Logspace self-reducibility

A definition of self-reducibility is proposed to deal with logarithmic space complexity classes. A general property derived from the definition is used to prove known results comparing uniform and nonuniform complexity classes below polynomial time, and to obtain novel ones regarding nondeterministic nonuniform classes and reducibility to context-free languages.Peer ReviewedPostprint (published version

New Insights on the (Non-)Hardness of Circuit Minimization and Related Problems

The Minimum Circuit Size Problem (MCSP) and a related problem (MKTP) that deals with time-bounded Kolmogorov complexity are prominent candidates for NP-intermediate status. We show that, under very modest cryptographic assumptions (such as the existence of one-way functions), the problem of approximating the minimum circuit size (or time-bounded Kolmogorov complexity) within a factor of n^{1 - o(1)} is indeed NP-intermediate. To the best of our knowledge, these problems are the first natural NP-intermediate problems under the existence of an arbitrary one-way function. We also prove that MKTP is hard for the complexity class DET under non-uniform NC^0 reductions. This is surprising, since prior work on MCSP and MKTP had highlighted weaknesses of "local" reductions such as NC^0 reductions. We exploit this local reduction to obtain several new consequences: * MKTP is not in AC^0[p]. * Circuit size lower bounds are equivalent to hardness of a relativized version MKTP^A of MKTP under a class of uniform AC^0 reductions, for a large class of sets A. * Hardness of MCSP^A implies hardness of MKTP^A for a wide class of sets A. This is the first result directly relating the complexity of MCSP^A and MKTP^A, for any A

Polynomial-Time Pseudodeterministic Construction of Primes

A randomized algorithm for a search problem is *pseudodeterministic* if it produces a fixed canonical solution to the search problem with high probability. In their seminal work on the topic, Gat and Goldwasser posed as their main open problem whether prime numbers can be pseudodeterministically constructed in polynomial time. We provide a positive solution to this question in the infinitely-often regime. In more detail, we give an *unconditional* polynomial-time randomized algorithm $B$ such that, for infinitely many values of $n$, $B(1^n)$ outputs a canonical $n$-bit prime $p_n$ with high probability. More generally, we prove that for every dense property $Q$ of strings that can be decided in polynomial time, there is an infinitely-often pseudodeterministic polynomial-time construction of strings satisfying $Q$. This improves upon a subexponential-time construction of Oliveira and Santhanam. Our construction uses several new ideas, including a novel bootstrapping technique for pseudodeterministic constructions, and a quantitative optimization of the uniform hardness-randomness framework of Chen and Tell, using a variant of the Shaltiel--Umans generator

On the Complexity of the Equivalence Problem for Probabilistic Automata

Checking two probabilistic automata for equivalence has been shown to be a key problem for efficiently establishing various behavioural and anonymity properties of probabilistic systems. In recent experiments a randomised equivalence test based on polynomial identity testing outperformed deterministic algorithms. In this paper we show that polynomial identity testing yields efficient algorithms for various generalisations of the equivalence problem. First, we provide a randomized NC procedure that also outputs a counterexample trace in case of inequivalence. Second, we show how to check for equivalence two probabilistic automata with (cumulative) rewards. Our algorithm runs in deterministic polynomial time, if the number of reward counters is fixed. Finally we show that the equivalence problem for probabilistic visibly pushdown automata is logspace equivalent to the Arithmetic Circuit Identity Testing problem, which is to decide whether a polynomial represented by an arithmetic circuit is identically zero.Comment: technical report for a FoSSaCS'12 pape

AND and/or OR: Uniform Polynomial-Size Circuits

We investigate the complexity of uniform OR circuits and AND circuits of polynomial-size and depth. As their name suggests, OR circuits have OR gates as their computation gates, as well as the usual input, output and constant (0/1) gates. As is the norm for Boolean circuits, our circuits have multiple sink gates, which implies that an OR circuit computes an OR function on some subset of its input variables. Determining that subset amounts to solving a number of reachability questions on a polynomial-size directed graph (which input gates are connected to the output gate?), taken from a very sparse set of graphs. However, it is not obvious whether or not this (restricted) reachability problem can be solved, by say, uniform AC^0 circuits (constant depth, polynomial-size, AND, OR, NOT gates). This is one reason why characterizing the power of these simple-looking circuits in terms of uniform classes turns out to be intriguing. Another is that the model itself seems particularly natural and worthy of study. Our goal is the systematic characterization of uniform polynomial-size OR circuits, and AND circuits, in terms of known uniform machine-based complexity classes. In particular, we consider the languages reducible to such uniform families of OR circuits, and AND circuits, under a variety of reduction types. We give upper and lower bounds on the computational power of these language classes. We find that these complexity classes are closely related to tallyNL, the set of unary languages within NL, and to sets reducible to tallyNL. Specifically, for a variety of types of reductions (many-one, conjunctive truth table, disjunctive truth table, truth table, Turing) we give characterizations of languages reducible to OR circuit classes in terms of languages reducible to tallyNL classes. Then, some of these OR classes are shown to coincide, and some are proven to be distinct. We give analogous results for AND circuits. Finally, for many of our OR circuit classes, and analogous AND circuit classes, we prove whether or not the two classes coincide, although we leave one such inclusion open.Comment: In Proceedings MCU 2013, arXiv:1309.104

Easiness Amplification and Uniform Circuit Lower Bounds

We present new consequences of the assumption that time-bounded algorithms can be "compressed" with non-uniform circuits. Our main contribution is an "easiness amplification" lemma for circuits. One instantiation of the lemma says: if n^{1+e}-time, tilde{O}(n)-space computations have n^{1+o(1)} size (non-uniform) circuits for some e > 0, then every problem solvable in polynomial time and tilde{O}(n) space has n^{1+o(1)} size (non-uniform) circuits as well. This amplification has several consequences: * An easy problem without small LOGSPACE-uniform circuits. For all e > 0, we give a natural decision problem, General Circuit n^e-Composition, that is solvable in about n^{1+e} time, but we prove that polynomial-time and logarithmic-space preprocessing cannot produce n^{1+o(1)}-size circuits for the problem. This shows that there are problems solvable in n^{1+e} time which are not in LOGSPACE-uniform n^{1+o(1)} size, the first result of its kind. We show that our lower bound is non-relativizing, by exhibiting an oracle relative to which the result is false. * Problems without low-depth LOGSPACE-uniform circuits. For all e > 0, 1 < d < 2, and e < d we give another natural circuit composition problem computable in tilde{O}(n^{1+e}) time, or in O((log n)^d) space (though not necessarily simultaneously) that we prove does not have SPACE[(log n)^e]-uniform circuits of tilde{O}(n) size and O((log n)^e) depth. We also show SAT does not have circuits of tilde{O}(n) size and log^{2-o(1)}(n) depth that can be constructed in log^{2-o(1)}(n) space. * A strong circuit complexity amplification. For every e > 0, we give a natural circuit composition problem and show that if it has tilde{O}(n)-size circuits (uniform or not), then every problem solvable in 2^{O(n)} time and 2^{O(sqrt{n log n})} space (simultaneously) has 2^{O(sqrt{n log n})}-size circuits (uniform or not). We also show the same consequence holds assuming SAT has tilde{O}(n)-size circuits. As a corollary, if n^{1.1} time computations (or O(n) nondeterministic time computations) have tilde{O}(n)-size circuits, then all problems in exponential time and subexponential space (such as quantified Boolean formulas) have significantly subexponential-size circuits. This is a new connection between the relative circuit complexities of easy and hard problems