[Proof Complexity] Paper announcement: Space Complexity in Polynomial Calculus

[Jakob Nordstrom]

Dear colleagues,

This is just to inform you that we have posted a full-length version at
http://eccc.hpi-web.de/report/2012/132/ of the paper "Space Complexity in
Polynomial Calculus" by Yuval Filmus, Massimo Lauria, Noga Ron-Zewi, Neil
Thapen and myself that appeared in CCC '12.

The abstract follows below. Any comments or questions are more than welcome.

Best regards,
Jakob Nordstrom

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ABSTRACT

During the last decade, an active line of research in proof complexity has
been to study space complexity and time-space trade-offs for proofs.
Besides being a natural complexity measure of intrinsic interest, space is
also an important issue in SAT solving, and so research has mostly focused
on weak systems that are used by SAT solvers.

There has been a relatively long sequence of papers on space in
resolution, which is now reasonably well understood from this point of
view.  For other natural candidates to study, however, such as polynomial
calculus or cutting planes, very little has been known. We are not aware
of any nontrivial space lower bounds for cutting planes, and for
polynomial calculus the only lower bound has been for CNF formulas of
unbounded width in [Alekhnovich et al. '02], where the space lower bound
is smaller than the initial width of the clauses in the formulas.  Thus,
in particular, it has been consistent with current knowledge that
polynomial calculus could be able to refute any k-CNF formula in constant
space.

In this paper, we prove several new results on space in polynomial
calculus (PC), and in the extended proof system polynomial calculus
resolution (PCR) studied in [Alekhnovich et al. '02]:

(1) We prove an Omega(n) space lower bound in PC for the canonical 3-CNF
version of the pigeonhole principle formulas PHP^m_n with m pigeons and n
holes, and show that this is tight.

(2) For PCR, we prove an Omega(n) space lower bound for a bitwise encoding
of the functional pigeonhole principle.  These formulas have width O(log
n), and hence this is an exponential improvement over [Alekhnovich et al.
'02] measured in the width of the formulas.

(3) We then present another encoding of the pigeonhole principle that has
constant width, and prove an Omega(n) space lower bound in PCR for these
formulas as well.

(4) Finally, we prove that any k-CNF formula can be refuted in PC in
simultaneous exponential size and linear space (which holds for resolution
and thus for PCR, but was not obviously the case for PC). We also
characterize a natural class of CNF formulas for which the space
complexity in resolution and PCR does not change when the formula is
transformed into 3-CNF in the canonical way, something that we believe can
be useful when proving PCR space lower bounds for other well-studied
formula families in proof complexity.


Jakob Nordström, Assistant Professor
KTH Royal Institute of Technology
Osquars backe 2, SE-100 44 Stockholm, Sweden
Phone: +46 8 790 69 19 (office), +46 70 742 21 98 (cell)
http://www.csc.kth.se/~jakobn/