We establish new, and surprisingly tight, connections between propositional proof complexity and finite model theory. Specifically, we show that the power of several propositional proof systems, such as Horn resolution, bounded width resolution, and the polynomial calculus of bounded degree, can be characterised in a precise sense by variants of fixed-point logics that are of fundamental importance in descriptive complexity theory. Our main results are that Horn resolution has the same

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... e power as least fixed-point logic, that bounded width resolution captures existential least fixed-point logic, and that the (monomial restriction of the) polynomial calculus of bounded degree solves precisely the problems definable in fixed-point logic with counting. The question whether there exists an efficient proof system by means of which the validity of arbitrary propositional formulas can be verified via proofs of polynomial size is equivalent to the closure of NP under complementation. Since Cook and Reckhow [14] made the notion of an efficient propositional proof system precise, a huge body of research on the power of various propositional proof system has been established. In particular, we now have super-polynomial lower bounds on the proof complexity for quite strong proof systems, see [7, 25] for surveys on propositional proof complexity. In this paper we study polynomial-time variants of propositional proof systems, which admit efficient proof search, resulting in proofs of polynomial size, such as restricted variants of resolution and the polynomial calculus. Recall that the resolution proof system Res takes as input a propositional formula ϕ in conjunctive normal form (CNF), and it refutes the satisfiability of ϕ if there is a derivation of the empty clause from ϕ. It is well-known that shortest resolution proofs can be of exponential size, so in general, we provably cannot search for resolution proofs in polynomial time. However, there are interesting restrictions of Res, such as Horn-Res (resolution restricted to Horn clauses) and bounded-width resolution * The third author was supported by a DFG grant (PA 2962/1-1).