52,756 research outputs found
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
Making proofs without Modus Ponens: An introduction to the combinatorics and complexity of cut elimination
This paper is intended to provide an introduction to cut elimination which is
accessible to a broad mathematical audience. Gentzen's cut elimination theorem
is not as well known as it deserves to be, and it is tied to a lot of
interesting mathematical structure. In particular we try to indicate some
dynamical and combinatorial aspects of cut elimination, as well as its
connections to complexity theory. We discuss two concrete examples where one
can see the structure of short proofs with cuts, one concerning feasible
numbers and the other concerning "bounded mean oscillation" from real analysis
Improved Soundness for QMA with Multiple Provers
We present three contributions to the understanding of QMA with multiple
provers:
1) We give a tight soundness analysis of the protocol of [Blier and Tapp,
ICQNM '09], yielding a soundness gap Omega(1/N^2). Our improvement is achieved
without the use of an instance with a constant soundness gap (i.e., without
using a PCP).
2) We give a tight soundness analysis of the protocol of [Chen and Drucker,
ArXiV '10], thereby improving their result from a monolithic protocol where
Theta(sqrt(N)) provers are needed in order to have any soundness gap, to a
protocol with a smooth trade-off between the number of provers k and a
soundness gap Omega(k^2/N), as long as k>=Omega(log N). (And, when
k=Theta(sqrt(N)), we recover the original parameters of Chen and Drucker.)
3) We make progress towards an open question of [Aaronson et al., ToC '09]
about what kinds of NP-complete problems are amenable to sublinear
multiple-prover QMA protocols, by observing that a large class of such examples
can easily be derived from results already in the PCP literature - namely, at
least the languages recognized by a non-deterministic RAMs in quasilinear time.Comment: 24 pages; comments welcom
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