Abstract
Combinatory logic is based on modus ponens and a schematic (polymorphic) interpretation of axioms. In this paper we propose to consider expressive combinatory logics under the restriction that axioms are not interpreted schematically but ,,literally”, corresponding to a monomorphic interpretation of types. We thereby arrive at finite combinatory logic, which is strictly finitely axiomatisable and based solely on modus ponens. We show that the provability (inhabitation) problem for finite combinatory logic with intersection types is Exptime-complete with or without subtyping. This result contrasts with the general case, where inhabitation is known to be Expspace-complete in rank 2 and undecidable for rank 3 and up. As a by-product of the considerations in the presence of subtyping, we show that standard intersection type subtyping is in Ptime. From an application standpoint, we can consider intersection types as an expressive specification formalism for which our results show that functional composition synthesis can be automated.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Barendregt, H., Coppo, M., Dezani-Ciancaglini, M.: A filter lambda model and the completeness of type assignment. Journal of Symbolic Logic 48(4), 931–940 (1983)
Ben-Yelles, C.: Type Assignment in the Lambda-Calculus: Syntax and Semantics. PhD thesis, Department of Pure Mathematics, University College of Swansea (September 1979)
Coppo, M., Dezani-Ciancaglini, M.: An extension of basic functionality theory for lambda-calculus. Notre Dame Journal of Formal Logic 21, 685–693 (1980)
Dezani-Ciancaglini, M., Hindley, J.R.: Intersection types for combinatory logic. Theoretical Computer Science 100(2), 303–324 (1992)
Comon, H., et al.: Tree Automata Techniques and Applications (2008), http://tata.gforge.inria.fr
Hindley, J.R.: The simple semantics for Coppo-Dezani-Sallé types. In: Dezani-Ciancaglini, M., Montanari, U. (eds.) International Symposium on Programming. LNCS, vol. 137, pp. 212–226. Springer, Heidelberg (1982)
Kobayashi, N., Luke Ong, C.-H.: A type system equivalent to the modal mu-calculus model checking of higher-order recursion schemes. In: LICS, pp. 179–188. IEEE Computer Society, Los Alamitos (2009)
Kozik, M.: A finite set of functions with an Exptime-complete composition problem. Theoretical Computer Science 407, 330–341 (2008)
Kurata, T., Takahashi, M.: Decidable properties of intersection type systems. In: Dezani-Ciancaglini, M., Plotkin, G. (eds.) TLCA 1995. LNCS, vol. 902, pp. 297–311. Springer, Heidelberg (1995)
Kuśmierek, D.: The inhabitation problem for rank two intersection types. In: Ronchi Della Rocca, S. (ed.) TLCA 2007. LNCS, vol. 4583, pp. 240–254. Springer, Heidelberg (2007)
Leivant, D.: Polymorphic type inference. In: Proc. 10th ACM Symp. on Principles of Programming Languages, pp. 88–98. ACM, New York (1983)
Lustig, Y., Vardi, M.Y.: Synthesis from component libraries. In: de Alfaro, L. (ed.) FOSSACS 2009. LNCS, vol. 5504, pp. 395–409. Springer, Heidelberg (2009)
Naik, M., Palsberg, J.: A type system equivalent to a model checker. In: Sagiv, M. (ed.) ESOP 2005. LNCS, vol. 3444, pp. 374–388. Springer, Heidelberg (2005)
Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)
Pottinger, G.: A type assignment for the strongly normalizable lambda-terms. In: Hindley, J., Seldin, J. (eds.) To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pp. 561–577. Academic Press, London (1980)
Rehof, J., Urzyczyn, P.: Finite combinatory logic with intersection types. Technical Report 834, Dept. of Computer Science, Technical University of Dortmund (2011), http://ls14-www.cs.tu-dortmund.de/index.php/Datei:TR-834.pdf
Salvati, S.: Recognizability in the simply typed lambda-calculus. In: Ono, H., Kanazawa, M., de Queiroz, R. (eds.) WoLLIC 2009. LNCS, vol. 5514, pp. 48–60. Springer, Heidelberg (2009)
Urzyczyn, P.: The emptiness problem for intersection types. Journal of Symbolic Logic 64(3), 1195–1215 (1999)
Urzyczyn, P.: Inhabitation of low-rank intersection types. In: Curien, P.-L. (ed.) TLCA 2009. LNCS, vol. 5608, pp. 356–370. Springer, Heidelberg (2009)
Urzyczyn, P.: The logic of persistent intersection. Fundamenta Informaticae 103, 303–322 (2010)
Venneri, B.: Intersection types as logical formulae. Journal of Logic and Computation 4(2), 109–124 (1994)
Wells, J.B., Yakobowski, B.: Graph-based proof counting and enumeration with applications for program fragment synthesis. In: Etalle, S. (ed.) LOPSTR 2004. LNCS, vol. 3573, pp. 262–277. Springer, Heidelberg (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Rehof, J., Urzyczyn, P. (2011). Finite Combinatory Logic with Intersection Types. In: Ong, L. (eds) Typed Lambda Calculi and Applications. TLCA 2011. Lecture Notes in Computer Science, vol 6690. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21691-6_15
Download citation
DOI: https://doi.org/10.1007/978-3-642-21691-6_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21690-9
Online ISBN: 978-3-642-21691-6
eBook Packages: Computer ScienceComputer Science (R0)