A Clausal Approach to Proof Analysis in Second-Order Logic

Abstract

This work defines an extension CERES2 of the first-order cut-elimination method CERES to the subclass of sequent calculus proofs in second-order logic using quantifier-free comprehension. This extension is motivated by the fact that cut-elimination can be used as a tool to extract information from real mathematical proofs, and often a crucial part of such proofs is the definition of sets by formulas. This is expressed by the comprehension axiom scheme, which is representable in second-order logic. At the core of CERES2 lies the production of a set of clauses CL(ϕ) from a proof ϕ that is always unsatisfiable. From a resolution refutation γ of CL(ϕ), a proof without essential cuts can be constructed. The main theoretical obstacle in the extension of CERES to second-order logic is the construction of this proof from γ. This issue is solved for the subclass considered in this paper. Moreover, we discuss the problems that have to be solved to extend CERES2 to the complete class of second-order proofs. Finally, the method is applied to a simple mathematical proof that involves induction and comprehension and the resulting proof is analyzed.