Space proof complexity for random $3$CNFs via a $(2\epsilon)$Hall's Theorem
Abstract
We investigate the space complexity of refuting $3$CNFs in Resolution and algebraic systems. No lower bound for refuting any family of $3$CNFs was previously known for the total space in resolution or for the monomial space in algebraic systems. We prove that every Polynomial Calculus with Resolution refutation of a random $3$CNF $\phi$ in $n$ variables requires, with high probability, $\Omega(n/\log n)$ distinct monomials to be kept simultaneously in memory. The same construction also proves that every Resolution refutation $\phi$ requires, with high probability, $\Omega(n/\log n)$ clauses each of width $\Omega(n/\log n)$ to be kept at the same time in memory. This gives a $\Omega(n^2/\log^2 n)$ lower bound for the total space needed in Resolution to refute $\phi$. The main technical innovation is a variant of Hall's theorem. We show that in bipartite graphs $G$ with bipartition $(L,R)$ and leftdegree at most 3, $L$ can be covered by certain families of disjoint paths, called $(2,4)$matchings, provided that $L$ expands in $R$ by a factor of $(2\epsilon)$, for $\epsilon < \frac{1}{23}$.
 Publication:

arXiv eprints
 Pub Date:
 November 2014
 DOI:
 10.48550/arXiv.1411.1619
 arXiv:
 arXiv:1411.1619
 Bibcode:
 2014arXiv1411.1619B
 Keywords:

 Computer Science  Computational Complexity;
 Mathematics  Combinatorics