112 research outputs found
On the Relative Strength of Pebbling and Resolution
The last decade has seen a revival of interest in pebble games in the context
of proof complexity. Pebbling has proven a useful tool for studying
resolution-based proof systems when comparing the strength of different
subsystems, showing bounds on proof space, and establishing size-space
trade-offs. The typical approach has been to encode the pebble game played on a
graph as a CNF formula and then argue that proofs of this formula must inherit
(various aspects of) the pebbling properties of the underlying graph.
Unfortunately, the reductions used here are not tight. To simulate resolution
proofs by pebblings, the full strength of nondeterministic black-white pebbling
is needed, whereas resolution is only known to be able to simulate
deterministic black pebbling. To obtain strong results, one therefore needs to
find specific graph families which either have essentially the same properties
for black and black-white pebbling (not at all true in general) or which admit
simulations of black-white pebblings in resolution. This paper contributes to
both these approaches. First, we design a restricted form of black-white
pebbling that can be simulated in resolution and show that there are graph
families for which such restricted pebblings can be asymptotically better than
black pebblings. This proves that, perhaps somewhat unexpectedly, resolution
can strictly beat black-only pebbling, and in particular that the space lower
bounds on pebbling formulas in [Ben-Sasson and Nordstrom 2008] are tight.
Second, we present a versatile parametrized graph family with essentially the
same properties for black and black-white pebbling, which gives sharp
simultaneous trade-offs for black and black-white pebbling for various
parameter settings. Both of our contributions have been instrumental in
obtaining the time-space trade-off results for resolution-based proof systems
in [Ben-Sasson and Nordstrom 2009].Comment: Full-length version of paper to appear in Proceedings of the 25th
Annual IEEE Conference on Computational Complexity (CCC '10), June 201
A Generalized Method for Proving Polynomial Calculus Degree Lower Bounds
We study the problem of obtaining lower bounds for polynomial calculus (PC)
and polynomial calculus resolution (PCR) on proof degree, and hence by
[Impagliazzo et al. '99] also on proof size. [Alekhnovich and Razborov '03]
established that if the clause-variable incidence graph of a CNF formula F is a
good enough expander, then proving that F is unsatisfiable requires high PC/PCR
degree. We further develop the techniques in [AR03] to show that if one can
"cluster" clauses and variables in a way that "respects the structure" of the
formula in a certain sense, then it is sufficient that the incidence graph of
this clustered version is an expander. As a corollary of this, we prove that
the functional pigeonhole principle (FPHP) formulas require high PC/PCR degree
when restricted to constant-degree expander graphs. This answers an open
question in [Razborov '02], and also implies that the standard CNF encoding of
the FPHP formulas require exponential proof size in polynomial calculus
resolution. Thus, while Onto-FPHP formulas are easy for polynomial calculus, as
shown in [Riis '93], both FPHP and Onto-PHP formulas are hard even when
restricted to bounded-degree expanders.Comment: Full-length version of paper to appear in Proceedings of the 30th
Annual Computational Complexity Conference (CCC '15), June 201
Tight Size-Degree Bounds for Sums-of-Squares Proofs
We exhibit families of -CNF formulas over variables that have
sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank)
but require SOS proofs of size for values of from
constant all the way up to for some universal constant.
This shows that the running time obtained by using the Lasserre
semidefinite programming relaxations to find degree- SOS proofs is optimal
up to constant factors in the exponent. We establish this result by combining
-reductions expressible as low-degree SOS derivations with the
idea of relativizing CNF formulas in [Kraj\'i\v{c}ek '04] and [Dantchev and
Riis'03], and then applying a restriction argument as in [Atserias, M\"uller,
and Oliva '13] and [Atserias, Lauria, and Nordstr\"om '14]. This yields a
generic method of amplifying SOS degree lower bounds to size lower bounds, and
also generalizes the approach in [ALN14] to obtain size lower bounds for the
proof systems resolution, polynomial calculus, and Sherali-Adams from lower
bounds on width, degree, and rank, respectively
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
Narrow Proofs May Be Maximally Long
We prove that there are 3-CNF formulas over n variables that can be refuted
in resolution in width w but require resolution proofs of size n^Omega(w). This
shows that the simple counting argument that any formula refutable in width w
must have a proof in size n^O(w) is essentially tight. Moreover, our lower
bound generalizes to polynomial calculus resolution (PCR) and Sherali-Adams,
implying that the corresponding size upper bounds in terms of degree and rank
are tight as well. Our results do not extend all the way to Lasserre, however,
where the formulas we study have proofs of constant rank and size polynomial in
both n and w
Narrow proofs may be maximally long
We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size n(Omega(w)). This shows that the simple counting argument that any formula refutable in width w must have a proof in size n(O(w)) is essentially tight. Moreover, our lower bound generalizes to polynomial calculus resolution and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. The lower bound does not extend all the way to Lasserre, however, since we show that there the formulas we study have proofs of constant rank and size polynomial in both n and w.Peer ReviewedPostprint (author's final draft
Separations of Matroid Freeness Properties
Properties of Boolean functions on the hypercube invariant with respect to
linear transformations of the domain are among the most well-studied properties
in the context of property testing. In this paper, we study the fundamental
class of linear-invariant properties called matroid freeness properties. These
properties have been conjectured to essentially coincide with all testable
linear-invariant properties, and a recent sequence of works has established
testability for increasingly larger subclasses. One question left open,
however, is whether the infinitely many syntactically different properties
recently shown testable in fact correspond to new, semantically distinct ones.
This is a crucial issue since it has also been shown that there exist
subclasses of these properties for which an infinite set of syntactically
different representations collapse into one of a small, finite set of
properties, all previously known to be testable.
An important question is therefore to understand the semantics of matroid
freeness properties, and in particular when two syntactically different
properties are truly distinct. We shed light on this problem by developing a
method for determining the relation between two matroid freeness properties P
and Q. Furthermore, we show that there is a natural subclass of matroid
freeness properties such that for any two properties P and Q from this
subclass, a strong dichotomy must hold: either P is contained in Q or the two
properties are "well separated." As an application of this method, we exhibit
new, infinite hierarchies of testable matroid freeness properties such that at
each level of the hierarchy, there are functions that are far from all
functions lying in lower levels of the hierarchy. Our key technical tool is an
apparently new notion of maps between linear matroids, called matroid
homomorphisms, that might be of independent interest
Graph Colouring is Hard for Algorithms Based on Hilbert's Nullstellensatz and Gr\"{o}bner Bases
We consider the graph -colouring problem encoded as a set of polynomial
equations in the standard way over -valued variables. We prove that there
are bounded-degree graphs that do not have legal -colourings but for which
the polynomial calculus proof system defined in [Clegg et al '96, Alekhnovich
et al '02] requires linear degree, and hence exponential size, to establish
this fact. This implies a linear degree lower bound for any algorithms based on
Gr\"{o}bner bases solving graph -colouring using this encoding. The same
bound applies also for the algorithm studied in a sequence of papers [De Loera
et al '08,'09,'11,'15] based on Hilbert's Nullstellensatz proofs for a slightly
different encoding, thus resolving an open problem mentioned in [De Loera et al
'08,'09,'11] and [Li '16]. We obtain our results by combining the polynomial
calculus degree lower bound for functional pigeonhole principle (FPHP) formulas
over bounded-degree bipartite graphs in [Mik\v{s}a and Nordstr\"{o}m '15] with
a reduction from FPHP to -colouring derivable by polynomial calculus in
constant degree
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
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