On the deduction rule and the number of proof lines

Proof systems for propositional logic called simple deduction Frege systems, general deduction Frege systems, and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule, are introduced. Upper bounds are given on the lengths of proofs in these systems compared to lengths in Frege proof systems. As an application, a near-linear simulation of the propositional Gentzen sequent calculus by Frege proofs is presented. It is shown that a general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege systems where by `nearly linear´ is meant that the ratio of proof lengths is O(α(n)), where α is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, and hence a Frege proof system can simulate the propositional sequent calculus with proof lengths bounded by O(nα(n)). As a technical tool, the serial transitive closure problem is introduced. Given a directed graph and a list of closure edges in the transitive closure of the graph, the problem is to derive all the closure edges. A nearly linear bound is given on the number of steps in such a derivation when the graph is treelike