Annals of Mathematics and Artificial Intelligence 37: 241–250, 2003.  2003 Kluwer Academic Publishers. Printed in the Netherlands. Resolution proofs of matching principles Alasdair Urquhart ∗ Department of Philosophy, University of Toronto, Toronto, Ontario, Canada M5S 1A1 E-mail: urquhart@theory.toronto.edu This paper discusses current techniques for proving lower bounds on the size of resolution proofs from sets of clauses expressing assertions about matchings in bipartite graphs (a class including the well-known pigeon-hole clauses). The techniques are illustrated by demon- strating an improved lower bound for some examples discussed by Kleine Büning. The final section discusses some problems where current techniques appear inadequate, and new ideas seem to be required. Keywords: complexity of proofs, resolution method, pigeon-hole principle, matchings in bipartite graphs 1. Introduction The most deeply investigated tautologies are those created by formalizing matching principles in graphs, and in particular, the pigeon-hole principle, which can be consid- ered as the assertion that the bipartite graph K(n,n - 1) has no perfect matching. In this paper, we consider techniques for proving lower bounds on resolution proofs based on sets of clauses expressing matching principles for bipartite graphs. These techniques have been simplified considerably as a result of work of Buss, Beame and Pitassi [2,6]. As an application of such techniques, we show how to improve a recent result of Kleine Büning. In his paper [4] (see also [5]), he shows that for all polynomials p and q there is a formula  in conjunctive normal form so that for any equivalent formula  whose size is less than q(||), there is a clause C so that C is a logical consequence of , but the shortest resolution proof of C from  requires more than p(||, |C|) steps. This result can be interpreted as saying that it is not possible to optimize formulas  for the resolution system, in the sense that we could replace  with an equivalent formula  of restricted length so that all questions of deducibility of clauses from  can be de- cided quickly by the resolution procedure. The main theorem of this paper improves Kleine Büning’s lower bound from super-polynomial to exponential, and also provides an explicit procedure for constructing the clause C from a given . Although the techniques so far developed for lower bounds are quite flexible, they do not seem to work for some examples that appear to be difficult for reso- ∗ The author gratefully acknowledges the support of the National Sciences and Engineering Research Council of Canada.