Proof Complexity and SAT Solving

  title={Proof Complexity and SAT Solving},
  author={Samuel R. Buss and Jakob Nordstr{\"o}m},
  booktitle={Handbook of Satisfiability},
This chapter gives an overview of proof complexity and connections to SAT solving, focusing on proof systems such as resolution, Nullstellensatz, polynomial calculus, and cutting planes (corresponding to conflict-driven clause learning, algebraic approaches using linear algebra or Gröbner bases, and pseudo-Boolean solving, respectively). There is also a discussion of extended resolution (which is closely related to DRAT proof logging) and Frege and extended Frege systems more generally. An… 

A Cardinal Improvement to Pseudo-Boolean Solving

This work presents a technique to recover cardinality constraints from CNF on the fly during search by collecting potential building blocks of cardinality constraint constraints during propagation and combining these blocks during conflict analysis.

The power of negative reasoning

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.

Certified Symmetry and Dominance Breaking for Combinatorial Optimisation

It is demonstrated that the cutting planes proof system can efficiently verify fully general symmetry breaking in Boolean satisfiability (SAT) solving, thus providing, for the first time, a unified method to certify a range of advanced SAT techniques that also includes XOR and cardinality reasoning.

Justifying All Differences Using Pseudo-Boolean Reasoning

This paper demonstrates that simple, clean, and efficient proof logging is still possible for the all-different constraint, through pseudo-Boolean reasoning.

Practical algebraic calculus and Nullstellensatz with the checkers Pacheck and Pastèque and Nuss-Checker

The practical algebraic calculus is presented as an instantiation of the polynomial calculus that can be checked efficiently and extension rules to simulate essential rewriting techniques required in practice are introduced.

Certifying Parity Reasoning Efficiently Using Pseudo-Boolean Proofs

This work shows how to instead use pseudo-Boolean inequalities with extension variables to concisely justify XOR reasoning, and points the way towards a unified approach for efficient machine-verifiable proofs for a rich class of combinatorial optimization paradigms.

Cutting to the Core of Pseudo-Boolean Optimization: Combining Core-Guided Search with Cutting Planes Reasoning

This work lifts core-guided search to pseudo-Boolean (PB) solvers, which deal with more general PB optimization problems and operate natively with cardinality constraints, and derives stronger cardinality constraint constraints, which yield better updates to solution bounds.

Watched Propagation of 0-1 Integer Linear Constraints

This work extends the CDCL paradigm from clausal constraints to \(0\)-\(1\) integer linear constraints, also known as (linear) PB constraints, and proposes a counter technique which watches all literals of a PB constraint.

Kernelization, Proof Complexity and Social Choice

A notable application of the framework is to impossibility results in computational social choice: it is shown that, for any fixed number of agents, propositional translations of the Arrow and Gibbard-Satterthwaite impossibility theorems have subexponential size Frege proofs.

Perfect Matching in Random Graphs is as Hard as Tseitin

We study the complexity of proving that a sparse random regular graph on an odd number of vertices does not have a perfect matching, and related problems involving each vertex being matched some



On the interplay between proof complexity and SAT solving

This paper is intended as an informal and accessible survey of proof complexity for non-experts, focusing on some comparatively weak proof systems of particular interest in connection with SAT

In Between Resolution and Cutting Planes: A Study of Proof Systems for Pseudo-Boolean SAT Solving

It is shown that even if general addition is allowed, this proof system is still polynomially simulated by resolution with respect to proof size as long as coefficients are polynOMially bounded.

Automating algebraic proof systems is NP-hard

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.

DRAT Proofs, Propagation Redundancy, and Extended Resolution

The proof complexity of RAT proofs and related systems including BC, SPR, and PR which use blocked clauses and (subset) propagation redundancy are studied and a new inference SR is introduced using substitution redundancy.

On Tackling the Limits of Resolution in SAT Solving

A new problem transformation is proposed, which enables reducing the decision problem for formulas in conjunctive normal form (CNF) to the problem of solving maximum satisfiability over Horn formulas, and proves a polynomial bound on the number of MaxSAT resolution steps for pigeonhole formulas.

Relating Proof Complexity Measures and Practical Hardness of SAT

A systematic study of the interconnections between theoretical complexity and practical SAT solver performance and concludes that the resolution space complexity of a formula would seem to be a more fine-grained indicator of whether the formula is hard or easy than the length or width needed in a resolution proof.

MaxSAT Resolution With the Dual Rail Encoding

This paper investigates the DRMaxSAT proof system, and shows that DR MaxSAT p-simulates general resolution, that AC0-Frege+PHP p-Simulates DRMax sAT, and that DRMax SAT can not p- Simulate AC0 -Frege-PHP or the cutting planes proof system.

Logical Foundations of Proof Complexity

The result is a uniform treatment of many systems in the literature, including Buss's theories for the polynomial hierarchy and many disparate systems for complexity classes such as AC0, AC0(m), TC0, NC1, L, NL, NC, and P.

Algebraic proof complexity: progress, frontiers and challenges

We survey recent progress in the proof complexity of strong proof systems and its connection to algebraic circuit complexity, showing how the synergy between the two gives rise to new approaches to

Proof Complexity Meets Algebra

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.