Log-space reduction

Abstract: The Minimum Circuit Size Problem (MCSP) is known to be hard for statistical zero knowledge via a BPP-Turing reduction (Allender and Das, 2014), whereas establishing NP-hardness of MCSP via a polynomial-time many-one reduction is difficult (Murray and Williams, 2015) in the sense that it implies ZPP ≠ EXP, which is a major open problem in computational complexity. In this paper, we provide strong evidence that current techniques cannot establish NP-hardness of MCSP, even under polynomial-time Turing reductions or randomized reductions: Specifically, we introduce the notion of oracle-independent reduction to MCSP, which captures all the currently known reductions. We say that a reduction to MCSP is oracle-independent if the reduction can be generalized to a reduction to MCSPA for any oracle A, where MCSPA denotes an oracle version of MCSP. We prove that no language outside P is reducible to MCSP via an oracle-independent polynomial-time Turing reduction. We also show that the class of languages reducible to MCSP via an oracle-independent randomized reduction that makes at most one query is contained in AM ∩ coAM. Thus, NP-hardness of MCSP cannot be established via such oracle-independent reductions unless the polynomial hierarchy collapses. We also extend the previous results to the case of more general reductions: We prove that establishing NP-hardness of MCSP via a polynomial-time nonadaptive reduction implies ZPP ≠ EXP, and that establishing NP-hardness of approximating circuit complexity via a polynomial-time Turing reduction also implies ZPP ≠ EXP. Along the way, we prove that approximating Levin's Kolmogorov complexity is provably not EXP-hard under polynomial-time Turing reductions, which is of independent interest.

Abstract: We show that the recognition problem of context-free languages can be reduced to membership in the language defined by a regular expression with intersection by a log space reduction with linear output length. We also show a matching upper bound improving the known fact that the membership problem for these regular expressions is in NC2. Together these results establish that the membership problem is complete in LOGCFL. For unary expressions we show hardness for the class NL and some related results.

Abstract: We present a simple deterministic gap-preserving reduction from SAT to the minimum distance of code problem over F-2. We also show how to extend the reduction to work over any fixed finite field. Previously, a randomized reduction was known due to Dumer, Micciancio, and Sudan, which was recently derandomized by Cheng and Wan. These reductions rely on highly nontrivial coding theoretic constructions, whereas our reduction is elementary. As an additional feature, our reduction gives hardness within a constant factor even for asymptotically good codes, i.e., having constant positive rate and relative distance. Previously, it was not known how to achieve a deterministic reduction for such codes.

Abstract: We present a simple deterministic gap-preserving reduction from SAT to the Minimum Distance of Code Problem over F2. We also show how to extend the reduction to work over any finite field (of constant size). Previously a randomized reduction was known due to Dumer, Micciancio, and Sudan [9], which was recently derandomized by Cheng and Wan [7, 8]. These reductions rely on highly non-trivial coding theoretic constructions whereas our reduction is elementary. As an additional feature, our reduction gives a constant factor hardness even for asymptotically good codes, i.e., having constant rate and relative distance. Previously it was not known how to achieve deterministic reductions for such codes.