, R. (2001) On the construction of stable models of λ-calculus. Theoretical Computer Science 269 23–46.CrossRefGoogle Scholar
Longo, G. (1983) Set-theoretical models of λ-calculus: theories, expansions and isomorphisms. Ann. Pure Applied Logic 24 153–188.CrossRefGoogle Scholar
Plotkin, G. (1993) Set-theoretical and other elementary models of the λ-calculus. Theoretical Computer Science 121 351–409.CrossRefGoogle Scholar
Salibra, A. (2001) A continuum of theories of lambda calculus without semantics. 16th Annual IEEE Symposium on Logic in Computer Science, IEEE Computer Society Press 334–343.Google Scholar
Salibra, A. (2003) Topological incompleteness and order incompleteness of the lambda calculus. (LICS'01 Special Issue). ACM Transactions on Computational Logic 4 379–401.CrossRefGoogle Scholar
Scott, D. S. (1972) Continuous lattices. In: Toposes, Algebraic geometry and Logic. Springer-Verlag Lecture Notes in Mathematics 274.CrossRefGoogle Scholar
Selinger, P. (2003) Order-incompleteness and finite lambda reduction models. Theoretical Computer Science 309 43–63.CrossRefGoogle Scholar