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 $\Delta$, ... more >>>

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

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 ... more >>>

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 >>>

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 ... more >>>

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 >>>

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 ... more >>>

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 ... more >>>

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 ... more >>>

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 ... more >>>

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 ... more >>>

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 >>>