Clause Set Cycles and Induction

TL;DR

This paper explores the relationship between clause set cycles in automated inductive theorem proving and various theories of induction, revealing their formal connections and differences.

Contribution

It introduces clause set cycles as a formalism and analyzes their relation to $orall_1$ and open induction, including a finite axiomatization and conjecture of incomparability.

Findings

Clause set cycles are contained in $orall_1$ induction.

Clause set cycles are not contained in open induction.

The paper conjectures that open induction and clause set cycles are incomparable.

Abstract

In this article we relate a family of methods for automated inductive theorem proving based on cycle detection in saturation-based provers to well-known theories of induction. To this end we introduce the notion of clause set cycles -- a formalism abstracting a certain type of cyclic dependency between clause sets. We first show that the formalism of clause set cycles is contained in the theory of $\exists_1$ induction. Secondly we consider the relation between clause set cycles and the theory of open induction. By providing a finite axiomatization of a theory of triangular numbers with open induction we show that the formalism of clause set cycles is not contained in the theory of open induction. Furthermore we conjecture that open induction and clause set cycles are incomparable. Finally, we transfer these results to a concrete method of automated inductive theorem proving called the n-clause calculus.