NP-hard Sets are P-Superterse Unless R = NP

A set A is p-terse (p-superterse) if, for all q, it is not possible to answer q queries to A by making only q ? 1 queries to A (any set X). Formally, let PF A q-tt be the class of functions reducible to A via a polynomial-time truth-table reduction of norm q, and let PF A q-T be the class of functions reducible to A via a polynomial-time Turing reduction that makes at most q queries. A set A is p-terse if PF A q-tt 6 6 PF A (q?1)-T for all constants q. A is p-superterse if PF A q-tt 6 6 PF X q-T for all constants q and sets X. We show that all NP-hard sets (under p tt-reductions) are p-superterse, unless it is possible to distinguish uniquely satissable formulas from satiss-able formulas in polynomial time. Consequently, all NP-complete sets are p-superterse unless P = UP (one-way functions fail to exist), R = NP (there exist randomized polynomial-time algorithms for all problems in NP), and the polynomial-time hierarchy collapses. This mostly solves the main open question in 4]. Allowing the number of queries q to be a function of the input length n, we obtain nontrivial partial answers to a question posed by Krentel: Namely, for all NP-complete sets A and all sets X, PF A q(n)-tt 6 6 PF X (q(n)?1)-T unless SAT 2 RTIME(n O(q(n))) This is interesting whenever q(n) = o(n= log n). As a special case, PF A log i n-tt 6 6 PF X (log i n?1)-T unless NTIME(n polylogn) = RTIME(n polylogn): 1 We extend the preceding results to tt-helpers; this allows us to isolate the key properties of NP-complete sets used in the proofs. Finally, we deene pbtt-cylinders, a natural class that includes all p m-complete sets for NP, PSPACE, and exponential time. For pbtt-cylinders, we show that p-terseness is equivalent to p-superterseness. We also prove a translational lemma for non-p-terse sets; this allows us to extend the preceding result to p btt-complete sets for NP, PSPACE, and exponential time.

[1]  Patrick C. Fischer,et al.  Translational methods and computational complexity , 1965, SWCT.

[2]  C. Jockusch.  Semirecursive sets and positive reducibility , 1968 .

[3]  Albert R. Meyer,et al.  The Equivalence Problem for Regular Expressions with Squaring Requires Exponential Space , 1972, SWAT.

[4]  Nancy A. Lynch,et al.  Comparison of polynomial-time reducibilities , 1974, STOC '74.

[5]  Larry J. Stockmeyer,et al.  The Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

[6]  A. K. Chandra,et al.  Alternation , 1976, 17th Annual Symposium on Foundations of Computer Science (sfcs 1976).

[7]  Leslie G. Valiant,et al.  Relative Complexity of Checking and Evaluating , 1976, Inf. Process. Lett..

[8]  Leonard M. Adleman,et al.  Two theorems on random polynomial time , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[9]  Ronald V. Book,et al.  Bounded Query Machines: On NP and PSPACE , 1981, Theor. Comput. Sci..

[10]  John Gill,et al.  Relative to a Random Oracle A, PA != NPA != co-NPA with Probability 1 , 1981, SIAM J. Comput..

[11]  Ronald V. Book,et al.  Bounded Query Machines: On NP( ) and NPQERY( ) , 1981, Theor. Comput. Sci..

[12]  Stuart A. Kurtz,et al.  On the random oracle hypothesis , 1982, STOC '82.

[13]  Alan L. Selman,et al.  Analogues of Semicursive Sets and Effective Reducibilities to the Study of NP Complexity , 1982, Inf. Control..

[14]  C. Rackoff.  Relativized Questions Involving Probabilistic Algorithms , 1982, JACM.

[15]  Chee-Keng Yap,et al.  Some Consequences of Non-Uniform Conditions on Uniform Classes , 1983, Theor. Comput. Sci..

[16]  Timothy J. Long,et al.  Quantitative Relativizations of Complexity Classes , 1984, SIAM J. Comput..

[17]  Alan L. Selman,et al.  Complexity Measures for Public-Key Cryptosystems , 1988, SIAM J. Comput..

[18]  David A. Russo.  Structural properties of complexity classes , 1985, Research in computer science.

[19]  Gerd Wechsung,et al.  On the Boolean closure of NP , 1985, FCT.

[20]  Leslie G. Valiant,et al.  NP is as easy as detecting unique solutions , 1985, STOC '85.

[21]  Mark W. Krentel.  The complexity of optimization problems , 1986, STOC '86.

[22]  Jin-Yi Cai.  On Some Most Probable Separations of Complexity Classes , 1986 .

[23]  Jin-Yi Cai,et al.  The Boolean Hierarchy: Hardware over NP , 1986, SCT.

[24]  Richard Beigel.  A structural theorem that depends quantitatively on the complexity of SAT , 1987, Computational Complexity Conference.

[25]  Klaus W. Wagner,et al.  The Difference and Truth-Table Hierarchies for NP , 1987, RAIRO Theor. Informatics Appl..

[26]  Ker-I Ko,et al.  On Helping by Robust Oracle Machines , 1987, Theor. Comput. Sci..

[27]  Klaus W. Wagner.  More Complicated Questions About Maxima and Minima, and Some Closures of NP , 1987, Theor. Comput. Sci..

[28]  Jim Kadin,et al.  P^(NP[O(log n)]) and Sparse Turing-Complete Sets for NP , 1989, J. Comput. Syst. Sci..

[29]  Ker-I Ko,et al.  On Sets Truth-Table Reducible to Sparse Sets , 1988, SIAM J. Comput..

[30]  Juris Hartmanis,et al.  The Boolean Hierarchy I: Structural Properties , 1988, SIAM J. Comput..

[31]  Klaus W. Wagner,et al.  Collapsing Oracle Hierarchies, Census Functions and Logarithmically Many Queries , 1988, STACS.

[32]  Jin-Yi Cai,et al.  Enumerative counting is hard , 1988, [1988] Proceedings. Structure in Complexity Theory Third Annual Conference.

[33]  R. Beigel.  Query-limited reducibilities , 1988 .

[34]  Amihood Amir,et al.  Polynomial Terse Sets , 1988, Inf. Comput..

[35]  Jim Kadin.  The Polynomial Time Hierarchy Collapses if the Boolean Hierarchy Collapses , 1988, SIAM J. Comput..

[36]  Richard Beigel,et al.  On the relativized power of additional accepting paths , 1989, [1989] Proceedings. Structure in Complexity Theory Fourth Annual Conference.

[37]  William I. Gasarch,et al.  On the Complexity of Finding the Chromatic Number of a Recursive Graph II: The Unbounded Case , 1989, Ann. Pure Appl. Log..

[38]  William I. Gasarch,et al.  Bounded query classes and the difference hierarchy , 1989, Arch. Math. Log..

[39]  William I. Gasarch,et al.  On the Complexity of Finding the Chromatic Number of a Recursive Graph I: The Bounded Case , 1989, Ann. Pure Appl. Log..

[40]  James C. Owings,et al.  A cardinality version of Beigel's nonspeedup theorem , 1989, Journal of Symbolic Logic.

[41]  William I. Gasarch,et al.  Nondeterministic Bounded Query Reducibilities , 1989, Ann. Pure Appl. Log..

[42]  Richard Beigel,et al.  Bi-Immunity Results for Cheatable Sets , 1990, Theor. Comput. Sci..