11 research outputs found
Total Space in Resolution Is at Least Width Squared
Given an unsatisfiable k-CNF formula phi we consider two complexity measures in Resolution: width and total space. The width is the minimal W such that there exists a Resolution refutation of phi with clauses of at most W literals. The total space is the minimal size T of a memory used to write down a Resolution refutation of phi where the size of the memory is measured as the total number of literals it can contain. We prove that T = Omega((W - k)^2)
From Small Space to Small Width in Resolution
In 2003, Atserias and Dalmau resolved a major open question about the
resolution proof system by establishing that the space complexity of CNF
formulas is always an upper bound on the width needed to refute them. Their
proof is beautiful but somewhat mysterious in that it relies heavily on tools
from finite model theory. We give an alternative, completely elementary proof
that works by simple syntactic manipulations of resolution refutations. As a
by-product, we develop a "black-box" technique for proving space lower bounds
via a "static" complexity measure that works against any resolution
refutation---previous techniques have been inherently adaptive. We conclude by
showing that the related question for polynomial calculus (i.e., whether space
is an upper bound on degree) seems unlikely to be resolvable by similar
methods
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
A Super-Polynomial Separation Between Resolution and Cut-Free Sequent Calculus
We show a quadratic separation between resolution and cut-free sequent calculus width. We use this gap to get, for the first time, first, a super-polynomial separation between resolution and cut-free sequent calculus for refuting CNF formulas, and secondly, a quadratic separation between resolution width and monomial space in polynomial calculus with resolution. Our super-polynomial separation between resolution and cut-free sequent calculus only applies when clauses are seen as disjunctions of unbounded arity; our examples have linear size cut-free sequent calculus proofs writing, in a particular way, their clauses using binary disjunctions. Interestingly, this shows that the complexity of sequent calculus depends on how disjunctions are represented
Understanding Space in Proof Complexity: Separations and Trade-offs via Substitutions
For current state-of-the-art DPLL SAT-solvers the two main bottlenecks are
the amounts of time and memory used. In proof complexity, these resources
correspond to the length and space of resolution proofs. There has been a long
line of research investigating these proof complexity measures, but while
strong results have been established for length, our understanding of space and
how it relates to length has remained quite poor. In particular, the question
whether resolution proofs can be optimized for length and space simultaneously,
or whether there are trade-offs between these two measures, has remained
essentially open.
In this paper, we remedy this situation by proving a host of length-space
trade-off results for resolution. Our collection of trade-offs cover almost the
whole range of values for the space complexity of formulas, and most of the
trade-offs are superpolynomial or even exponential and essentially tight. Using
similar techniques, we show that these trade-offs in fact extend to the
exponentially stronger k-DNF resolution proof systems, which operate with
formulas in disjunctive normal form with terms of bounded arity k. We also
answer the open question whether the k-DNF resolution systems form a strict
hierarchy with respect to space in the affirmative.
Our key technical contribution is the following, somewhat surprising,
theorem: Any CNF formula F can be transformed by simple variable substitution
into a new formula F' such that if F has the right properties, F' can be proven
in essentially the same length as F, whereas on the other hand the minimal
number of lines one needs to keep in memory simultaneously in any proof of F'
is lower-bounded by the minimal number of variables needed simultaneously in
any proof of F. Applying this theorem to so-called pebbling formulas defined in
terms of pebble games on directed acyclic graphs, we obtain our results.Comment: This paper is a merged and updated version of the two ECCC technical
reports TR09-034 and TR09-047, and it hence subsumes these two report
Lifting with Simple Gadgets and Applications to Circuit and Proof Complexity
We significantly strengthen and generalize the theorem lifting
Nullstellensatz degree to monotone span program size by Pitassi and Robere
(2018) so that it works for any gadget with high enough rank, in particular,
for useful gadgets such as equality and greater-than. We apply our generalized
theorem to solve two open problems:
* We present the first result that demonstrates a separation in proof power
for cutting planes with unbounded versus polynomially bounded coefficients.
Specifically, we exhibit CNF formulas that can be refuted in quadratic length
and constant line space in cutting planes with unbounded coefficients, but for
which there are no refutations in subexponential length and subpolynomial line
space if coefficients are restricted to be of polynomial magnitude.
* We give the first explicit separation between monotone Boolean formulas and
monotone real formulas. Specifically, we give an explicit family of functions
that can be computed with monotone real formulas of nearly linear size but
require monotone Boolean formulas of exponential size. Previously only a
non-explicit separation was known.
An important technical ingredient, which may be of independent interest, is
that we show that the Nullstellensatz degree of refuting the pebbling formula
over a DAG G over any field coincides exactly with the reversible pebbling
price of G. In particular, this implies that the standard decision tree
complexity and the parity decision tree complexity of the corresponding
falsified clause search problem are equal