$\begingroup$

Do all propositional tautologies have polynomial-size Frege proofs?

Arguably the major open problem of proof complexity: demonstrate super-polynomial size lower bounds on propositional proofs (called also Frege proofs).

Informally, a Frege proof system is just a standard propositional proof system for proving propositional tautologies (one learns in a basic logic course), having axioms and deduction rules, where proof-lines are written as formulas. The size of a Frege proof is the number of symbols it takes to write down the proof.

The problem then asks whether there is a family $(F_n)_{n=1}^\infty$ of propositional tautological formulas for which there is no polynomial $ p $ such that the minimal Frege proof size of $ F_n $ is at most $ p(|F_n|)$, for all $ n=1,2,\ldots$ (where $ |F_n| $ denotes the size of the formula $ F_n $).

Formal definition of a Frege proof system

Definition (Frege rule) A Frege rule is a sequence of propositional formulas $ A_0(\overline x),\ldots,A_k(\overline x) $, for $ k \le 0 $, written as $ \frac{A_1(\overline x), \ldots,A_k(\overline x)}{A_0(\overline x)}$. In case $ k = 0 $, the Frege rule is called an axiom scheme. A formula $ F_0 $ is said to be derived by the rule from $ F_1,\ldots,F_k $ if $ F_0,\ldots,F_k $ are all substitution instances of $ A_1,\ldots,A_k $, for some assignment to the $ \overline x $ variables (that is, there are formulas $B_1,\ldots,B_n $ such that $F_i = A_i(B_1/x_1,\ldots,B_n/x_n), $ for all $ i=0,\ldots,k $. The Frege rule is said to be sound if whenever an assignment satisfies the formulas in the upper side $A_1,\ldots,A_k $, then it also satisfies the formula in the lower side $ A_0 $.

Definition (Frege proof) Given a set of Frege rules, a Frege proof is a sequence of formulas such that every proof-line is either an axiom or was derived by one of the given Frege rules from previous proof-lines. If the sequence terminates with the formula $ A $, then the proof is said to be a proof of $ A $. The size of a Frege proof is the the total sizes of all the formulas in the proof.

A proof system is said to be implicationally complete if for all set of formulas $ T $, if $ T $ semantically implies $ F $, then there is a proof of $ F $ using (possibly) axioms from $ T $. A proof system is said to be sound if it admits proofs of only tautologies (when not using auxiliary axioms, like in the $ T $ above).

Definition (Frege proof system) Given a propositional language and a finite set $ P $ of sound Frege rules, we say that $ P $ is a Frege proof system if $ P $ is implicationally complete.

Note that a Frege proof is always sound since the Frege rules are assumed to be sound. We do not need to work with a specific Frege proof system, since a basic result in proof complexity states that every two Frege proof systems, even over different languages, are polynomially equivalent [Reckhow, PhD thesis, University of Toronto, 1976].

Establishing lower bounds on Frege proofs could be viewed as a step towards proving $NP

eq coNP$, since if this is true then no propositional proof system (including Frege) can have polynomial size proofs for all tautologies.