Easy lambda-terms are not always simple

Carraro, Alberto; Salibra, Antonino

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 46 (2012) no. 2, pp. 291-314.

A closed λ-term M is easy if, for any other closed term N, the lambda theory generated by M = N is consistent. Recently, it has been introduced a general technique to prove the easiness of λ-terms through the semantical notion of simple easiness. Simple easiness implies easiness and allows to prove consistency results via construction of suitable filter models of λ-calculus living in the category of complete partial orderings: given a simple easy term M and an arbitrary closed term N, it is possible to build (in a canonical way) a non-trivial filter model which equates the interpretation of M and N. The question whether easiness implies simple easiness constitutes Problem 19 in the TLCA list of open problems. In this paper we negatively answer the question providing a non-empty co-r.e. (complement of a recursively enumerable) set of easy, but not simple easy, λ-terms.

[1] F. Alessi and M. Dezani-Ciancaglini, TLCA list of open problems : Problem 19, http://tlca.di.unito.it/opltlca/problem19.pdf (2002).

[2] F. Alessi and S. Lusin, Simple easy terms, ITRS '02, Intersection Types and Related Systems (FLoC Satellite Event). Electron. Notes Theoret. Comput. Sci. 70 (2003) 1-18. | Zbl

[3] F. Alessi, M. Dezani-Ciancaglini and F. Honsell, Filter models and easy terms, in ICTCS '01 : Proc. of the 7th Italian Conference on Theoretical Computer Science. Springer-Verlag, London, UK (2001) 17-37. | MR | Zbl

[4] F. Alessi, M. Dezani-Ciancaglini and S. Lusin, Intersection types and domain operators. Theoret. Comput. Sci. 316 (2004) 25-47. | MR | Zbl

[5] J.C.M. Baeten and B. Boerboom, Omega can be anything it should not be. Proc. of Indagationes Mathematicae 82 (1979) 111-120. | MR | Zbl

[6] H.P. Barendregt, Some extensional term models for combinatory logics and λ-calculi. Ph.D. thesis, University of Utrecht (1971).

[7] H.P. Barendregt, The Lambda calculus : Its syntax and semantics. North-Holland, Amsterdam (1984). | MR | Zbl

[8] H.P. Barendregt, M. Coppo and M. Dezani-Ciancaglini, A filter lambda model and the completeness of type assignment. J. Symbolic Logic 48 (1983) 931-940. | MR | Zbl

[9] O. Bastonero, A. Pravato and S. Ronchi Della Rocca, Structures for lazy semantics, in Programming Concepts and Methods (PROCOMET'98), edited by D. Gries and W.P. de Roever. Chaptman & Hall (1998) 30-48.

[10] A. Berarducci, Infinite lambda-calculus and non-sensible models, in Logic and Algebra (Pontignano, 1994). Lect. Notes Pure Appl. Math. Ser. 180 (1996) 339-378. | MR | Zbl

[11] A. Berarducci and B. Intrigila, Some new results on easy lambda-terms. Theoret. Comput. Sci. 121 (1993) 71-88. | MR | Zbl

[12] C. Berline, From computation to foundations via functions and application : the λ-calculus and its webbed models. Theoret. Comput. Sci. 249 (2000) 81-161. | MR | Zbl

[13] C. Berline, Graph models of λ-calculus at work, and variations. Math. Struct. Comput. Sci. 16 (2006) 185-221. | MR | Zbl

[14] C. Berline and A. Salibra, Easiness in graph models. Theoret. Comput. Sci. 354 (2006) 4-23. | MR | Zbl

[15] C. Berline, G. Manzonetto and A. Salibra, Effective λ-models versus recursively enumerable λ-theories. Math. Struct. Comput. Sci. 19 (2009) 897-942. | MR | Zbl

[16] G. Berry, Stable models of typed lambda-calculi, in Proc. of ICALP'78. Lect. Notes Comput. Sci. 62 (1978).

[17] C. Böhm, Alcune proprietá delle forme β-η-normali nel λ-K-calcolo. Technical Report 696, CNR (1968).

[18] A. Bucciarelli and T. Ehrhard, Sequentiality and strong stability, in Proc. of LICS'91. IEEE Computer Society Press (1991) 138-145.

