Nathan Segerlind

We demonstrate a family of propositional formulas in conjunctive normal form

so that a formula of size $N$ requires size $2^{\Omega(\sqrt[7]{N/logN})}$

to refute using the tree-like OBDD refutation system of

Atserias, Kolaitis and Vardi

with respect to all variable orderings.

All known symbolic quantifier elimination algorithms for satisfiability

generate ...
more >>>

Rahul Santhanam, Srikanth Srinivasan

Impagliazzo, Paturi and Zane (JCSS 2001) proved a sparsification lemma for $k$-CNFs:

every k-CNF is a sub-exponential size disjunction of $k$-CNFs with a linear

number of clauses. This lemma has subsequently played a key role in the study

of the exact complexity of the satisfiability problem. A natural question is

more >>>

Rahul Santhanam

I discuss recent progress in developing and exploiting connections between

SAT algorithms and circuit lower bounds. The centrepiece of the article is

Williams' proof that $NEXP \not \subseteq ACC^0$, which proceeds via a new

algorithm for $ACC^0$-SAT beating brute-force search. His result exploits

a formal connection from non-trivial SAT algorithms ...
more >>>

Rahul Santhanam, Ryan Williams

We revisit the complexity of the satisfiability problem for quantified Boolean formulas. We show that satisfiability

of quantified CNFs of size $\poly(n)$ on $n$ variables with $O(1)$

quantifier blocks can be solved in time $2^{n-n^{\Omega(1)}}$ by zero-error

randomized algorithms. This is the first known improvement over brute force search in ...
more >>>

Ruiwen Chen, Rahul Santhanam, Srikanth Srinivasan

We show average-case lower bounds for explicit Boolean functions against bounded-depth threshold circuits with a superlinear number of wires. We show that for each integer d > 1, there is \epsilon_d > 0 such that Parity has correlation at most 1/n^{\Omega(1)} with depth-d threshold circuits which have at most

n^{1+\epsilon_d} ...
more >>>

Igor Carboni Oliveira, Ruiwen Chen, Rahul Santhanam

In a seminal work, Williams [Wil14] showed that NEXP (non-deterministic exponential time) does not have polynomial-size ACC^0 circuits. Williams' technique inherently gives a worst-case lower bound, and until now, no average-case version of his result was known.

We show that there is a language L in NEXP (resp. EXP^NP) ... more >>>

Vikraman Arvind, Venkatesan Guruswami

We introduce the problem of finding a satisfying assignment to a CNF formula that must further belong to a prescribed input subspace. Equivalent formulations of the problem include finding a point outside a union of subspaces (the Union-of-Subspace Avoidance (USA) problem), and finding a common zero of a system of ... more >>>

Bruno Pasqualotto Cavalar, Zhenjian Lu

Comparator circuits are a natural circuit model for studying bounded fan-out computation whose power sits between nondeterministic branching programs and general circuits. Despite having been studied for nearly three decades, the first superlinear lower bound against comparator circuits was proved only recently by Gál and Robere (ITCS 2020), who established ... more >>>