Symmetric Exponential Time Requires Near-Maximum Circuit Size: Simplified, Truly Uniform

In a recent breakthrough, Chen, Hirahara and Ren prove that $\mathsf{S_2E}/_1 \not\subset \mathsf{SIZE}[2^n/n]$ by giving a single-valued $\mathsf{FS_2P}$ algorithm for the Range Avoidance Problem ($\mathsf{Avoid}$) that works for infinitely many input size $n$. Building on their work, we present a simple single-valued $\mathsf{FS_2P}$ algorithm for $\mathsf{Avoid}$ that works for all input size $n$. As a result, we obtain the circuit lower bound $\mathsf{S_2E} \not\subset \mathsf{SIZE}[2^n/n]$ and many other corollaries: 1. Near-maximum circuit lower bound for $\mathsf{\Sigma_2E} \cap \mathsf{\Pi_2E}$ and $\mathsf{ZPE}^{\mathsf{NP}}$. 2. Pseudodeterministic $\mathsf{FZPP}^{\mathsf{NP}}$ constructions for: Ramsey graphs, rigid matrices, pseudorandom generators, two-source extractors, linear codes, hard truth tables, and $K^{poly}$-random strings

Derandomization with Minimal Memory Footprint

Existing proofs that deduce BPL = ? from circuit lower bounds convert randomized algorithms into deterministic algorithms with large constant overhead in space. We study space-bounded derandomization with minimal footprint, and ask what is the minimal possible space overhead for derandomization. We show that BPSPACE[S] ? DSPACE[c ? S] for c ? 2, assuming space-efficient cryptographic PRGs, and, either: (1) lower bounds against bounded-space algorithms with advice, or: (2) lower bounds against certain uniform compression algorithms. Under additional assumptions regarding the power of catalytic computation, in a new setting of parameters that was not studied before, we are even able to get c ? 1. Our results are constructive: Given a candidate hard function (and a candidate cryptographic PRG) we show how to transform the randomized algorithm into an efficient deterministic one. This follows from new PRGs and targeted PRGs for space-bounded algorithms, which we combine with novel space-efficient evaluation methods. A central ingredient in all our constructions is hardness amplification reductions in logspace-uniform TC?, that were not known before

Limits on Representing Boolean Functions by Linear Combinations of Simple Functions: Thresholds, ReLUs, and Low-Degree Polynomials

We consider the problem of representing Boolean functions exactly by "sparse" linear combinations (over $\mathbb{R}$) of functions from some "simple" class ${\cal C}$. In particular, given ${\cal C}$ we are interested in finding low-complexity functions lacking sparse representations. When ${\cal C}$ is the set of PARITY functions or the set of conjunctions, this sort of problem has a well-understood answer, the problem becomes interesting when ${\cal C}$ is "overcomplete" and the set of functions is not linearly independent. We focus on the cases where ${\cal C}$ is the set of linear threshold functions, the set of rectified linear units (ReLUs), and the set of low-degree polynomials over a finite field, all of which are well-studied in different contexts. We provide generic tools for proving lower bounds on representations of this kind. Applying these, we give several new lower bounds for "semi-explicit" Boolean functions. For example, we show there are functions in nondeterministic quasi-polynomial time that require super-polynomial size: $\bullet$ Depth-two neural networks with sign activation function, a special case of depth-two threshold circuit lower bounds. $\bullet$ Depth-two neural networks with ReLU activation function. $\bullet$ $\mathbb{R}$-linear combinations of $O(1)$-degree $\mathbb{F}_p$-polynomials, for every prime $p$ (related to problems regarding Higher-Order "Uncertainty Principles"). We also obtain a function in $E^{NP}$ requiring $2^{\Omega(n)}$ linear combinations. $\bullet$ $\mathbb{R}$-linear combinations of $ACC \circ THR$ circuits of polynomial size (further generalizing the recent lower bounds of Murray and the author). (The above is a shortened abstract. For the full abstract, see the paper.

