# Narrow Proofs May Be Maximally Long

@article{Atserias2014NarrowPM, title={Narrow Proofs May Be Maximally Long}, author={Albert Atserias and Massimo Lauria and Jakob Nordstr{\"o}m}, journal={2014 IEEE 29th Conference on Computational Complexity (CCC)}, year={2014}, pages={286-297} }

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 nΩ(w). This shows that the simple counting argument that any formula refutable in width w must have a proof in size nO(w) is essentially tight. Moreover, our lower bounds can be generalized to polynomial calculus resolution (PCR) and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. Our results do…

## 62 Citations

### From Small Space to Small Width in Resolution

- MathematicsACM Trans. Comput. Log.
- 2015

It is shown that the related question for polynomial calculus seems unlikely to be resolvable by similar methods, and a “black-box” technique for proving space lower bounds via a "static” complexity measure that works against any resolution refutation is developed.

### A Tradeoff Between Length and Width in Resolution

- Computer ScienceTheory Comput.
- 2014

It is shown that, for a family of CNF formulas in n variables, this decrease in width comes at the expense of an increase in size, and any such narrow refutations must have exponential length.

### Separations in Proof Complexity and TFNP

- Computer ScienceElectron. Colloquium Comput. Complex.
- 2022

It is shown that PPADS, PPAD, SOPL, and Reversible Resolution are captured by unary-SA, unARY-NS, and reversible Resolution, respectively, relative to an oracle.

### Short Proofs Are Hard to Find

- Computer Science, MathematicsICALP
- 2019

A streamlined proof is obtained of an important result by Alekhnovich and Razborov, showing that it is hard to automatize both tree-like and general Resolution.

### Proof Complexity Meets Algebra

- Mathematics, Computer ScienceICALP
- 2017

It is shown that, for the most studied propositional, algebraic, and semialgebraic proof systems, the classical constructions of pp-interpretability, homomorphic equivalence, and addition of constants to a core preserve the proof complexity of the CSP.

### An Ultimate Trade-Off in Propositional Proof Complexity

- Computer ScienceElectron. Colloquium Comput. Complex.
- 2015

This work shows that for any parameter k = k(n) there are unsatisfiable k-CNFs that possess refutations of width O(k), but such that any tree-like refutation of width n1− /k must necessarily have double exponential size exp(nΩ(k)).

### A New Kind of Tradeoffs in Propositional Proof Complexity

- Computer ScienceJ. ACM
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This work shows that for any parameter k = k(n), there are unsatisfiable k-CNFs that possess refutations of width O(k), but such that any tree-like refutation must necessarily have doubly exponential size exp (nΩ(k)).

### MaxSAT Resolution and Subcube Sums

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 2020

We study the MaxRes rule in the context of certifying unsatisfiability. We show that it can be exponentially more powerful than tree-like resolution, and when augmented with weakening (the system…

### Circular (Yet Sound) Proofs

- Mathematics, Computer ScienceSAT
- 2019

Surprisingly, as proof systems for deriving clauses from clauses, Circular Resolution turns out to be equivalent to Sherali-Adams, a proof system for reasoning through polynomial inequalities that has linear programming at its base.

### The power of negative reasoning

- MathematicsComputational Complexity Conference
- 2021

It is shown for Nullstellensatz, polynomial calculus, Sherali-Adams, and sums-of-squares that adding formal variables for negative literals makes the proof systems exponentially stronger, with respect to the number of terms in the proofs.

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