On the number of steps in proofs

Abstract

In this paper we prove some results about the complexity of proofs. We consider proofs in Hilbert-style formal systems such as in [17]. Thus a proof is a sequence offormulas satisfying certain conditions. We can view the formulas as being strings of symbols; hence the whole proof is a string too. We consider the following measures of complexity of proofs: length ( \\= the number of symbols in the proof), depth ( \\= the maximal depth of a formula in the proof)and number of steps ( \\= the number of formulas in the proof) For a particular formal system and a given formula A we consider the shortest length of a proof of A, the minimal depth of a proof of A and the minimal number of steps in a proof of A. The main results are the following: (1) a bound on the depth in terms of the number of steps: Theorem 2.2, (2) a bound on the depth in terms of the length: Theorem 2.3, (3) a bound on the length in terms of the number of steps for restricted systems: Theorem 3.1. These results are applied to obtain several corollaries. In particular we show: (1) a bound on the number of steps in a cut-free proof, (2) some speed-up results, (3) bounds on the number of steps in proofs of Paris-Harrington sentences. Some papers related to our research are listed in the references. We were especially influenced by Parikh's paper [17] on the famous conjecture of Kreisel (cf. [3], problem 34]). Many important problems in this field remain open. We hope that our paper will contribute to progress in this area. It should be noted that some results of this paper can be equally well obtained using unification theory (cf. [5]), by translating a complexity-of-proof problem into a unification problem. A unification algorithm solving the unification problem can then be constructed and the complexity of the unification algorithm analyzed. As pointed out by the referee this method can be used to solve the problem stated in Section 3 for some non-simple (defined below) schematic systems.