Stronger Connections Between Circuit Analysis and Circuit Lower Bounds, via PCPs of Proximity

We considerably sharpen the known connections between circuit-analysis algorithms and circuit lower bounds, show intriguing equivalences between the analysis of weak circuits and (apparently difficult) circuits, and provide strong new lower bounds for approximately computing Boolean functions with depth-two neural networks and related models. - We develop approaches to proving THR o THR lower bounds (a notorious open problem), by connecting algorithmic analysis of THR o THR to the provably weaker circuit classes THR o MAJ and MAJ o MAJ, where exponential lower bounds have long been known. More precisely, we show equivalences between algorithmic analysis of THR o THR and these weaker classes. The epsilon-error CAPP problem asks to approximate the acceptance probability of a given circuit to within additive error epsilon; it is the "canonical" derandomization problem. We show: - There is a non-trivial (2^n/n^{omega(1)} time) 1/poly(n)-error CAPP algorithm for poly(n)-size THR o THR circuits if and only if there is such an algorithm for poly(n)-size MAJ o MAJ. - There is a delta > 0 and a non-trivial SAT (delta-error CAPP) algorithm for poly(n)-size THR o THR circuits if and only if there is such an algorithm for poly(n)-size THR o MAJ. Similar results hold for depth-d linear threshold circuits and depth-d MAJORITY circuits. These equivalences are proved via new simulations of THR circuits by circuits with MAJ gates. - We strengthen the connection between non-trivial derandomization (non-trivial CAPP algorithms) for a circuit class C, and circuit lower bounds against C. Previously, [Ben-Sasson and Viola, ICALP 2014] (following [Williams, STOC 2010]) showed that for any polynomial-size class C closed under projections, non-trivial (2^{n}/n^{omega(1)} time) CAPP for OR_{poly(n)} o AND_{3} o C yields NEXP does not have C circuits. We apply Probabilistic Checkable Proofs of Proximity in a new way to show it would suffice to have a non-trivial CAPP algorithm for either XOR_2 o C, AND_2 o C or OR_2 o C. - A direct corollary of the first two bullets is that NEXP does not have THR o THR circuits would follow from either: - a non-trivial delta-error CAPP (or SAT) algorithm for poly(n)-size THR o MAJ circuits, or - a non-trivial 1/poly(n)-error CAPP algorithm for poly(n)-size MAJ o MAJ circuits. - Applying the above machinery, we extend lower bounds for depth-two neural networks and related models [R. Williams, CCC 2018] to weak approximate computations of Boolean functions. For example, for arbitrarily small epsilon > 0, we prove there are Boolean functions f computable in nondeterministic n^{log n} time such that (for infinitely many n) every polynomial-size depth-two neural network N on n inputs (with sign or ReLU activation) must satisfy max_{x in {0,1}^n}|N(x)-f(x)|>1/2-epsilon. That is, short linear combinations of ReLU gates fail miserably at computing f to within close precision. Similar results are proved for linear combinations of ACC o THR circuits, and linear combinations of low-degree F_p polynomials. These results constitute further progress towards THR o THR lower bounds

Unconditional Relationships within Zero Knowledge

Zero-knowledge protocols enable one party, called a prover, to "convince" another party, called a verifier, the validity of a mathematical statement such that the verifier "learns nothing" other than the fact that the proven statement is true. The different ways of formulating the terms "convince" and "learns nothing" gives rise to four classes of languages having zero-knowledge protocols, which are: statistical zero-knowledge proof systems, computational zero-knowledge proof systems, statistical zero-knowledge argument systems, and computational zero-knowledge argument systems. We establish complexity-theoretic characterization of the classes of languages in NP having zero-knowledge argument systems. Using these characterizations, we show that for languages in NP: -- Instance-dependent commitment schemes are necessary and sufficient for zero-knowledge protocols. Instance-dependent commitment schemes for a given language are commitment schemes that can depend on the instance of the language, and where the hiding and binding properties are required to hold only on the YES and NO instances of the language, respectively. -- Computational zero knowledge and computational soundness (a property held by argument systems) are symmetric properties. Namely, we show that the class of languages in NP intersect co-NP having zero-knowledge arguments is closed under complement, and that a language in NP has a statistical zero-knowledge **argument** system if and only if its complement has a **computational** zero-knowledge proof system. -- A method of transforming any zero-knowledge protocol that is secure only against an honest verifier that follows the prescribed protocol into one that is secure against malicious verifiers. In addition, our transformation gives us protocols with desirable properties like having public coins, being black-box simulatable, and having an efficient prover. The novelty of our results above is that they are **unconditional**, meaning that they do not rely on any unproven complexity assumptions such as the existence of one-way functions. Moreover, in establishing our complexity-theoretic characterizations, we give the first construction of statistical zero-knowledge argument systems for NP based on any one-way function