Linear-size Circuits Research Articles

On Existential MSO and Its Relation to ETH

Impagliazzo et al. proposed a framework, based on the logic fragment defining the complexity class SNP, to identify problems that are equivalent to k -CNF- Sat modulo subexponential-time reducibility (serf-reducibility). The subexponential-time solvability of any of these problems implies the failure of the Exponential Time Hypothesis (ETH). In this article, we extend the framework of Impagliazzo et al. and identify a larger set of problems that are equivalent to k -CNF- Sat modulo serf-reducibility. We propose a complexity class, referred to as Linear Monadic NP, that consists of all problems expressible in existential monadic second-order logic whose expressions have a linear measure in terms of a complexity parameter, which is usually the universe size of the problem. This research direction can be traced back to Fagin’s celebrated theorem stating that NP coincides with the class of problems expressible in existential second-order logic. Monadic NP, a well-studied class in the literature, is the restriction of the aforementioned logic fragment to existential monadic second-order logic. The proposed class Linear Monadic NP is then the restriction of Monadic NP to problems whose expressions have linear measure in the complexity parameter. We show that Linear Monadic NP includes many natural complete problems such as the satisfiability of linear-size circuits, dominating set, independent dominating set, and perfect code. Therefore, for any of these problems, its subexponential-time solvability is equivalent to the failure of ETH. We prove, using logic games, that the aforementioned problems are inexpressible in the monadic fragment of SNP, and hence, are not captured by the framework of Impagliazzo et al. Finally, we show that Feedback Vertex Set is inexpressible in existential monadic second-order logic, and hence is not in Linear Monadic NP, and investigate the existence of certain reductions between Feedback Vertex Set (and variants of it) and 3-CNF-Sat .

Optimal Rate Code Constructions for Computationally Simple Channels

We consider coding schemes for computationally bounded channels, which can introduce an arbitrary set of errors as long as (a) the fraction of errors is bounded with high probability by a parameter p and (b) the process that adds the errors can be described by a sufficiently “simple” circuit. Codes for such channel models are attractive since, like codes for standard adversarial errors, they can handle channels whose true behavior is unknown or varying over time. For two classes of channels, we provide explicit, efficiently encodable/decodable codes of optimal rate where only in efficiently decodable codes were previously known. In each case, we provide one encoder/decoder that works for every channel in the class. The encoders are randomized, and probabilities are taken over the (local, unknown to the decoder) coins of the encoder and those of the channel. Unique decoding for additive errors: We give the first construction of a polynomial-time encodable/decodable code for additive (a.k.a. oblivious ) channels that achieve the Shannon capacity 1 − H ( p ). These are channels that add an arbitrary error vector e ∈ {0, 1} N of weight at most pN to the transmitted word; the vector e can depend on the code but not on the randomness of the encoder or the particular transmitted word. Such channels capture binary symmetric errors and burst errors as special cases. List decoding for polynomial-time channels: For every constant c > 0, we construct codes with optimal rate (arbitrarily close to 1 − H ( p )) that efficiently recover a short list containing the correct message with high probability for channels describable by circuits of size at most N c . Our construction is not fully explicit but rather Monte Carlo (we give an algorithm that, with high probability, produces an encoder/decoder pair that works for all time N c channels). We are not aware of any channel models considered in the information theory literature other than purely adversarial channels, which require more than linear-size circuits to implement. We justify the relaxation to list decoding with an impossibility result showing that, in a large range of parameters ( p > 1/4), codes that are uniquely decodable for a modest class of channels (online, memoryless, nonuniform channels) cannot have positive rate.

Correlation bounds and #SAT algorithms for small linear-size circuits

We revisit the gate elimination method, generalize it to prove correlation bounds of boolean circuits with Parity, and also derive deterministic satisfiability counting algorithms for small linear-size circuits. Let B2 be the full binary basis, and let U2=B2∖{⊕,≡}. We prove that, for circuits over U2 of size 3n−nϵ for any constant ϵ>0.5, the correlation with Parity is at most 2−nΩ(1), and there is a #SAT algorithm (which counts the number of satisfying assignments) running in time 2n−nΩ(1); for circuit size 3n−ϵn for ϵ>0, the correlation with Parity is at most 2−Ω(n), and there is a #SAT algorithm running in time 2n−Ω(n). Similar correlation bounds and algorithms are also proved for circuits over B2 of size almost 2.5n.

On linear-size pseudorandom generators and hardcore functions

We consider the question of constructing pseudorandom generators that simultaneously have linear circuit complexity (in the output length), exponential security (in the seed length), and a large stretch (linear or polynomial in the seed length). We refer to such a pseudorandom generator as an asymptotically optimal PRG. We present a simple construction of an asymptotically optimal PRG from any one-way function f:{0,1}n→{0,1}n which satisfies the following requirements:1.f can be computed by linear-size circuits;2.f is 2βn-hard to invert, for some constant β>0;3.f either has high entropy, in the sense that the min-entropy of f(x) on a random input x is at least γn where β/3+γ>1, or alternatively it is regular in the sense that the preimage size of every output of f is fixed. Known constructions of PRGs from one-way functions can do without the entropy or regularity requirements, but they achieve slightly sub-exponential security (Vadhan and Zheng (2012) [27]).Our construction relies on a technical result about hardcore functions that may be of independent interest. We obtain a family of hardcore functions H={h:{0,1}n→{0,1}αn} that can be computed by linear-size circuits for any 2βn-hard one-way function f:{0,1}n→{0,1}n where β>3α. Our construction of asymptotically optimal PRGs uses such hardcore functions, which are obtained via linear-size computable affine hash functions (Ishai et al. (2008) [24]).

