The Value of Help Bits in Randomized and Average-Case Complexity

"Help bits" are some limited trusted information about an instance or instances of a computational problem that may reduce the computational complexity of solving that instance or instances. In this paper, we study the value of help bits in the settings of randomized and average-case complexity. Amir, Beigel, and Gasarch (1990) show that for constant $k$, if $k$ instances of a decision problem can be efficiently solved using less than $k$ bits of help, then the problem is in P/poly. We extend this result to the setting of randomized computation: We show that the decision problem is in P/poly if using $\ell$ help bits, $k$ instances of the problem can be efficiently solved with probability greater than $2^{\ell-k}$. The same result holds if using less than $k(1 - h(\alpha))$ help bits (where $h(\cdot)$ is the binary entropy function), we can efficiently solve $(1-\alpha)$ fraction of the instances correctly with non-vanishing probability. We also extend these two results to non-constant but logarithmic $k$. In this case however, instead of showing that the problem is in P/poly we show that it satisfies "$k$-membership comparability," a notion known to be related to solving $k$ instances using less than $k$ bits of help. Next we consider the setting of average-case complexity: Assume that we can solve $k$ instances of a decision problem using some help bits whose entropy is less than $k$ when the $k$ instances are drawn independently from a particular distribution. Then we can efficiently solve an instance drawn from that distribution with probability better than $1/2$. Finally, we show that in the case where $k$ is super-logarithmic, assuming $k$-membership comparability of a decision problem, one cannot prove that the problem is in P/poly by a "black-box proof.

On the reducibility of sets inside NP to sets with low information content

This paper studies for various natural problems in NP whether they can be reduced to sets with low information content, such as branches, P-selective sets, and membership comparable sets. The problems that are studied include the satisfiability problem, the graph automorphism problem, the undirected graph accessibility problem, the determinant function, and all logspace self-reducible languages. Some of these are complete for complexity classes within NP, but for others an exact complexity theoretic characterization is not known. Reducibility of these problems is studied in a general framework introduced in this paper: prover–verifier protocols with low-complexity provers. It is shown that all these natural problems indeed have such protocols. This fact is used to show, for certain reduction types, that these problems are not reducible to sets with low information content unless their complexity is much less than what it is currently believed to be. The general framework is also used to obtain a new characterization of the complexity class L : L is the class of all logspace self-reducible sets in L L-sel

On the Reducibility of Sets Inside NP to Sets with Low Information Content

We study whether sets inside NP can be reduced to sets with low information content but possibly still high computational complexity. Examples of sets with low information content are tally sets, sparse sets, P-selective sets and membership comparable sets. For the graph automorphism and isomorphism problems GA and GI, for the directed graph reachability problem GAP, for the determinant function det, and for logspace self-reducible languages we establish the following results: o If GA is polynomial-time truth-table reducible to a P-selective set, then GA is in P. o If GI is O(log n)-membership comparable, then GI is in RP. o If GAP is logspace O(1)-membership comparable, then GAP is in L. o If det is logspace Turing reducible to an L-selective set, then det is in FL. o If a language A is logspace self-reducible and logspace Turing reducible to an L-selective set, then A is in L. The last result is a strong logspace version of the characterisation of P as the class of self-reducible P-selective languages. As P and NL have logspace self-reducible complete sets, it also establishes a logspace analogue of the conjecture that if SAT is polynomial-time Turing reducible to a P-selective set, then SAT is in P

On the Reducibility of Sets Inside NP to Sets with Low Information Content

We study whether sets inside NP can be reduced to sets with low information content but possibly still high computational complexity. Examples of sets with low information content are tally sets, sparse sets, P-selective sets and membership comparable sets. For the graph automorphism..