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REPORTS > AUTHORS > JAKOB NORDSTRÖM:
All reports by Author Jakob Nordström:

TR24-185 | 21st November 2024
Susanna F. de Rezende, Noah Fleming, Duri Andrea Janett, Jakob Nordström, Shuo Pang

Truly Supercritical Trade-offs for Resolution, Cutting Planes, Monotone Circuits, and Weisfeiler-Leman

We exhibit supercritical trade-off for monotone circuits, showing that there are functions computable by small circuits for which any circuit must have depth super-linear or even super-polynomial in the number of variables, far exceeding the linear worst-case upper bound. We obtain similar trade-offs in proof complexity, where we establish the ... more >>>


TR21-006 | 18th January 2021
Susanna de Rezende, Jakob Nordström, Marc Vinyals

How Limited Interaction Hinders Real Communication (and What It Means for Proof and Circuit Complexity)

We obtain the first true size-space trade-offs for the cutting planes proof system, where the upper bounds hold for size and total space for derivations with constant-size coefficients, and the lower bounds apply to length and formula space (i.e., number of inequalities in memory) even for derivations with exponentially large ... more >>>


TR20-099 | 6th July 2020
Susanna de Rezende, Or Meir, Jakob Nordström, Toniann Pitassi, Robert Robere

KRW Composition Theorems via Lifting

Revisions: 1

One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}^1$). Karchmer, Raz, and Wigderson (Computational Complexity 5(3/4), 1995) suggested to approach this problem by proving that depth complexity behaves “as expected” with respect to the composition of functions $f ... more >>>


TR20-064 | 2nd May 2020
Mika Göös, Jakob Nordström, Toniann Pitassi, Robert Robere, Dmitry Sokolov, Susanna de Rezende

Automating Algebraic Proof Systems is NP-Hard

Revisions: 2

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 >>>


TR20-001 | 31st December 2019
Or Meir, Jakob Nordström, Robert Robere, Susanna de Rezende

Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling

Revisions: 2

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 >>>


TR19-186 | 31st December 2019
Or Meir, Jakob Nordström, Toniann Pitassi, Robert Robere, Susanna de Rezende

Lifting with Simple Gadgets and Applications to Circuit and Proof Complexity

Revisions: 4

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 >>>


TR19-174 | 2nd December 2019
Susanna de Rezende, Jakob Nordström, Kilian Risse, Dmitry Sokolov

Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs

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 >>>


TR16-203 | 21st December 2016
Christoph Berkholz, Jakob Nordström

Supercritical Space-Width Trade-offs for Resolution

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 >>>


TR16-135 | 31st August 2016
Christoph Berkholz, Jakob Nordström

Near-Optimal Lower Bounds on Quantifier Depth and Weisfeiler-Leman Refinement Steps

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 >>>


TR15-078 | 4th May 2015
Mladen Mikša, Jakob Nordström

A Generalized Method for Proving Polynomial Calculus Degree Lower Bounds

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 >>>


TR15-053 | 7th April 2015
Massimo Lauria, Jakob Nordström

Tight Size-Degree Bounds for Sums-of-Squares Proofs

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 >>>


TR14-118 | 9th September 2014
Albert Atserias, Massimo Lauria, Jakob Nordström

Narrow Proofs May Be Maximally Long

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 >>>


TR14-081 | 13th June 2014
Yuval Filmus, Massimo Lauria, Mladen Mikša, Jakob Nordström, Marc Vinyals

From Small Space to Small Width in Resolution

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 >>>


TR12-132 | 21st October 2012
Yuval Filmus, Massimo Lauria, Jakob Nordström, Noga Ron-Zewi, Neil Thapen

Space Complexity in Polynomial Calculus

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 >>>


TR10-136 | 26th August 2010
Arnab Bhattacharyya, Elena Grigorescu, Jakob Nordström, Ning Xie

Separations of Matroid Freeness Properties

Revisions: 1

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 >>>


TR10-125 | 11th August 2010
Eli Ben-Sasson, Jakob Nordström

Understanding Space in Proof Complexity: Separations and Trade-offs via Substitutions

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 >>>


TR10-045 | 15th March 2010
Jakob Nordström

On the Relative Strength of Pebbling and Resolution

Revisions: 1

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 >>>


TR09-100 | 16th October 2009
Jakob Nordström, Alexander Razborov

On Minimal Unsatisfiability and Time-Space Trade-offs for $k$-DNF Resolution

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 >>>


TR09-047 | 20th April 2009
Eli Ben-Sasson, Jakob Nordström

A Space Hierarchy for k-DNF Resolution

Comments: 1

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 >>>


TR09-034 | 25th March 2009
Eli Ben-Sasson, Jakob Nordström

Understanding Space in Resolution: Optimal Lower Bounds and Exponential Trade-offs

Comments: 1

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 >>>


TR09-002 | 23rd November 2008
Eli Ben-Sasson, Jakob Nordström

Short Proofs May Be Spacious: An Optimal Separation of Space and Length in Resolution

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 >>>


TR08-026 | 28th February 2008
Jakob Nordström, Johan Håstad

Towards an Optimal Separation of Space and Length in Resolution

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 >>>


TR07-114 | 28th September 2007
Jakob Nordström

A Simplified Way of Proving Trade-off Results for Resolution

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 >>>


TR05-066 | 4th June 2005
Jakob Nordström

Narrow Proofs May Be Spacious: Separating Space and Width in Resolution

Revisions: 2 , Comments: 1

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 >>>




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