Eli Ben-Sasson, Nicola Galesi

We study the space complexity of refuting unsatisfiable random $k$-CNFs in

the Resolution proof system. We prove that for any large enough $\Delta$,

with high probability a random $k$-CNF over $n$ variables and $\Delta n$

clauses requires resolution clause space of

$\Omega(n \cdot \Delta^{-\frac{1+\epsilon}{k-2-\epsilon}})$,

for any $0<\epsilon<1/2$. For constant $\Delta$, ...
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Michele Zito

We prove that, with high probability, the space complexity of refuting

a random unsatisfiable boolean formula in $k$-CNF on $n$

variables and $m = \Delta n$ clauses is

$O(n \cdot \Delta^{-\frac{1}{k-2}})$.

Juan Luis Esteban, Jacobo Toran

We show that the Player-Adversary game from a paper

by Pudlak and Impagliazzo played over

CNF propositional formulas gives

an exact characterization of the space needed

in treelike resolution refutations. This

characterization is purely combinatorial

and independent of the notion of resolution.

We use this characterization to give ...
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Jakob Nordström

The width of a resolution proof is the maximal number of literals in any clause of the proof. The space of a proof is the maximal number of memory cells used if the proof is only allowed to resolve on clauses kept in memory. Both of these measures have previously ... more >>>

Oded Lachish, Ilan Newman, Asaf Shapira

Combinatorial property testing deals with the following relaxation

of decision problems: Given a fixed property and an input $x$, one

wants to decide whether $x$ satisfies the property or is ``far''

from satisfying it. The main focus of property testing is in

identifying large families of properties that can be ...
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Philipp Hertel

Satisfiability algorithms have become one of the most practical and successful approaches for solving a variety of real-world problems, including hardware verification, experimental design, planning and diagnosis problems. The main reason for the success is due to highly optimized algorithms for SAT based on resolution. The most successful of these ... more >>>

Jakob Nordström

We present a greatly simplified proof of the length-space

trade-off result for resolution in Hertel and Pitassi (2007), and

also prove a couple of other theorems in the same vein. We point

out two important ingredients needed for our proofs to work, and

discuss possible conclusions to be drawn regarding ...
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Jakob Nordström, Johan Hastad

Most state-of-the-art satisfiability algorithms today are variants of

the DPLL procedure augmented with clause learning. The main bottleneck

for such algorithms, other than the obvious one of time, is the amount

of memory used. In the field of proof complexity, the resources of

time and memory correspond to the length ...
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Eli Ben-Sasson, Jakob Nordström

For current state-of-the-art satisfiability algorithms based on the

DPLL procedure and clause learning, the two main bottlenecks are the

amounts of time and memory used. Understanding time and memory

consumption, and how they are related to one another, is therefore a

question of considerable practical importance. In the field of ...
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Eli Ben-Sasson, Jakob Nordström

The k-DNF resolution proof systems are a family of systems indexed by

the integer k, where the kth member is restricted to operating with

formulas in disjunctive normal form with all terms of bounded arity k

(k-DNF formulas). This family was introduced in [Krajicek 2001] as an

extension of the ...
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Jakob Nordström

The last decade has seen a revival of interest in pebble games in the

context of proof complexity. Pebbling has proven to be 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 ...
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Eli Ben-Sasson, Jakob Nordström

For current state-of-the-art satisfiability algorithms based on the DPLL procedure and clause learning, the two main bottlenecks are the amounts of time and memory used. In the field of proof complexity, these resources correspond to the length and space of resolution proofs for formulas in conjunctive normal form (CNF). There ... more >>>

Yuval Filmus, Massimo Lauria, Jakob Nordström, Noga Ron-Zewi, Neil Thapen

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 ... more >>>

Yuval Filmus, Massimo Lauria, Mladen Mikša, Jakob Nordström, Marc Vinyals

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 ... more >>>

Nicola Galesi, Pavel Pudlak, Neil Thapen

We study the space complexity of the cutting planes proof system, in which the lines in a proof are integral linear inequalities. We measure the space used by a refutation as the number of inequalities that need to be kept on a blackboard while verifying it. We show that any ... more >>>

Chin Ho Lee, Emanuele Viola

We exhibit $\epsilon$-biased distributions $D$

on $n$ bits and functions $f\colon \{0,1\}^n

\to \{0,1\}$ such that the xor of two independent

copies ($D+D$) does not fool $f$, for any of the

following choices:

1. $\epsilon = 2^{-\Omega(n)}$ and $f$ is in P/poly;

2. $\epsilon = 2^{-\Omega(n/\log n)}$ and $f$ is ... more >>>

Christoph Berkholz, Jakob Nordström

We show that there are CNF formulas which can be refuted in resolution

in both small space and small width, but for which any small-width

proof must have space exceeding by far the linear worst-case upper

bound. This significantly strengthens the space-width trade-offs in

[Ben-Sasson '09]}, and provides one more ...
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Nicola Galesi, Leszek Kolodziejczyk, Neil Thapen

We show that if a $k$-CNF requires width $w$ to refute in resolution, then it requires space $\sqrt w$ to refute in polynomial calculus, where the space of a polynomial calculus refutation is the number of monomials that must be kept in memory when working through the proof. This is ... more >>>