TY - GEN

T1 - Space complexity in polynomial calculus

AU - Filmus, Yuval

AU - Lauria, Massimo

AU - Nordström, Jakob

AU - Thapen, Neil

AU - Ron-Zewi, Noga

PY - 2012

Y1 - 2012

N2 - 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. For the polynomial calculus proof system, the only previously known space lower bound is for CNF formulas of unbounded width in [Alekhnovich et. al.'02], where the 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 refute any k-CNF formula in constant space. 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]. - For PCR, we prove an Omega(n) space lower bound for a bit wise encoding of the functional pigeonhole principle with m pigeons and n holes. 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. - 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. - 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. - 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 known to be 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.

AB - 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. For the polynomial calculus proof system, the only previously known space lower bound is for CNF formulas of unbounded width in [Alekhnovich et. al.'02], where the 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 refute any k-CNF formula in constant space. 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]. - For PCR, we prove an Omega(n) space lower bound for a bit wise encoding of the functional pigeonhole principle with m pigeons and n holes. 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. - 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. - 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. - 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 known to be 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.

KW - k-CNF

KW - pcr

KW - polynomial calculus

KW - proof complexity

KW - resolution

KW - space

UR - http://www.scopus.com/inward/record.url?scp=84866503368&partnerID=8YFLogxK

U2 - 10.1109/CCC.2012.27

DO - 10.1109/CCC.2012.27

M3 - Article in proceedings

AN - SCOPUS:84866503368

SN - 9780769547084

T3 - Proceedings of the Annual IEEE Conference on Computational Complexity

SP - 334

EP - 344

BT - Proceedings - 2012 IEEE 27th Conference on Computational Complexity, CCC 2012

T2 - IEEE Computer Society Technical Committee on Mathematical Foundations of Computing

Y2 - 26 June 2012 through 29 June 2012

ER -