LPC(ID): A Sequent Calculus Proof System for Propositional Logic Extended with Inductive Definitions

TL;DR

This paper introduces a sequent calculus proof system for propositional logic extended with inductive definitions, enhancing understanding of proof methods for FO(ID) and its fragment PC(ID).

Contribution

It develops a sound and complete sequent calculus proof system for PC(ID), integrating propositional calculus with inductive definitions.

Findings

Proof system is sound and complete for a fragment of PC(ID)

Provides complexity results for PC(ID)

Enhances proof-theoretic understanding of FO(ID)

Abstract

The logic FO(ID) uses ideas from the field of logic programming to extend first order logic with non-monotone inductive definitions. Such logic formally extends logic programming, abductive logic programming and datalog, and thus formalizes the view on these formalisms as logics of (generalized) inductive definitions. The goal of this paper is to study a deductive inference method for PC(ID), which is the propositional fragment of FO(ID). We introduce a formal proof system based on the sequent calculus (Gentzen-style deductive system) for this logic. As PC(ID) is an integration of classical propositional logic and propositional inductive definitions, our sequent calculus proof system integrates inference rules for propositional calculus and definitions. We present the soundness and completeness of this proof system with respect to a slightly restricted fragment of PC(ID). We also provide some complexity results for PC(ID). By developing the proof system for PC(ID), it helps us to enhance the understanding of proof-theoretic foundations of FO(ID), and therefore to investigate useful proof systems for FO(ID).