43 research outputs found
Random Resolution Refutations
We study the random resolution refutation system definedin [Buss et al. 2014]. This attempts to capture the notion of a resolution refutation that may make mistakes but is correct most of the time. By proving the equivalence of several different definitions, we show that this concept is robust. On the other hand, if P does not equal NP, then random resolution cannot be polynomially simulated by any proof system in which correctness of proofs is checkable in polynomial time.
We prove several upper and lower bounds on the width and size of random resolution refutations of explicit and random unsatisfiable CNF formulas. Our main result is a separation between polylogarithmic width random resolution and quasipolynomial size resolution, which solves the problem stated in [Buss et al. 2014]. We also prove exponential size lower bounds on random resolution refutations of the pigeonhole principle CNFs, and of a family of CNFs which have polynomial size refutations in constant depth Frege
Space complexity in polynomial calculus
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 Ω(n) space lower bound in PC for the canonical 3-CNF version of the pigeonhole
principle formulas PHPm
n with m pigeons and n holes, and show that this is tight.
2. For PCR, we prove an Ω(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 Ω(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
The weak pigeonhole principle in models of bounded arithmetic
Available from British Library Document Supply Centre- DSC:DN056606 / BLDSC - British Library Document Supply CentreSIGLEGBUnited Kingdo
The strength of replacement in weak arithmetic
The replacement (or collection or choice) axiom scheme BB(Γ) asserts bounded quantifier exchange as follows: ∀i<|a | ∃x<aφ(i,x) → ∃w ∀i<|a|φ(i,[w]i) proves the scheme where φ is in the class Γ of formulas. The theory S1 2 BB(Σb 1), and thus in S1 2 every Σb1 formula is equivalent to a strict Σb1 formula (in which all non-sharply-bounded quantifiers are in front). Here we prove (sometimes subject to an assumption) that certain theories weaker than S1 2 do not prove either BB(Σb1) or BB(Σb0). We show (unconditionally) that V 0 does not prove BB(ΣB 0), where V 0 (essentially IΣ 1,b 0) is the two-sorted theory associated with the complexity class AC0. We show that PV does not prove BB(Σb 0), assumin
On semantic cutting planes with very small coefficients
Cutting planes proofs for integer programs can naturally be defined both in a syntactic and in a semantic fashion. Filmus et al. (STACS 2016) proved that semantic cutting planes proofs may be exponentially stronger than syntactic ones, even if they use the semantic rule only once. We show that when semantic cutting planes proofs are restricted to have coefficients bounded by a function growing slowly enough, syntactic cutting planes can simulate them efficiently. Furthermore if we strengthen the restriction to a constant bound, then the simulating syntactic proof even has polynomially small coefficients
Digital Object Identifier (DOI):
Abstract. We investigate the theories LA, LAP, ∀LAP of linear algebra, which were originally defined to study the question of whether commutativity of matrix inverses has polysize Frege proofs. We give sentences separating quantified versions of these theories, and define a fragment ∃LA of ∀LAP in which we can interpret a weak theory V 1 of bounded arithmetic and carry out polynomial time reasoning about matrices- for example, we can formalize the Gaussian elimination algorithm. We show that, even if we restrict our language, ∃LA proves the commutativity of inverses. 1