THE COMPLEXITY OF PROPOSITIONAL PROOFS

NATHAN SEGERLIND

Abstract. Propositional proof complexity is the study of the sizes of proposi- tional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational com- plexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.

Contents

Part 1. A tour of propositional proof complexity 2 1. Is there a way to prove every tautology with a short proof? 2 2. Satisfiability algorithms and theories of arithmetic 4 3. A menagerie of Frege-like proof systems 11 4. Reverse mathematics of propositional principles 18 5. Feasible interpolation 22 6. Further connections with satisfiability algorithms 22 7. Beyond the Frege systems 25

Part 2. Some lower bounds on refutation sizes 28 8. The size-width trade-off for resolution 29 9. The small restriction switching lemma 34 10. Expansion clean-up and random 3-CNFs 43 11. Resolution pseudowidth and very weak pigeonhole principles 51

Part 3. Open problems, further reading, acknowledgments 56 Appendix A. Notation 65

THIS SURVEY ARTICLE IS BASED UPON THE AUTHOR’S SACK PRIZE WINNING PHD DISSERTATION, HOWEVER, THE EMPHASIS HERE IS ON CONTEXT. THE VAST MAJORITY OF RESULTS DISCUSSED ARE NOT THE AUTHOR’S, AND SEVERAL RESULTS FROM THE AUTHOR’S DISSER- TATION ARE OMITTED.

1 2 NATHAN SEGERLIND Part 1. A tour of propositional proof complexity §1. Is there a way to prove every tautology with a short proof? One way to certify that a propositional formula is a tautology is to present a proof of the formula in a , such as the system below: F

Definition 1.1. The formulas of are the well-formed formulas over the connectives , , and . TheF inference rule of is modus ponens (from A and A ∧ B∨infer→ B),¬ and its axioms are all substitutionF instances of: → 1. A (B A) 2. A B B 3. (A→ B)→ (A B C) (A C) 4. A ∧ B → A 5. A →A B→ → → → → 6. A ∧ B→ A B 7. (A→ B∨) (A B) A 8. B → A →B ∧ 9. (A → C) → (B →¬C) →¬(A B C) 10. →A ∨A → → → → ∨ → ¬¬ → Let τ be a propositional formula. An -proof of τ is a sequence of F formulas F1, ..., Fm so that Fm = τ, and each Fi is either an axiom, or follows from the application of modus ponens to two formulas Fj and Fk, with j, k < i. The completeness theorem for guarantees that every tautology has an -proof. Moreover, most proofsF of the completeness theorem give quantitativeF bounds on proof sizes: Every tautology τ on n variables has an -proof in which there are at most 2O(n) formulas, each of which has sizeF polynomial in the size of τ. Of course, for many tautologies, much smaller proofs are possible. Does every tautology have an -proof signif- icantly smaller than the exponential length derivation? MoF re generally, does there exist a propositional proof system in which every tautology has a small proof? This question requires a clarification of what is meant by “propositional proof sytem”. For example, any algorithm for deciding satisfiability of a Boolean formula can be viewed as a proof system, with an execution trace for a run that declares ψ to be unsatisfiable being viewed as a proof that ψ is a tautology. Another possibility would be to formalize the defini- tions¬ of propositional formulas and tautologies in ZFC and present a proof in formal ZFC that the formula in question is a tautology. This might seem extreme but by using high-level mathematics, some proofs might be shorter than possible with a more commonplace system such as . These methods and the proof system share three properties that seemF necessary for any method of certifyingF tautologies: Every tautology has a proof, only tautologies have proofs, and valid proofs are computationally easy t