A Sequent Calculus for General Inductive Definitions

2026-04-21

AI Summary

Existing proof systems struggle to accommodate non-monotonic inductive definitions due to syntactic limitations. This work addresses this gap by extending Brotherston and Simpson’s sequent calculus LKID to SCFO(ID), a system tailored for first-order logic with inductive definitions (FO(ID)). Grounded in the principle of mathematical induction, SCFO(ID) is the first sequent calculus capable of effectively handling general— including non-monotonic—inductive definitions. The system integrates ideas from stable model semantics, seamlessly combining classical first-order logic with inductive definitions to yield a formal reasoning framework endowed with robust mathematical properties. Its expressiveness and correctness are demonstrated through several illustrative examples.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Inductive definitions are an important form of knowledge. The logic FO(ID) is an extension of classical first-order logic FO with general non-monotone inductive definitions. Most existing proof systems for inductive definitions impose syntactic constraints on their definitions, thereby excluding many useful and natural definitions. We extend an existing sequent calculus LKID by Brotherston and Simpson, founded on the principle of mathematical induction, to a sequent calculus SCFO(ID) for FO(ID). The main challenge in this extension is the accommodation of non-monotone inductive definitions. To overcome this challenge, we draw inspiration from the stable semantics, which is a commonly used semantics in logic programming that is closely related to the well-founded semantics behind FO(ID). We corroborate SCFO(ID) by establishing several proof-theoretical properties and through demonstration on various examples. In conclusion, SCFO(ID) is a theoretically substantiated sequent calculus for FO(ID), enabling formal proofs of theorems involving general inductive definitions.

Problem

• inductive definitions
• non-monotone
• sequent calculus
• FO(ID)
• proof system

Innovation

• sequent calculus
• non-monotone inductive definitions
• FO(ID)
• stable semantics
• proof theory