Competing provers yield improved Karp-Lipton collapse results

  • Authors:

  • Jin-Yi Cai;Venkatesan T. Chakaravarthy;Lane A. Hemaspaandra;Mitsunori Ogihara

  • Affiliations:

  • Computer Sciences Department, University of Wisconsin, Madison, WI and Department of Computer Science, Tsinghua University, Beijing, China;Computer Sciences Department, University of Wisconsin, Madison, WI;Department of Computer Science, University of Rochester, Rochester, NY;Department of Computer Science, University of Rochester, Rochester, NY

  • Venue:
  • Information and Computation
  • Year:
  • 2005

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Abstract

Via competing provers, we show that if a language A is self-reducible and has polynomial-size circuits then S2A = S2. Building on this, we strengthen the Kämper-AFK theorem, namely, we prove that if NP ⊆ (NP ∩ coNP)/poly then the polynomial hierarchy collapses to S2NP∩coNP. We also strengthen Yap's theorem, namely, we prove that if NP ⊆ coNP/poly then the polynomial hierarchy collapses to S2NP. Under the same assumptions, the best previously known collapses were to ZPPNP and ZPPNPNP, respectively ([SIAM Journal on Computing 28 (1) (1998) 311; Journal of Computer and System Sciences 52 (3) (1996) 421], building on [Proceedings of the 12th ACM Symposium on Theory of Computing, ACM Press, New York, 1980, pp. 302-309; Journal of Computer and System Sciences 39 (1989) 21; Theoretical Computer Science 85 (2) (1991) 305; Theoretical Computer Science 26 (3) (1983) 287]). It is known that S2 ⊆ ZPPNP [Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Silver Spring, MD, 2001, pp. 620-629]. That result and its relativized version show that our new collapses indeed improve the previously known results. The Kämper-AFK theorem and Yap's theorem are used in the literature as bridges in a variety of results--ranging from the study of unique solutions to issues of approximation--and so our results implicitly strengthen those results.