Interpolants, cut elimination and flow graphs for the propositional calculus

Abstract

We analyse the structure of propositional proofs in the sequent calculus focusing on the well-known procedures of Interpolation and Cut Elimination. We are motivated in part by the desire to understand why a tautology might be ‘hard to prove’. Given a proof we associate to it a logical graph tracing the flow of formulas in it (Buss, 1991). We show some general facts about logical graphs such as acyclicity of cut-free proofs and acyclicity of contraction-free proofs (possibly containing cuts), and we give a proof of a strengthened version of the Craig Interpolation Theorem based on flows of formulas. We show that tautologies having minimal interpolants of non-linear size (i.e. number of symbols) must have proofs with certain precise structural properties. We then show that given a proof Π and a cut-free form Π′ associated to it (obtained by a particular cut elimination procedure), certain subgraphs of Π′ which are logical graphs (i.e. graphs of proofs) correspond to subgraphs of Π which are logical graphs for the same sequent. This locality property of cut elimination leads to new results on the complexity of interpolants, which cannot follow from the known constructions proving the Craig Interpolation Theorem.