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 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
  complexity.