### Table 1: Hardness versus randomness trade-o s for AM. If the hardness condition on the left-hand side of Table 1 holds for in nitely many input lengths, then the corresponding derandomization on the right-hand side works for in nitely many input lengths. Similar weak interpretations hold for all our results. As a corollary to the weak version of Table 1, we obtain that every language in AM, and graph nonisomorphism in particular, has subexponential size proofs for in nitely many input sizes unless the polynomial-time hierarchy collapses.Using other hardness measures and various oracles B, we get derandomization results for other complexity classes. We summarize the situation in Table 2. lower bound on: derandomizes:

1998

"... In PAGE 2: ... We show that the existence of an exponential-time decidable language with high worst-case nonuniform complexity when the circuits have access to an oracle for satis ability, implies nontrivial derandomizations of AM-games. The trade-o s are presented in Table1 . We use CB to denote circuit complexity given access to oracle B, and similarly ~ CB given only parallel access to the oracle.... In PAGE 2: ... See Section 2 for precise de nitions. The parameter s in Table1 can be any space constructible function, the interesting range lying between logarithmic and subpolynomial, e.... In PAGE 8: ...4 and 3.13 to the nondeterministic setting, and thus relax the hardness assumptions in Table1 from circuit complexity for parallel access to SAT to nondeterministic circuit complexity .... ..."

Cited by 2

### Table 3 Polynomial-Time Algorithms Take Better Advantage of Technology

"... In PAGE 12: ... Even more illuminating is the effect that a technological breakthrough improving computer speed would have. Table3 (taken from Papadimitriou, Steiglitz 1982, p. 165) demonstrates how the size of the largest instance solvable increases when a computer (or an algorithm) with the tenfold speed becomes available: The most striking insight from such a comparison is that for a polynomial function this size multiplies by some factor while for an exponential function... ..."

### Table 5 Polynomial-Time Algorithms Take Better Advantage of Technology

"... In PAGE 10: ... Even more illuminating is the effect that a technological breakthrough improving computer speed would have. Table5 (taken from Papadimitriou, Steiglitz 1982, p. 165) demonstrates... ..."

### Table 1: Wilkinson polynomial timings

1998

"... In PAGE 20: ... 7.1 Univariate Root Finding Table1 presents some timing results for univariate root nding. The exam- ples are from the family of Wilkinson polynomials W n (x)= Y i=1... ..."

Cited by 2

### Table 3: Hardness versus randomness trade-o s for MA. Theorem 4.2 If NEXP \ coNEXP 6 P=poly, then MA \ gt;0i.o.-NTIME[2n ]. Consequently, if MA does not have subexponential size proofs for in nitely many lengths, then NEXP \ coNEXP P=poly, which implies that EXP = p 2 \ p 2, hence that the polynomial-time hierarchy collapses to the second level. Finally, by looking at oracle circuits instead of parallel oracle circuits, we can derandomize BPPB for any oracle B, given an exponential-time computable predicate with high circuit complexity relative to B. Replacing ~ C by C, ~ H by H, and dropping the parallel in Theorems 3.2 and 3.11 and their proofs, yields the derandomization results of Table 4. Both A and B represent arbitrary oracles.

1998

Cited by 2

### Table 1: Existence of Polynomial Time Learning Algorithms

1993

"... In PAGE 25: ...quivalence queries consist of arbitrary read-once formulas. Q.E.D. 9 Summary and remarks Table1 summarizes what is known of the computational di culty of learning monotone and arbitrary read-once formulas according to six types of learning protocols. The entries are discussed in order below.... ..."

Cited by 107

### Table 1. Problems solvable in polynomial time

1998

Cited by 10

### Table 1: This table summarizes the results known about learning monomials and k- DNF formulas under various models of noise. Note that the upper bounds correspond to polynomial-time algorithms that can tolerate the given noise rate, and the lower bounds correspond to information-theoretic proofs that no algorithm can tolerate the given noise rate.

1995

"... In PAGE 6: ... In this case he shows that a large amount of noise can be handled where the irrelevant attributes are a ected by arbitrary adversarial noise and the relevant attributes are a ected by random noise independently of one another. In Table1 we summarize the results (from previous work and this paper) about PAC learning monomials and k-DNF formulas from the noise oracles discussed in the Section 3.... ..."

Cited by 30

### Table 3: The maximal subalgebras of V which have a polynomial-time satis- ability problem.

"... In PAGE 16: ...emma 1.2 V-SAT(V23) is in P. Proof: Follows immediately from the de nition of V23 and the previous proposition. 2 Before we can show that the other algebras in Table3 have polynomial-time satis ability problems, we need an auxiliary de nition. De nition 1.... ..."