[Proof Complexity] New paper

Neil Thapen thapen at math.cas.cz
Tue Nov 4 10:58:16 CET 2014

Dear colleagues,

A new paper, "The space complexity of cutting planes refutations", by 
Nicola Galesi, Pavel Pudlak and myself, is available on ECCC, at
Any comments or corrections are appreciated.

The abstract is: We study the space complexity of the cutting planes proof 
system, in which the lines in a proof are integral linear inequalities. We 
measure the space used by a refutation as the number of inequalities that 
need to be kept on a blackboard while verifying it. We show that any 
unsatisfiable set of inequalities has a cutting planes refutation in space 
five. This is in contrast to the weaker resolution proof system, for which 
the analogous space measure has been well-studied and many optimal lower 
bounds are known.

Motivated by this result we consider a natural restriction of cutting 
planes, in which all coefficients have size bounded by a constant. We show 
that there is a CNF which requires super-constant space to refute in this 
system. The system nevertheless already has an exponential speed-up over 
resolution with respect to size, and we additionally show that it is 
stronger than resolution with respect to space, by constructing 
constant-space cutting planes proofs of the pigeonhole principle with 
coefficients bounded by two.

We also consider variable space for cutting planes, where we count the 
number of instances of variables on the blackboard, and total space, where 
we count the total number of symbols.



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