45 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
Hardness of Approximation in PSPACE and Separation Results for Pebble Games
We consider the pebble game on DAGs with bounded fan-in introduced in
[Paterson and Hewitt '70] and the reversible version of this game in [Bennett
'89], and study the question of how hard it is to decide exactly or
approximately the number of pebbles needed for a given DAG in these games. We
prove that the problem of eciding whether ~pebbles suffice to reversibly
pebble a DAG is PSPACE-complete, as was previously shown for the standard
pebble game in [Gilbert, Lengauer and Tarjan '80]. Via two different graph
product constructions we then strengthen these results to establish that both
standard and reversible pebbling space are PSPACE-hard to approximate to within
any additive constant. To the best of our knowledge, these are the first
hardness of approximation results for pebble games in an unrestricted setting
(even for polynomial time). Also, since [Chan '13] proved that reversible
pebbling is equivalent to the games in [Dymond and Tompa '85] and [Raz and
McKenzie '99], our results apply to the Dymond--Tompa and Raz--McKenzie games
as well, and from the same paper it follows that resolution depth is
PSPACE-hard to determine up to any additive constant. We also obtain a
multiplicative logarithmic separation between reversible and standard pebbling
space. This improves on the additive logarithmic separation previously known
and could plausibly be tight, although we are not able to prove this. We leave
as an interesting open problem whether our additive hardness of approximation
result could be strengthened to a multiplicative bound if the computational
resources are decreased from polynomial space to the more common setting of
polynomial time
Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling
We establish an exactly tight relation between reversible pebblings of graphs
and Nullstellensatz refutations of pebbling formulas, showing that a graph
can be reversibly pebbled in time and space if and only if there is a
Nullstellensatz refutation of the pebbling formula over in size and
degree (independently of the field in which the Nullstellensatz refutation
is made). We use this correspondence to prove a number of strong size-degree
trade-offs for Nullstellensatz, which to the best of our knowledge are the
first such results for this proof system
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
Linear-time algorithms for testing the satisfiability of propositional horn formulae
AbstractNew algorithms for deciding whether a (propositional) Horn formula is satisfiable are presented. If the Horn formula A contains K distinct propositional letters and if it is assumed that they are exactly P1,âŠ, PK, the two algorithms presented in this paper run in time O(N), where N is the total number of occurrences of literals in A. By representing a Horn proposition as a graph, the satisfiability problem can be formulated as a data flow problem, a certain type of pebbling. The difference between the two algorithms presented here is the strategy used for pebbling the graph. The first algorithm is based on the principle used for finding the set of nonterminals of a context-free grammar from which the empty string can be derived. The second algorithm is a graph traversal and uses a âcall-by-needâ strategy. This algorithm uses an attribute grammar to translate a propositional Horn formula to its corresponding graph in linear time. Our formulation of the satisfiability problem as a data flow problem appears to be new and suggests the possibility of improving efficiency using parallel processors
LIPIcs
We study space complexity and time-space trade-offs with a focus not on peak memory usage but on overall memory consumption throughout the computation. Such a cumulative space measure was introduced for the computational model of parallel black pebbling by [Alwen and Serbinenko â15] as a tool for obtaining results in cryptography. We consider instead the non- deterministic black-white pebble game and prove optimal cumulative space lower bounds and trade-offs, where in order to minimize pebbling time the space has to remain large during a significant fraction of the pebbling. We also initiate the study of cumulative space in proof complexity, an area where other space complexity measures have been extensively studied during the last 10â15 years. Using and extending the connection between proof complexity and pebble games in [Ben-Sasson and Nordström â08, â11] we obtain several strong cumulative space results for (even parallel versions of) the resolution proof system, and outline some possible future directions of study of this, in our opinion, natural and interesting space measure
Covering Numbers of the Cubes
How many triangles does it take to make a square? The answer is simple: two. This problem has a direct analogue in dimensions three and higher, but the answers are much harder to find. We provide new lower bounds in dimensions 4 through 13, an asymptotic lower bound which is inferior to the best known bound in high dimensions, and some new ideas which produce good upper bounds in both low and high dimensions
Understanding space in resolution: optimal lower bounds and exponential trade-offs
We continue the study of tradeoffs between space and length of
resolution proofs and focus on two new results:
begin{enumerate}
item
We show that length and space in resolution are uncorrelated. This
is proved by exhibiting families of CNF formulas of size that
have proofs of length but require space . Our
separation is the strongest possible since any proof of length
can always be transformed into a proof in space , and
improves previous work reported in [Nordstr"{o}m 2006, Nordstr"{o}m and
H{aa}stad 2008].
item We prove a number of trade-off results for space in the range
from constant to , most of them superpolynomial or even
exponential. This is a dramatic improvement over previous results in
[Ben-Sasson 2002, Hertel and Pitassi 2007, Nordstr"{o}m 2007].
end{enumerate}
The key to our results is the following, somewhat surprising, theorem:
Any CNF formula can be transformed by simple substitution
transformation into a new formula such that if has the right
properties, can be proven in resolution in essentially the same
length as but the minimal space needed for is lower-bounded
by the number of variables that have to be mentioned simultaneously in
any proof for . Applying this theorem to so-called pebbling
formulas defined in terms of pebble games over directed acyclic graphs
and analyzing black-white pebbling on these graphs yields our results
Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling
We establish an exactly tight relation between reversible pebblings of graphs and Nullstellensatz refutations of pebbling formulas, showing that a graph G can be reversibly pebbled in time t and space s if and only if there is a Nullstellensatz refutation of the pebbling formula over G in size t+1 and degree s (independently of the field in which the Nullstellensatz refutation is made). We use this correspondence to prove a number of strong size-degree trade-offs for Nullstellensatz, which to the best of our knowledge are the first such results for this proof system