Separation results for the size of constant-depth propositional proofs

This paper proves exponential separations between depth d -LK and depth (d + 1 2 )-LK for every d ∈ 1 2 N utilizing the order induction principle. As a consequence, we obtain an exponential separation between depth d -LK and depth (d+1)-LK for d ∈ N . We investigate the relationship between the sequence-size, tree-size and height of depth d -LK-derivations for d ∈ 1 2 N , and describe transformations between them. We define a general method to lift principles requiring exponential tree-size (d + 1 2 )-LK-refutations for d ∈ N to principles requiring exponential sequence-size d -LK-refutations, which will be described for the Ramsey principle and d = 0. From this we also deduce width lower bounds for resolution refutations of the Ramsey principle. Constant-depth propositional proof systems have been extensively studied because of their connection with the complexity of constant-depth circuits and fragments of bounded arithmetic (c.f. [2, 9, 13, 14, 17]). Kraj́ıček [9] defined an alternative notion of constant-depth proofs: a formula is defined to have Σ-depth d iff if is depth d+1 and the bottommost level of connectives have fanin ≤ logS , where S is a size parameter. A proof is defined to have Σ-depth d provided every formula in the proof has Σ-depth d where S is the size of the proof. Partially supported by a Marie Curie Individual Fellowship #HPMF-CT-2000-00803 from the European Commission, and by FWF-grants #P16264-N05 and #P16539-N04 of the Austrian Science Fund. Institute of Algebra and Computational Mathematics, Vienna University of Technology, A-1040 Vienna, Austria. Email: [email protected] Supported in part by NSF grant DMS-0100589. Dept. of Mathematics, Univ. of California, San Diego, La Jolla, CA 92093-0112. Email: [email protected].