Abstract
The linear lambda calculus is very weak in terms of expressive power: in particular, all functions terminate in linear time. In this paper we consider a simple extension with Booleans, natural numbers and a linear iterator. We show properties of this linear version of Gödel’s System \(\mathcal{T}\) and study the class of functions that can be represented. Surprisingly, this linear calculus is extremely expressive: it is as powerful as System \(\mathcal{T}\)
Research partially supported by the Treaty of Windsor Grant: “Linearity: Programming Languages and Implementations”, and by funds granted to LIACC through the Programa de Financiamento Plurianual, FCT and FEDER/POSI.
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References
Abramsky, S.: Computational Interpretations of Linear Logic. Theoretical Computer Science 111, 3–57 (1993)
Asperti, A.: Light affine logic. In: Proc. Logic in Computer Science (LICS 1998). IEEE Computer Society Press, Los Alamitos (1998)
Asperti, A., Roversi, L.: Intuitionistic light affine logic. ACM Transactions on Computational Logic (2002)
Baillot, P., Mogbil, V.: Soft lambda-calculus: A language for polynomial time computation. In: Walukiewicz, I. (ed.) FOSSACS 2004. LNCS, vol. 2987, pp. 27–41. Springer, Heidelberg (2004)
Barendregt, H.P.: The Lambda Calculus: Its Syntax and Semantics. Studies in Logic and the Foundations of Mathematics, 2nd revised edn., vol. 103. North-Holland Publishing Company, Amsterdam (1984)
Fernández, M., Mackie, I., Sinot, F.-R.: Closed reduction: explicit substitutions without alpha conversion. Mathematical Structures in Computer Science 15(2), 343–381 (2005)
Girard, J.: Light linear logic. Information and Computation (1998)
Girard, J.-Y.: Linear Logic. Theoretical Computer Science 50(1), 1–102 (1987)
Girard, J.-Y.: Towards a geometry of interaction. In: Gray, J.W., Scedrov, A. (eds.) Categories in Computer Science and Logic: Proc. of the Joint Summer Research Conference, pp. 69–108. American Mathematical Society, Providence (1989)
Girard, J.-Y., Lafont, Y., Taylor, P.: Proofs and Types. Cambridge Tracts in Theoretical Computer Science, vol. 7. Cambridge University Press, Cambridge (1989)
Girard, J.-Y., Scedrov, A., Scott, P.J.: Bounded linear logic: A modular approach to polynomial time computability. Theoretical Computer Science 97, 1–66 (1992)
Hindley, J.: BCK-combinators and linear lambda-terms have types. Theoretical Computer Science (TCS) 64(1), 97–105 (1989)
Hofmann, M.: Linear types and non-size-increasing polynomial time computation. In: Proc. Logic in Computer Science (LICS 1999). IEEE Computer Society Press, Los Alamitos (1999)
Holmström, S.: Linear functional programming. In: Johnsson, T., Peyton Jones, S.L., Karlsson, K. (eds.) Proceedings of the Workshop on Implementation of Lazy Functional Languages, pp. 13–32 (1988)
Lafont, Y.: Soft linear logic and polynomial time. Theoretical Computer Science (2004)
Lago, U.D.: The geometry of linear higher-order recursion. In: Panangaden, P. (ed.) Proceedings of the Twentieth Annual IEEE Symp. on Logic in Computer Science, LICS 2005, pp. 366–375. IEEE Computer Society Press, Los Alamitos (2005)
Lambek, J.: From lambda calculus to cartesian closed categories. In: Seldin, J.P., Hindley, J.R. (eds.) To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pp. 363–402. Academic Press, London (1980)
Lambek, J., Scott, P.J.: Introduction to Higher Order Categorical Logic. Cambridge Studies in Advanced Mathematics, vol. 7. Cambridge University Press, Cambridge (1986)
Mackie, I.: Lilac: A functional programming language based on linear logic. Journal of Functional Programming 4(4), 395–433 (1994)
Mackie, I., Román, L., Abramsky, S.: An internal language for autonomous categories. Journal of Applied Categorical Structures 1(3), 311–343 (1993)
Newman, M.: On theories with a combinatorial definition of “equivalence”. Annals of Mathematics 43(2), 223–243 (1942)
Paulin-Mohring, C.: Inductive Definitions in the System Coq - Rules and Properties. In: Bezem, M., Groote, J.F. (eds.) TLCA 1993. LNCS, vol. 664. Springer, Heidelberg (1993)
Phillips, I.: Recursion theory. In: Abramsky, S., Gabbay, D., Maibaum, T. (eds.) Handbook of Logic in Computer Science, vol. 1, pp. 79–187. Oxford University Press, Oxford (1992)
Terui, K.: Affine lambda-calculus and polytime strong normalization. In: Proc. Logic in Computer Science (LICS 2001). IEEE Computer Society Press, Los Alamitos (2001)
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Alves, S., Fernández, M., Florido, M., Mackie, I. (2006). The Power of Linear Functions. In: Ésik, Z. (eds) Computer Science Logic. CSL 2006. Lecture Notes in Computer Science, vol 4207. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11874683_8
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DOI: https://doi.org/10.1007/11874683_8
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