Pseudorandomness and Average-Case Complexity via Uniform Reductions

Impagliazzo and Wigderson (36th FOCS, 1998) gave the first construction of pseudorandom generators from a uniform complexity assumption on EXP (namely EXP [not equal to] BPP). Unlike results in the nonuniform setting, their result does not provide a continuous trade-off between worst-case hardness and pseudorandomness, nor does it explicitly establish an average-case hardness result. In this paper: 1. We obtain an optimal worst-case to average-case connection for EXP: if EXP is not a subset of BPTIME(t(n)), EXP has problems that cannot be solved on a fraction 1/2 +1/t'(n) of the inputs by BPTIME(t'(n)) algorithms, for t'=t^{\Omega(1)}. 2. We exhibit a PSPACE-complete self-correctible and downward self-reducible problem. This slightly simplifies and strengthens the proof of Impaglaizzo and Wigderson, which used a a #P-complete problem with these properties. 3. We argue that the results of Impagliazzo and Wigderson, and the ones in this paper, cannot be proved via "black-box" uniform reductions.Engineering and Applied Science

Pseudodeterministic constructions in subexponential time

We study pseudodeterministic constructions, i.e., randomized algorithms which output the same solution on most computation paths. We establish unconditionally that there is an infinite sequence {pn}n∈N of increasing primes and a randomized algorithm A running in expected sub-exponential time such that for each n, on input 1|pn|, A outputs pn with probability 1. In other words, our result provides a pseudodeterministic construction of primes in sub-exponential time which works infinitely often. This result follows from a much more general theorem about pseudodeterministic constructions. A property Q ⊆ {0, 1}* is γ-dense if for large enough n, |Q ⋂ {0, 1}n| ≥ γ2n. We show that for each c > 0 at least one of the following holds: (1) There is a pseudodeterministic polynomial time construction of a family {Hn} of sets, Hn ⊆ {0, 1}n, such that for each (1=nc)-dense property Q ∈ DTIME(n^c) and every large enough n, Hn ⋂ Q ≠ ∅; or (2) There is a deterministic sub-exponential time construction of a family {H'n} of sets, H'n ⊆ {0, 1}n, such that for each (1/n^c)-dense property Q ∈ DTIME(n^c) and for infinitely many values of n, H'n ⋂ Q ≠ ∅. We provide further algorithmic applications that might be of independent interest. Perhaps intriguingly, while our main results are unconditional, they have a non-constructive element, arising from a sequence of applications of the hardness versus randomness paradigm.</p

On Oracles and Algorithmic Methods for Proving Lower Bounds

This paper studies the interaction of oracles with algorithmic approaches to proving circuit complexity lower bounds, establishing new results on two different kinds of questions. 1) We revisit some prominent open questions in circuit lower bounds, and provide a clean way of viewing them as circuit upper bound questions. Let Missing-String be the (total) search problem of producing a string that does not appear in a given list L containing M bit-strings of length N, where M < 2?. We show in a generic way how algorithms and uniform circuits (from restricted classes) for Missing-String imply complexity lower bounds (and in some cases, the converse holds as well). We give a local algorithm for Missing-String, which can compute any desired output bit making very few probes into the input, when the number of strings M is small enough. We apply this to prove a new nearly-optimal (up to oracles) time hierarchy theorem with advice. We show that the problem of constructing restricted uniform circuits for Missing-String is essentially equivalent to constructing functions without small non-uniform circuits, in a relativizing way. For example, we prove that small uniform depth-3 circuits for Missing-String would imply exponential circuit lower bounds for ?? EXP, and depth-3 lower bounds for Missing-String would imply non-trivial circuits (relative to an oracle) for ?? EXP problems. Both conclusions are longstanding open problems in circuit complexity. 2) It has been known since Impagliazzo, Kabanets, and Wigderson [JCSS 2002] that generic derandomizations improving subexponentially over exhaustive search would imply lower bounds such as NEXP ? ? ?/poly. Williams [SICOMP 2013] showed that Circuit-SAT algorithms running barely faster than exhaustive search would imply similar lower bounds. The known proofs of such results do not relativize (they use techniques from interactive proofs/PCPs). However, it has remained open whether there is an oracle under which the generic implications from circuit-analysis algorithms to circuit lower bounds fail. Building on an oracle of Fortnow, we construct an oracle relative to which the circuit approximation probability problem (CAPP) is in ?, yet EXP^{NP} has polynomial-size circuits. We construct an oracle relative to which SAT can be solved in "half-exponential" time, yet exponential time (EXP) has polynomial-size circuits. Improving EXP to NEXP would give an oracle relative to which ?? ? has "half-exponential" size circuits, which is open. (Recall it is known that ?? ? is not in "sub-half-exponential" size, and the proof relativizes.) Moreover, the running time of the SAT algorithm cannot be improved: relative to all oracles, if SAT is in "sub-half-exponential" time then EXP does not have polynomial-size circuits

Enumerating Error Bounded Polytime Algorithms Through Arithmetical Theories

We consider a minimal extension of the language of arithmetic, such that the bounded formulas provably total in a suitably-defined theory \`a la Buss (expressed in this new language) precisely capture polytime random functions. Then, we provide two new characterizations of the semantic class BPP obtained by internalizing the error-bound check within a logical system: the first relies on measure-sensitive quantifiers, while the second is based on standard first-order quantification. This leads us to introduce a family of effectively enumerable subclasses of BPP, called BPP_T and consisting of languages captured by those probabilistic Turing machines whose underlying error can be proved bounded in the theory T. As a paradigmatic example of this approach, we establish that polynomial identity testing is in BPP_T where T=$\mathrm{I}\Delta_0+\mathrm{Exp}$ is a well-studied theory based on bounded induction

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum