Abstract
We present RSLR, an implicit higher-order characterization of the class PP of those problems which can be decided in probabilistic polynomial time with error probability smaller than \(\frac{1}{2}\). Analogously, a (less implicit) characterization of the class BPP can be obtained. RSLR is an extension of Hofmann’s SLR with a probabilistic primitive, which enjoys basic properties such as subject reduction and confluence. Polynomial time soundness of RSLR is obtained by syntactical means, as opposed to the standard literature on SLR-derived systems, which use semantics in an essential way.
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References
Arora, S., Barak, B.: Computational Complexity, A Modern Approach. Cambridge University Press (2009)
Bellantoni, S.J., Niggl, K.H., Schwichtenberg, H.: Higher type recursion, ramification and polynomial time. Annals of Pure and Applied Logic 104(1-3), 17–30 (2000)
Bellantoni, S.: Predicative recursion and the polytime hierarchy. In: Clote, P., Remmel, J.B. (eds.) Feasible Mathematics II, pp. 15–29. Birkhäuser (1995)
Bellantoni, S., Cook, S.A.: A new recursion-theoretic characterization of the polytime functions. Computational Complexity 2, 97–110 (1992)
Bonfante, G., Kahle, R., Marion, J.-Y., Oitavem, I.: Recursion Schemata for \({\mathit NC}^k\). In: Kaminski, M., Martini, S. (eds.) CSL 2008. LNCS, vol. 5213, pp. 49–63. Springer, Heidelberg (2008)
Dal Lago, U., Martini, S., Sangiorgi, D.: Light logics and higher-order processes. In: Fröschle, S.B., Valencia, F.D. (eds.) Proceedings of the 17th International Workshop Expressiveness in Concurrency. EPTCS, vol. 41 (2010)
Dal Lago, U., Masini, A., Zorzi, M.: Quantum implicit computational complexity. Theoretical Computer Science 411(2), 377–409 (2010)
Dal Lago, U., Parisen Toldin, P.: An higher-order characterization of probabilistic polynomial time (long version), http://arxiv.org/abs/1202.3317
Hofmann, M.: A Mixed Modal/Linear Lambda Calculus with Applications to Bellantoni-Cook Safe Recursion. In: Nielsen, M., Thomas, W. (eds.) CSL 1997. LNCS, vol. 1414, pp. 275–294. Springer, Heidelberg (1998)
Mitchell, J.C., Mitchell, M., Scedrov, A.: A linguistic characterization of bounded oracle computation and probabilistic polynomial time. In: Proceedings of the 39th Annual Symposium Foundations of Computer Science, pp. 725–733. IEEE Computer Society (1998)
Jones, N.D.: Logspace and ptime characterized by programming languages. Theoretical Computer Science 228, 151–174 (1999)
Leivant, D.: Stratified functional programs and computational complexity. In: Proceedings of the 20th International Symposium Principles of Programming Languages, pp. 325–333. ACM (1993)
Leivant, D., Marion, J.-Y.: Ramified Recurrence and Computational Complexity II: Substitution and Poly-Space. In: Pacholski, L., Tiuryn, J. (eds.) CSL 1994. LNCS, vol. 933, pp. 486–500. Springer, Heidelberg (1995)
Schwichtenberg, H., Bellantoni, S.: Feasible computation with higher types. In: Proof and System-Reliability, pp. 399–415. Kluwer Academic Publishers (2001)
Zhang, Y.: The computational SLR: a logic for reasoning about computational indistinguishability. Mathematical Structures in Computer Science 20(5), 951–975 (2010)
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Dal Lago, U., Parisen Toldin, P. (2012). A Higher-Order Characterization of Probabilistic Polynomial Time. In: Peña, R., van Eekelen, M., Shkaravska, O. (eds) Foundational and Practical Aspects of Resource Analysis. FOPARA 2011. Lecture Notes in Computer Science, vol 7177. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32495-6_1
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