[19] A. Carraro and A. Salibra, Reflexive Scott domains are not complete for the extensional lambda calculus, in Proc. of the Twenty-Fourth Annual IEEE Symposium on Logic in Computer Science (LICS 2009). IEEE Computer Society Press (2009) 91-100. | MR

[20] M. Coppo and M. Dezani-Ciancaglini, An extension of the basic functionality theory for the λ-calculus. Notre-Dame J. Form. Log. 21 (1980) 685-693. | MR | Zbl

[21] M. Coppo, M. Dezani-Ciancaglini, F. Honsell and G. Longo, Extended type structures and filter lambda models, in Logic Colloquium 82, edited by G. Lolli, G. Longo and A. Marcja. Amsterdam, The Netherlands, North-Holland (1984) 241-262. | MR | Zbl

[22] M. Coppo, M. Dezani-Ciancaglini, F. Honsell and G. Longo, Extended type structures and filter lambda models, in Logic Colloquium 82, edited by G. Lolli, G. Longo and A. Marcja. Amsterdam, The Netherlands, North-Holland (1984) 241-262. | MR | Zbl

[23] E. Engeler, Algebras and combinators. Algebra Univers. 13 (1981) 389-392. | MR | Zbl

[24] F. Honsell and S. Ronchi Della Rocca, Reasoning about interpretations in qualitative lambda models, in Proc. of the IFIP Working Group 2.2/2.3, edited by M. Broy and C.B. Jones. North-Holland (1990) 505-521.

[25] F. Honsell and S. Ronchi Della Rocca, An approximation theorem for topological lambda models and the topological incompleteness of lambda calculus. J. Comput. System Sci. 45 (1992) 49-75. | MR | Zbl

[26] B. Intrigila, A problem on easy terms in lambda-calculus. Fundamenta Informaticae 15 (1991) 99-106. | MR | Zbl

[27] G. Jacopini, A condition for identifying two elements of whatever model of combinatory logic, in Proceedings of the Symposium on Lambda-Calculus and Computer Science Theory. Springer-Verlag, London, UK (1975) 213-219. | MR | Zbl

[28] G. Jacopini and M. Venturini-Zilli, Equating for recurrent terms of λ-calculus and combinatory logic. Technical Report 85, CNR (1978). | Zbl

[29] G. Jacopini and M. Venturini-Zilli, Easy terms in the lambda-calculus. Fundam. Inform. 8 (1985) 225-233. | MR | Zbl

[30] R. Kerth, Isomorphism et équivalence équationnelle entre modèles du λ-calcul. Master's thesis, Université de Paris 7 (1995).

[31] R. Kerth, Isomorphism and equational equivalence of continuous lambda models. Stud. Log. 61 (1998) 403-415. | MR | Zbl

[32] R. Kerth, On the construction of stable models of λ-calculus. Theoret. Comput. Sci. 269 (2001) 23-46. | MR | Zbl

[33] C.P.J. Koymans, Models of the lambda calculus. Inform. Control 52 (1982) 306-332. | MR | Zbl

[34] J. Kuper, On the Jacopini technique. Inform. Comput. 138 (1997) 101-123. | MR | Zbl

[35] G. Longo, Set-theoretical models of λ-calculus : theories, expansions, isomorphisms. Ann. Pure Appl. Logic 24 (1983) 153-188. | MR | Zbl

[36] A.R. Meyer, What is a model of the lambda calculus? Inform. Control 52 (1982) 87-122. | MR | Zbl

[37] G.D. Plotkin, Set-theoretical and other elementary models of the λ-calculus. Theoret. Comput. Sci. 121 (1993) 351-409. | MR | Zbl

[38] A. Salibra, Topological incompleteness and order incompleteness of the lambda calculus. ACM Trans. Comput. Log. 4 (2003) 379-401. | MR

[39] D.S. Scott, Continuous lattices, in Toposes, algebraic geometry and logic. Springer-Verlag (1972). | MR | Zbl

[40] D.S. Scott, Data types as lattices, in ISILC Logic Conference, edited by G. Müller, A. Oberschelp and K. Potthoff. Lect. Notes Math. 499 (1975) 579-651. | MR | Zbl

[41] D.S. Scott, Lambda calculus : some models, some philosophy, in The Kleene Symposium. Amsterdam, The Netherlands, North-Holland (1980). | MR | Zbl

[42] D.S. Scott, Domains for denotational semantics, in Proc. of ICALP '82. ACM Press. New York, NY, USA (1982) 95-104. | MR | Zbl

[43] A. Visser, Numerations, λ-calculus, and arithmetic, in To H.B. Curry : Essays on Combinatory Logic, Lambda Calculus and Formalism, edited by J.R. Hindley and J.P. Seldin (1980) 259-284. | MR

[44] C. Zylberajch, Syntaxe et sémantique de la facilité en λ-calcul. Thèse de Doctorat D'État, Université Paris 7 (1991).