Proof Complexity in Propositional Logic
Summary
Proof complexity investigates the inherent difficulty of demonstrating that a given propositional formula is unsatisfiable. By analysing measures such as proof length, size, width and degree across various proof systems, researchers reveal fundamental limits on automated reasoning and the efficiency of SAT solvers. Central proof systems include resolution, which operates by deriving new clauses until a contradiction is reached; polynomial calculus, which encodes Boolean formulas as polynomial equations; and algebraic or semi-algebraic systems such as Sherali–Adams and Nullstellensatz. Proof complexity connects combinatorial graph parameters—such as treewidth—with the complexity of refutations, and explores trade-offs between time, space and size. These insights underpin advances in verification, cryptography and complexity theory by clarifying which logical principles admit succinct proofs and which inherently resist efficient proof search.
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Proof Complexity in Propositional Logic publication trend
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Technical terms
Proof complexity: The study of the resources required to establish unsatisfiability in propositional logic, including proof size, length, width and degree.
Propositional logic: A formal language of Boolean variables combined with logical connectives, where formulae evaluate to true or false under variable assignments.
Resolution: A rule-based proof system that derives contradictions by resolving pairs of clauses on complementary literals.
Polynomial calculus: An algebraic proof system translating propositional clauses into polynomial equations and deriving contradictions through polynomial manipulations.
Tseitin formulas: Canonical families of unsatisfiable formulas encoding parity constraints on graph vertices, often used as hard instances in proof complexity.
Treewidth: A graph-theoretic parameter measuring how close a graph is to a tree, which influences the complexity of regular resolution proofs.
Clause space: The maximum number of clauses that must be stored concurrently during a resolution refutation, reflecting memory usage.
References
- A Generalized Method for Proving Polynomial Calculus Degree Lower Bounds. Journal of the ACM (2024).
- Characterizing Tseitin-Formulas with Short Regular Resolution Refutations. Journal of Artificial Intelligence Research (2023).
- Reversible Pebble Games and the Relation Between Tree-Like and General Resolution Space. computational complexity (2021).
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