Abstract
We present a quantitative analysis of random combinatory logic terms. Our main goal is to investigate likelihood of semantic properties of random combinators. We show that asymptotically almost all weakly normalizing terms are not strongly normalizing. Moreover, we present a proof that asymptotically almost all strongly normalizing terms are not in normal form. We also prove that asymptotically almost all normal forms in combinatory logic are not typeable.
This work was supported within the grant 2013/11/B/ST6/0095 funded by the Polish National Science Center.
K. Grygiel—This author was supported by funding from the Jagiellonian University within the SET project. The project is co-financed by the European Union.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Barendregt, H.P., Bergstra, J., Klop, J.W., Volken, H.: Some notes on lambda reduction, in: Degrees, reductions and representability in the lambda calculus. Preprint no. 22, University of Utrecht, Department of mathematics, pp. 13–53 (1976)
Chauvin, B., Flajolet, P., Gardy, D., Gittenberger, B.: And/Or trees revisited. Comb. Probab. Comput. 13(4–5), 475–497 (2004)
Curry, H., Feys, R.: Combinatory Logic, vol. I. North Holland, Amsterdam (1958)
David, R., Grygiel, K., Kozik, J., Raffalli, C., Theyssier, G., Zaionc, M.: Asymptotically almost all \(\lambda \)-terms are strongly normalizing. Logical Methods Comput. Sci. 9, 1–30 (2013)
Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)
Fournier, H., Gardy, D., Genitrini, A., Zaionc, M.: Classical and intuitionistic logic are asymptotically identical. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 177–193. Springer, Heidelberg (2007)
Gardy, D. Random boolean expressions. In: Discrete Mathematics and Theoretical Computer Science Proceedings AF, pp.1–36 (2005)
Gardy, D., Woods, A.: And/Or tree probabilities of boolean functions. Discrete Math. Theor. Comput. Sci. 6, 139–146 (2005)
Genitrini, A., Kozik, J.: In the full propositional logic, 5/8 of classical tautologies are intuitionistically valid. Ann. Pure Appl. Logic 163(7), 875–887 (2012)
Grygiel, K., and Lescanne, P. Counting terms in the binary lambda calculus. In: DMTCS 25th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (2014)
Kostrzycka, Z., Zaionc, M.: Statistics of intuitionistic versus classical logic. Stud. Logica 76(3), 307–328 (2004)
Kozik, J.: Subcritical pattern languages for And/Or trees. In: DMTCS Proceedings from Fifth Colloquium on Mathematics and Computer Science Algorithms Trees, Combinatorics and Probabilities, pp. 437–448 (2008)
Lefmann, H., Savický, P.: Some typical properties of large And/Or Boolean formulas. Random Struct. Algorithms 10, 337–351 (1997)
Moczurad, M., Tyszkiewicz, J., Zaionc, M.: Statistical properties of simple types. Math. Struct. Comput. Sci. 10(5), 575–594 (2000)
Olivier, B., Danièle, G., Bernhard, G., Alice, J.: Enumeration of generalized BCI lambda-terms. Electr. J. Comb. 20, 4 (2013)
Szegö, G.: Orthogonal polynomials. Am. Math. Soc. Colloquium Ser. Publ. 23, 413–421 (1967)
Tromp, J. Binary lambda calculus and combinatory logic. Unpublished manuscript (2014). http://tromp.github.io/cl/LC.pdf.
Wilf, H.: Generating Functionology. Academic Press, Boston (1994)
Woods, A.: On the probability of absolute truth for And/Or formulas. Bull. Symbolic Logic 12, 3 (2006)
Zaionc, M.: On the asymptotic density of tautologies in logic of implication and negation. Rep. Math. Logic 39, 67–87 (2005)
Zaionc, M.: Probability distribution for simple tautologies. Theor. Comput. Sci. 355(2), 243–260 (2006)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Bendkowski, M., Grygiel, K., Zaionc, M. (2015). Asymptotic Properties of Combinatory Logic. In: Jain, R., Jain, S., Stephan, F. (eds) Theory and Applications of Models of Computation. TAMC 2015. Lecture Notes in Computer Science(), vol 9076. Springer, Cham. https://doi.org/10.1007/978-3-319-17142-5_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-17142-5_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-17141-8
Online ISBN: 978-3-319-17142-5
eBook Packages: Computer ScienceComputer Science (R0)