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 and resolution-based proof systems, and it is probably fair to say that resolution is 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 with m pigeons and
n holes. These formulas have width O(log(n)), and so this is an
exponential improvement over [Alekhnovich et al.'02] measured in
the width of the formulas.
3. We then present another encoding of a version 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 a 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