We show that algebraic proofs are hard to find: Given an unsatisfiable CNF formula $F$, it is NP-hard to find a refutation of $F$ in the Nullstellensatz, Polynomial Calculus, or Sherali--Adams proof systems in time polynomial in the size of the shortest such refutation. Our work extends, and gives a ... more >>>

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 $t+1$ ... more >>>

We significantly strengthen and generalize the theorem lifting Nullstellensatz degree to monotone span program size by Pitassi and Robere (2018) so that it works for any gadget with high enough rank, in particular, for useful gadgets such as equality and greater-than. We apply our generalized theorem to solve two open ... more >>>

We show exponential lower bounds on resolution proof length for pigeonhole principle (PHP) formulas and perfect matching formulas over highly unbalanced, sparse expander graphs, thus answering the challenge to establish strong lower bounds in the regime between balanced constant-degree expanders as in [Ben-Sasson and Wigderson '01] and highly unbalanced, dense ... more >>>

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 ... more >>>

We prove near-optimal trade-offs for quantifier depth versus number of variables in first-order logic by exhibiting pairs of n-element structures that can be distinguished by a k-variable first-order sentence but where every such sentence requires quantifier depth at least n^?(k/log k). Our trade-offs also apply to first-order counting logic, and ... more >>>

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 ... more >>>

We exhibit families of 4-CNF formulas over n variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) d but require SOS proofs of size n^{Omega(d)} for values of d = d(n) from constant all the way up to n^{delta} for some universal constant delta. This shows that ... more >>>

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. ... more >>>

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 >>>

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 >>>

Properties of Boolean functions on the hypercube that are invariant
with respect to linear transformations of the domain are among some of
the most well-studied properties in the context of property testing.
In this paper, we study a particular natural class of linear-invariant
properties, called matroid freeness properties. These properties ... 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 >>>

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 >>>

In the context of proving lower bounds on proof space in $k$-DNF
resolution, [Ben-Sasson and Nordström 2009] introduced the concept of
minimally unsatisfiable sets of $k$-DNF formulas and proved that a
minimally unsatisfiable $k$-DNF set with $m$ formulas can have at most
$O((mk)^{k+1})$ variables. They also gave an example 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 >>>

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 >>>

A number of works have looked at the relationship between length and space of resolution proofs. A notorious question has been whether the existence of a short proof implies the existence of a proof that can be verified using limited space.

In this paper we resolve the question by answering ... 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 >>>

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 >>>

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 >>>