ON THE HARDNESS AGAINST CONSTANT-DEPTH LINEAR-SIZE CIRCUITS

The notion of average-case hardness is a fundamental one in complexity theory. In particular, it plays an important role in the research on derandomization, as there are general derandomization results which are based on the assumption that average-case hard functions exist. However, to achieve a complete derandomization, one usually needs a function which is extremely hard against a complexity class, in the sense that any algorithm in the class fails to compute the function on 1/2 - 2-Ω(n) fraction of its n-bit inputs. Unfortunately, lower bound results are very rare and they are only known for very restricted complexity classes, and achieving such extreme hardness seems even more difficult. Motivated by this, we study the hardness against linear-size circuits of constant depth in this paper. We show that the parity function is extremely hard for them: any such circuit must fail to compute the parity function on at least 1/2 - 2-Ω(n) fraction of inputs.

Algebrization

Any proof of P ≠ NP will have to overcome two barriers: relativization and natural proofs . Yet over the last decade, we have seen circuit lower bounds (e.g., that PP does not have linear-size circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in complexity theory. In this article, we present such a barrier, which we call algebraic relativization or algebrization . The idea is that, when we relativize some complexity class inclusion, we should give the simulating machine access not only to an oracle A , but also to a low-degree extension of A over a finite field or ring. We systematically go through basic results and open problems in complexity theory to delineate the power of the new algebrization barrier. First, we show that all known nonrelativizing results based on arithmetization---both inclusions such as IP = PSPACE and MIP = NEXP, and separations such as MA EXP ⊄ P/poly---do indeed algebrize. Second, we show that almost all of the major open problems---including P versus NP, P versus RP, and NEXP versus P/poly---will require non-algebrizing techniques . In some cases, algebrization seems to explain exactly why progress stopped where it did: for example, why we have superlinear circuit lower bounds for PromiseMA but not for NP. Our second set of results follows from lower bounds in a new model of algebraic query complexity , which we introduce in this article and which is interesting in its own right. Some of our lower bounds use direct combinatorial and algebraic arguments, while others stem from a surprising connection between our model and communication complexity. Using this connection, we are also able to give an MA-protocol for the Inner Product function with O (√n log n ) communication (essentially matching a lower bound of Klauck), as well as a communication complexity conjecture whose truth would imply NL ≠ NP.

Circuit Complexity of Regular Languages

We survey the current state of knowledge on the circuit complexity of regular languages and we prove that regular languages that are in AC0 and ACC0 are all computable by almost linear size circuits, extending the result of Chandra et al. (J. Comput. Syst. Sci. 30:222–234, 1985). As a consequence we obtain that in order to separate ACC0 from NC1 it suffices to prove for some e>0 an Ω(n 1+e ) lower bound on the size of ACC0 circuits computing certain NC1-complete functions.

A linear-size quantum circuit for addition with no ancillary qubits

We construct a quantum circuit for addition of two n-bit binary numbers that uses no ancillary qubits. The circuit is based on the ripple-carry approach. The depth and size of the circuit are O(n)....

Randomness versus Fault-Tolerance

We investigate the relations between two major properties of multiparty protocols: fault tolerance (or resilience ) and randomness . Fault-tolerance is measured in terms of the maximum number of colluding faulty parties, t , that a protocol can withstand and still maintain the privacy of the inputs and the correctness of the outputs (of the honest parties). Randomness is measured in terms of the total number of random bits needed by the parties in order to execute the protocol. Previously, the upper bound on the amount of randomness required by general constructions for securely computing any nontrivial function f was polynomial both in n , the total number of parties, and the circuit-size C(f) . This was the state of knowledge even for the special case t=1 (i.e., when there is at most one faulty party). In this paper we show that for any linear-size circuit, and for any number t < n/3 of faulty

Characterizing Linear Size Circuits in Terms of Privacy

In this paper we prove a perhaps unexpected relationship between the complexity class of the boolean functions that have linear size circuits andn-party private protocols. Specifically, letfbe a boolean function. We show thatfhas a linear size circuit if and only iffhas a 1-privaten-party protocol in which the total number of random bits used byallplayers is constant. From the point of view of complexity theory, our result gives a characterization of the class of linear size circuits in terms of another class of a very different nature. From the point of view of privacy, this result provides 1-private protocols that use a constant number of random bits, for many important functions for which no such protocol was previously known. On the other hand, our result suggests that proving, for anyNPfunction, that it has no 1-private constant-random protocol, might be difficult.

Linear Circuits over $\operatorname{GF}(2)$

For $n=2^k $, let S be an $n \times n$ matrix whose rows and columns are indexed by $\operatorname{GF}(2)^k $ and, for $i, j \in \operatorname{GF}(2)^k , S_{i.j}=\langle i, j \rangle $, the standard inner product. Size-depth trade-oils are investigated for computing $S{\bf x}$ with circuits using only linear operations. In particular, linear size circuits with depth bounded by the inverse of an Ackerman function are constructed, and it is shown that depth two circuits require $\Omega (n \log n)$ size. The lower bound applies to any Hadamard matrix.