🤖 AI Summary
This work addresses the problem of second-order quantifier elimination by proposing a method that explicitly constructs a logically equivalent first-order formula together with its corresponding second-order witness function. Building upon the clause-set-based SCAN saturation algorithm, we extend it for the first time to support the automatic computation of witness functions, thereby enabling effective witness generation for a broad class of applicable formulas while preserving the original soundness guarantees of logical equivalence. This advancement not only broadens the applicability of the SCAN algorithm within computational logic but also demonstrates the feasibility and practical utility of the proposed approach through an implemented prototype system.
📝 Abstract
Second-order quantifier elimination is the problem of finding, given a formula with second-order quantifiers, a logically equivalent first-order formula. While such formulas are not computable in general, there are practical algorithms and subclasses with applications throughout computational logic. One of the most prominent algorithms for second-order quantifier elimination is the saturation-based SCAN algorithm. In this paper we show how the SCAN algorithm on clause sets can be extended to solve a more general problem: namely, finding a witness for the second-order quantifiers that results in a logically equivalent first-order formula. In addition, we provide a prototype implementation of the proposed method.