Some connections between bounded query classes and non-uniform complexity

  • Authors:

  • Amihood Amir;Richard Beigel;William Gasarch

  • Affiliations:

  • Department of Math and Comp. Sci., Bar-Ilan University, Ramat Gan 52900, Israel;Department of CIS, Temple University, Wachman Hall, 1805 N Broad Street, Philadelphia, PA;Department of C.S. and Inst. for Adv. Comp. Stud., University of Maryland, Room 3213, A. V. Williams Building, College Park, MD

  • Venue:
  • Information and Computation
  • Year:
  • 2003

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Abstract

Let A(x) be the characteristic function of A. Consider the function CkA(x1,...,xk) = A(x1)...A(xk). We show that if CkA can be computed in polynomial time with fewer than k queries to some set X then A ∈ P/poly. A generalization of this result has applications to bounded query classes, circuits, and enumerability. In particular we obtain the following. (1) Assuming Σ3p ≠ Φ3p, there are functions computable using f(n) + 1 queries to SAT that are not computable using f(n) queries to SAT, for f(n) = O(log n). (2) If CkA, restricted to length n inputs, can be computed by an unbounded fanin oracle circuit of size s(n) and depth d(n), with k - 1 queries to some set X, then A can be computed with an unbounded fanin (non-oracle) circuit of size nO(k) s(n) and depth d(n) + O(1). (3) Assuming that PH ≠ Σ4p ∩ Φ4p, and ε 1, #SAT is not 2nε-enumerable.