Abstract
We consider three complexity-theoretic hypotheses that have been studied in different contexts and shown to have many plausible consequences.
–The Measure Hypothesis: NP does not have p-measure 0.
–The pseudo-NP Hypothesis: there is an NP Language L such that any DTIME \(2^{{n^\epsilon}}\) Language L’ can be distinguished from L by an NP refuter.
–The NP-Machine Hypothesis: there is an NP machine accepting 0* for which no \(2^{{n^\epsilon}}\)-time machine can find infinitely many accepting computations.
We show that the NP-machine hypothesis is implied by each of the first two. Previously, no relationships were known among these three hypotheses. Moreover, we unify previous work by showing that several derandomization and circuit-size lower bounds that are known to follow from the first two hypotheses also follow from the NP-machine hypothesis. We also consider UP versions of the above hypotheses as well as related immunity and scaled dimension hypotheses.
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References
Allender, E.: When the worlds collide: derandomization, lower bounds and kolmogorov complexity. In: Foundations of Software Technology and Theoretical Computer Science, pp. 1–15. Springer, Heidelberg (2001)
Allender, E., Strauss, M.: Measure on small complexity classes with applications for BPP. In: Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pp. 807–818. IEEE Computer Society, Los Alamitos (1994)
Ambos-Spies, K., Mayordomo, E.: Resource-bounded measure and randomness. In: Sorbi, A. (ed.) Complexity, Logic and Recursion Theory. Lecture Notes in Pure and Applied Mathematics, pp. 1–47. Marcel Dekker, New York (1997)
Balcázar, J., Schöning, U.: Bi-immune sets for complexity classes. Mathematical Systems Theory 18(1), 1–18 (1985)
Bshouty, N., Cleve, R., Kannan, S., Gavalda, R., Tamon, C.: Oracles and queries that are sufficient for exact learning. Journal of Computer and System Sciences 52, 421–433 (1996)
Fenner, S., Fortnow, L., Naik, A., Rogers, J.: Inverting onto functions. Information and Computation 186(1), 90–103 (2003)
Glaßer, C., Pavan, A., Selman, A.L., Sengupta, S.: Properties of NP-complete sets. In: Proceedings of the 19th IEEE Conference on Computational Complexity, pp. 184–197. IEEE Computer Society, Los Alamitos (2004)
Hemaspaandra, L.A., Rothe, J., Wechsung, G.: Easy sets and hard certificate schemes. Acta Informatica 34(11), 859–879 (1997)
Hitchcock, J.M.: Small spans in scaled dimension. SIAM Journal on Computing.(to appear)
Hitchcock, J.M.: MAX3SAT is exponentially hard to approximate if NP has positive dimension. Theoretical Computer Science 289(1), 861–869 (2002)
Hitchcock, J.M., Lutz, J.H., Mayordomo, E.: Scaled dimension and nonuniform complexity. Journal of Computer and System Sciences 69(2), 97–122 (2004)
Impagliazzo, R., Kabanets, V., Wigderson, A.: In search of an easy witness: Exponential time vs. probabilistic polynomial time. Journal of Computer and System Sciences 65, 672–694 (2002)
Impagliazzo, R., Moser, P.: A zero-one law for RP. In: Proceedings of the 18th IEEE Conference on Computational Complexity, pp. 43–47. IEEE Computer Society, Los Alamitos (2003)
Impagliazzo, R., Wigderson, A.: P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In: Proceedings of the 29th ACM Symposium on Theory of Computing, pp. 220–229 (1997)
Kabanets, V.: Easiness assumptions and hardness tests: Trading time for zero error. Journal of Computer and System Sciences 63, 236–252 (2001)
Kabanets, V.: Derandomization: A brief overview. Bulletin of the EATCS 76, 88–103 (2002)
Kannan, R.: Circuit-size lower bounds and non-reducibility to sparse sets. Information and Control 55, 40–56 (1982)
Klivans, A., van Melkebeek, D.: Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM Journal on Computing 31, 1501–1526 (2002)
Köbler, J., Watanabe, O.: New collapse consequences of NP having small circuits. SIAM Journal on Computing 28(1), 311–324 (1998)
Lu, C.-J.: Derandomizing Arthur-Merlin games under uniform assumptions. Computational Complexity 10(3), 247–259 (2001)
Lutz, J.H.: Almost everywhere high nonuniform complexity. Journal of Computer and System Sciences 44(2), 220–258 (1992)
Lutz, J.H.: Observations on measure and lowness for \(\Delta^{\mathrm{P}}_2\). Theory of Computing Systems 30(4), 429–442 (1997)
Lutz, J.H.: The quantitative structure of exponential time. In: Hemaspaandra, L.A., Selman, A.L. (eds.) Complexity Theory Retrospective II, pp. 225–260. Springer, New York (1997)
Lutz, J.H., Mayordomo, E.: Measure, stochasticity, and the density of hard languages. SIAM Journal on Computing 23(4), 762–779 (1994)
Lutz, J.H., Mayordomo, E.: Cook versus Karp-Levin: Separating completeness notions if NP is not small. Theoretical Computer Science 164, 141–163 (1996)
Miltersen, P.B.: Derandomizing complexity classes. In: Handbook of Randomized Computing, vol. II, pp. 843–934. Kluwer, Dordrecht (2001)
Pavan, A., Selman, A.L.: Separation of NP-completeness notions. SIAM Journal on Computing 31(3), 906–918 (2002)
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Hitchcock, J.M., Pavan, A. (2004). Hardness Hypotheses, Derandomization, and Circuit Complexity. In: Lodaya, K., Mahajan, M. (eds) FSTTCS 2004: Foundations of Software Technology and Theoretical Computer Science. FSTTCS 2004. Lecture Notes in Computer Science, vol 3328. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30538-5_28
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DOI: https://doi.org/10.1007/978-3-540-30538-5_28
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