Eli Ben-Sasson

Short proofs are narrow—resolution made simple

The widthof a Resolution proof is defined to be the maximal number of literals in any clause of the proof. In this paper, we relate proof width to proof length (=size), in both general Resolution, and its tree-like variant. The following consequences of these relations reveal width as a crucial “resource” of Resolution proofs. In one direction, the relations allow us to give simple, unified proofs for almost all known exponential lower bounds on size of resolution proofs, as well as several interesting new ones. They all follow from width lower bounds, and we show how these follow from natural expansion property of clauses of the input tautology. In the other direction, the width-size relations naturally suggest a simple dynamic programming procedure for automated theorem proving—one which simply searches for small width proofs. This relation guarantees that the runnuing time (and thus the size of the produced proof) is at most quasi-polynomial in the smallest tree-like proof. This algorithm is never much worse than any of the recursive automated provers (such as DLL) used in practice. In contrast, we present a family of tautologies on which it is exponentially faster.

Robust PCPs of Proximity, Shorter PCPs, and Applications to Coding

We continue the study of the trade-off between the length of probabilistically checkable proofs (PCPs) and their query complexity, establishing the following main results (which refer to proofs of satisfiability of circuits of size

SNARKs for C : verifying program executions succinctly and in zero knowledge

n

Some 3CNF Properties Are Hard to Test

): 1. We present PCPs of length

Randomness-efficient low degree tests and short PCPs via epsilon-biased sets

\exp(o(\log\log n)^2)\cdot n

Near Optimal Separation Of Tree-Like And General Resolution

that can be verified by making

Short PCPs with Polylog Query Complexity

o(\log\log n)

Space complexity of random formulae in resolution

Boolean queries. 2. For every \epsilon>0, we present PCPs of length

Short Proofs May Be Spacious: An Optimal Separation of Space and Length in Resolution

\exp(\log^\epsilon n)\cdot n

Size-Space Tradeoffs for Resolution

that can be verified by making a constant number of Boolean queries. In both cases, false assertions are rejected with constant probability (which may be set to be arbitrarily close to 1). The multiplicative overhead on the length of the proof, introduced by transforming a proof into a probabilistically checkable one, is just quasi polylogarithmic in the first case (of query complexity