# Automating Resolution is NP-Hard

@article{Atserias2019AutomatingRI, title={Automating Resolution is NP-Hard}, author={Albert Atserias and Moritz Christian M{\"u}ller}, journal={2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)}, year={2019}, pages={498-509} }

We show that the problem of finding a Resolution refutation that is at most polynomially longer than a shortest one is NP-hard. In the parlance of proof complexity, Resolution is not automatizable unless P = NP. Indeed, we show that it is NP-hard to distinguish between formulas that have Resolution refutations of polynomial length and those that do not have subexponential length refutations. This also implies that Resolution is not automatizable in subexponential time or quasi-polynomial time…

## 36 Citations

### Automating Regular or Ordered Resolution is NP-Hard

- MathematicsElectron. Colloquium Comput. Complex.
- 2020

It is shown that it is hard to find regular or even ordered Resolution proofs, extending the breakthrough result for general Resolution from Atserias and Müller to these restricted forms.

### Automating algebraic proof systems is NP-hard

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2020

This work extends, and gives a simplified proof of, the recent breakthrough of Atserias and Müller that established an analogous result for Resolution, and shows that algebraic proofs are hard to find.

### Automating OBDD proofs is NP-hard

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2022

It is proved that the proof system OBDD( ∧, weakening) is not automatable unless P = NP, where n is the number of variables in the non-lifted formula.

### Failure of Feasible Disjunction Property for k-DNF Resolution and NP-hardness of Automating It

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2020

It is shown that if NP is not included in P (resp. QP, SUBEXP) then for every integer $k \geq 1$, Res ($k$) is not automatable in polynomial time.

### Resolution Lower Bounds for Refutation Statements

- MathematicsElectron. Colloquium Comput. Complex.
- 2019

For any unsatisfiable CNF formula we give an exponential lower bound on the size of resolution refutations of a propositional statement that the formula has a resolution refutation. We describe three…

### Lower Bounds for Refutation Statements

- Mathematics, Computer Science
- 2019

For any unsatisfiable CNF formula we give an exponential lower bound on the size of resolution refutations of a propositional statement that the formula has a resolution refutation. We describe three…

### 37 : 2 Resolution Lower Bounds for Refutation Statements

- Computer Science, Mathematics
- 2019

For any unsatisfiable CNF formula we give an exponential lower bound on the size of resolution refutations of a propositional statement that the formula has a resolution refutation. We describe three…

### Automating Tree-Like Resolution in Time no(log

- Computer Science
- 2021

We show that tree-like resolution is not automatable in time no(log n) unless ETH is false. This implies that, under ETH, the algorithm given by Beame and Pitassi (FOCS 1996) that automates tree-like…

### Automating cutting planes is NP-hard

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

Two new lifting theorems are relied on: Dag-like lifting for gadgets with many output bits and tree- like lifting that simulates an r-round protocol with gadgets of query complexity O(r).

### 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.

## References

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### Automating algebraic proof systems is NP-hard

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2020

This work extends, and gives a simplified proof of, the recent breakthrough of Atserias and Müller that established an analogous result for Resolution, and shows that algebraic proofs are hard to find.

### Failure of Feasible Disjunction Property for k-DNF Resolution and NP-hardness of Automating It

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2020

It is shown that if NP is not included in P (resp. QP, SUBEXP) then for every integer $k \geq 1$, Res ($k$) is not automatable in polynomial time.

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It is shown that if there is a polynomial-time (superpolynomial-time or subexponential time, respectively) approximation algorithm that finds a nearly shortest proof of length up to S + O(n d ), where S is the length of the shortest proof and d may be any constant, then this immediately gives a positive answer to the open problem asking whether finding a shortest Resolution proof is NP-hard.

### Resolution Lower Bounds for Refutation Statements

- MathematicsElectron. Colloquium Comput. Complex.
- 2019

### Automating cutting planes is NP-hard

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

Two new lifting theorems are relied on: Dag-like lifting for gadgets with many output bits and tree- like lifting that simulates an r-round protocol with gadgets of query complexity O(r).

### 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.

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Abstract.In this paper, we show how to extend the argument due to Bonet, Pitassi and Raz to show that bounded-depth Frege proofs do not have feasible interpolation, assuming that factoring of Blum…

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We show that neither Resolution nor tree-like Resolution is automatizable unless the class W[P] from the hierarchy of parameterized problems is fixed-parameter tractable by randomized algorithms with…

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New lower bounds on the lengths of resolution proofs for the pigeonhole principle and for randomly generated formulas are given and the range of formula sizes for which non-trivial lower bounds are known is extended.

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If there exists an optimal proof system for propositional logic in which every tautology has a short proof, it can be formed by augmenting the usual text-book calculus by the extension rule and by an additional, polynomial time recognizable, set of tautologies as extra axioms.