Abstract
We define a new cost model for the call-by-value lambda-calculus satisfying the invariance thesis. That is, under the proposed cost model, Turing machines and the call-by-value lambda-calculus can simulate each other within a polynomial time overhead. The model only relies on combinatorial properties of usual beta-reduction, without any reference to a specific machine or evaluator. In particular, the cost of a single beta reduction is proportional to the difference between the size of the redex and the size of the reduct. In this way, the total cost of normalizing a lambda term will take into account the size of all intermediate results (as well as the number of steps to normal form).
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References
Asperti, A.: On the complexity of beta-reduction. In: Proc. 23rd ACM SIGPLAN Symposium on Principles of Programming Languages, pp. 110–118 (1996)
Asperti, A., Guerrini, S.: The Optimal Implementation of Functional Programming Languages. Cambridge Tracts in Theoretical Computer Science, vol. 45. Cambridge University Press, Cambridge (1998)
Dal Lago, U., Hofmann, M.: Quantitative models and implicit complexity. In: Ramanujam, R., Sen, S. (eds.) FSTTCS 2005. LNCS, vol. 3821, pp. 189–200. Springer, Heidelberg (2005)
Dal Lago, U., Martini, S.: An invariant cost model for the lambda calculus (2005) (extended Version) Available at: http://arxiv.org/cs.LO/0511045
de Bruijn, N.G.: Lambda calculus with nameless dummies, a tool for automatic formula manipulation, with application to the church-rosser theorem. Indagationes Mathematicae 34(5), 381–392 (1972)
Dezani-Ciancaglini, M., della Rocca, S.R., Saitta, L.: Complexity of lambda-terms reductions. RAIRO Informatique Theorique 13(3), 257–287 (1979)
Frandsen, G.S., Sturtivant, C.: What is an efficient implementation of the lambda-calculus? In: Proc. 5th ACM Conference on Functional Programming Languages and Computer Architecture, pp. 289–312 (1991)
Lamping, J.: An algorithm for optimal lambda calculus reduction. In: Proc. 17th ACM SIGPLAN Symposium on Principles of Programming Languages, pp. 16–30 (1990)
Lawall, J.L., Mairson, H.G.: Optimality and inefficiency: What isn’t a cost model of the lambda calculus? In: Proc. 1996 ACM SIGPLAN International Conference on Functional Programming, pp. 92–101 (1996)
Lawall, J.L., Mairson, H.G.: On global dynamics of optimal graph reduction. In: Proc. 1997 ACM SIGPLAN International Conference on Functional Programming, pp. 188–195 (1997)
Lévy, J.-J.: Réductions corrected et optimales dans le lambda-calcul. Université Paris 7, Thèses d’Etat (1978)
Della Rocca, S.R., Paolini, L.: The parametric lambda-calculus. Texts in Theoretical Computer Science: An EATCS Series. Springer, Heidelberg (2004)
van Emde Boas, P.: Machine models and simulation. In: Handbook of Theoretical Computer Science. Algorithms and Complexity (A), vol. A, pp. 1–66. MIT Press, Cambridge (1990)
Wadsworth, C.: Some unusual λ-calculus numeral systems. In: Seldin, J.P., Hindley, J.R. (eds.) To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism. Academic Press, London (1980)
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Dal Lago, U., Martini, S. (2006). An Invariant Cost Model for the Lambda Calculus. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds) Logical Approaches to Computational Barriers. CiE 2006. Lecture Notes in Computer Science, vol 3988. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11780342_11
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DOI: https://doi.org/10.1007/11780342_11
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