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 monomial 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 expressive power as least fixed-point logic, that bounded-width resolution captures existential least fixed-point logic, and that the polynomial calculus with bounded degree over the rationals solves precisely the problems definable in fixed-point logic with counting. We also study the bounded-degree polynomial calculus. Over the rationals, it captures fixed-point logic with counting if we restrict the bit-complexity of the coefficients. For unrestricted coefficients, we can only say that the bounded-degree polynomial calculus is at most as powerful as bounded variable infinitary counting logic, but a precise logical characterisation of its power remains an open problem. These connections between logics and proof systems allow us to establish finite-model-theoretic tools for proving lower bounds for the polynomial calculus over the rationals and also over finite fields. This is a corrected version of the paper (arXiv:1802.09377) published originally on January 23, 2019.

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