Minimum propositional proof length is NP-hard to linearly approximate

We prove that the problem of determining the minimum propositional proof length is NP-hard to approximate within any constant factor. These results hold for all Frege systems, for all extended Frege systems, for resolution and Horn resolution, and for the sequent calculus and the cut-free sequent calculus. Also, if NP is not in \(QP = DTIME(n^{log^{O(1)} n} )\), then it is impossible to approximate minimum propositional proof length within a factor of \(2^{log^{(1 - \varepsilon )} n}\) for any є > 0. All these hardness of approximation results apply to proof length measured either by number of symbols or by number of inferences, for tree-like or dag-like proofs. We introduce the Monotone Minimum (Circuit) Satisfying Assignment problem and prove the same hardness results for Monotone Minimum (Circuit) Satisfying Assignment.