Abstract
Simply-typed lambda terms are often used in the internal language of compilers and proof assistants, for which generation of large, uniformly distributed random terms is instrumental for testing correctness and scalability. Recently, Boltzmann samplers have enabled uniform random generation of large terms belonging to several families of combinatorial objects that have a regular structure, amenable to methods from analytic combinatorics. Unfortunately, no closed formula or generating function facilitating such methods is known for closed simply-typed lambda terms. Moreover, given their asymptotic sparsity in the family of closed lambda terms, filtering simply-typed terms in the much larger set of terms generated by a Boltzmann sampler becomes quickly intractable. By taking advantage of the synergy between logic variables, unification with occurs check and efficient backtracking in today’s Prolog systems we advance this technique to term sizes interesting not only for correctness but also for scalability tests, by deriving Boltzmann samplers returning in a few seconds simply-typed random lambda terms of size 120 and above. We also apply our techniques to the generation of uniformly random closed simply-typed normal forms and give some hints on pushing them further via parallel execution algorithms.
The first two authors have been partially supported by the Polish National Science Center grant 2013/11/B/ST6/00975. The third author has been supported by NSF grant 1423324.
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Notes
- 1.
A lambda term is called closed if it has no free variables and open otherwise. A term is called plain if it is either closed or open.
References
Hindley, J.R., Seldin, J.P.: Lambda-Calculus and Combinators: An Introduction, vol. 13. Cambridge University Press, Cambridge (2008)
Barendregt, H.P.: Lambda calculi with types. In: Handbook of Logic in Computer Science, vol. 2. Oxford University Press (1991)
Grygiel, K., Lescanne, P.: Counting and generating lambda terms. J. Funct. Program. 23(5), 594–628 (2013)
Bodini, O., Gardy, D., Gittenberger, B.: Lambda terms of bounded unary height. In: 2011 Proceedings of the Eighth Workshop on Analytic Algorithmics and Combinatorics (ANALCO), pp. 23–32 (2011)
David, R., Grygiel, K., Kozik, J., Raffalli, C., Theyssier, G., Zaionc, M.: Asymptotically almost all \(\lambda \)-terms are strongly normalizing. Logical Meth. Comput. Sci. 9(1:02), 1–30 (2013)
Flajolet, P., Sedgewick, R.: Analytic Combinatorics, 1st edn. Cambridge University Press, New York (2009)
Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences (2014). https://oeis.org/
Tarau, P.: On logic programming representations of lambda terms: de Bruijn indices, compression, type inference, combinatorial generation, normalization. In: Pontelli, E., Son, T.C. (eds.) PADL 2015. LNCS, vol. 9131, pp. 115–131. Springer, Cham (2015). doi:10.1007/978-3-319-19686-2_9
Grygiel, K., Lescanne, P.: Counting and generating terms in the binary lambda calculus. J. Funct. Program. 25, e24 (2015)
Bendkowski, M., Grygiel, K., Lescanne, P., Zaionc, M.: A natural counting of lambda terms. In: Freivalds, R.M., Engels, G., Catania, B. (eds.) SOFSEM 2016. LNCS, vol. 9587, pp. 183–194. Springer, Heidelberg (2016). doi:10.1007/978-3-662-49192-8_15
de Bruijn, N.G.: Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem. Indagationes Math. 34, 381–392 (1972)
Duchon, P., Flajolet, P., Louchard, G., Schaeffer, G.: Boltzmann samplers for the random generation of combinatorial structures. Comb. Probab. Comput. 13(4–5), 577–625 (2004)
Claessen, K., Hughes, J.: QuickCheck: a lightweight tool for random testing of Haskell programs. In: Proceedings of the Fifth ACM SIGPLAN International Conference on Functional Programming, ICFP 2000, pp. 268–279. ACM, New York (2000)
Koopman, P., Alimarine, A., Tretmans, J., Plasmeijer, R.: Gast: generic automated software testing. In: Peña, R., Arts, T. (eds.) IFL 2002. LNCS, vol. 2670, pp. 84–100. Springer, Berlin (2003). doi:10.1007/3-540-44854-3_6
Palka, M.H., Claessen, K., Russo, A., Hughes, J.: Testing an optimising compiler by generating random lambda terms. In: Proceedings of the 6th International Workshop on Automation of Software Test, AST 2011, pp. 91–97. ACM, New York (2011)
Koopman, P., Plasmeijer, R.: Systematic synthesis of functions, pp. 68–83. The University of Nottingham (2006)
Tarau, P.: On Type-directed Generation of Lambda Terms. In: De Vos, M., Eiter, T., Lierler, Y., Toni, F. (eds.) 31st International Conference on Logic Programming (ICLP 2015), Technical Communications, Cork, Ireland, CEUR (2015). http://ceur-ws.org/Vol-1433/
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Bendkowski, M., Grygiel, K., Tarau, P. (2017). Boltzmann Samplers for Closed Simply-Typed Lambda Terms. In: Lierler, Y., Taha, W. (eds) Practical Aspects of Declarative Languages. PADL 2017. Lecture Notes in Computer Science(), vol 10137. Springer, Cham. https://doi.org/10.1007/978-3-319-51676-9_8
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