Automating Induction by Reflection

TL;DR

This paper introduces a novel method for formalizing schematic inductive definitions within multi-sorted first-order logic, enabling automated inductive reasoning with theorem provers.

Contribution

It presents a new approach inspired by axiomatic theories of truth to express inductive schemas in standard first-order logic, bridging a gap in automated theorem proving.

Findings

Method successfully formalizes schematic inductive definitions.

Compared favorably to native induction techniques in theorem provers.

Demonstrates practical feasibility with state-of-the-art provers.

Abstract

Despite recent advances in automating theorem proving in full first-order theories, inductive reasoning still poses a serious challenge to state-of-the-art theorem provers. The reason for that is that in first-order logic induction requires an infinite number of axioms, which is not a feasible input to a computer-aided theorem prover requiring a finite input. Mathematical practice is to specify these infinite sets of axioms as axiom schemes. Unfortunately these schematic definitions cannot be formalized in first-order logic, and therefore not supported as inputs for first-order theorem provers. In this work we introduce a new method, inspired by the field of axiomatic theories of truth, that allows to express schematic inductive definitions, in the standard syntax of multi-sorted first-order logic. Further we test the practical feasibility of the method with state-of-the-art theorem provers, comparing it to solvers' native techniques for handling induction.