[Proof Complexity] Preprint

thapen thapen at math.cas.cz
Wed Apr 10 16:48:38 CEST 2019

Dear colleagues,

Nicola Galesi, Leszek Kolodziejczyk and I have a new preprint, 
"Polynomial calculus space and resolution width".

It is on ECCC: https://eccc.weizmann.ac.il/report/2019/052/

The abstract is: We show that if a k-CNF requires width w to refute in 
resolution, then it requires space \sqrt{w} to refute in polynomial 
calculus, where the space of a polynomial calculus refutation is the 
number of monomials that must be kept in memory when working through the 
proof. This is the first analogue, in polynomial calculus, of Atserias 
and Dalmau's result lower-bounding clause space in resolution by 
resolution width.

As a by-product of our new approach to space lower bounds we give a 
simple proof of Bonacina's recent result that total space in resolution 
(the total number of variable occurrences that must be kept in memory) 
is lower-bounded by the width squared. As corollaries of the main result 
we obtain some new lower bounds on the PCR space needed to refute 
specific formulas, as well as partial answers to some open problems 
about relations between space, size, and degree for polynomial calculus.

Comments appreciated,



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