A study for recovering the cut-elimination property in cyclic proof systems by restricting the arity of inductive predicates

TL;DR

This paper investigates why the cut-elimination property fails in cyclic proof systems for logics with inductive definitions, even under restrictions to unary predicates and functions, providing sharper insights than previous results.

Contribution

It demonstrates that restricting to unary inductive predicates and functions does not restore cut-elimination in cyclic proof systems, refining understanding of the property’s failure.

Findings

Cut-elimination fails even with unary restrictions

Provides a sharper counterexample than previous work

Clarifies limitations of cyclic proof systems for inductive logics

Abstract

The framework of cyclic proof systems provides a reasonable proof system for logics with inductive definitions. It also offers an effective automated proof search procedure for such logics without finding induction hypotheses. Recent researches have shown that the cut-elimination property, one of the most fundamental properties in proof theory, of cyclic proof systems for several logics does not hold. These results suggest that a naive proof search, which avoids the Cut rule, is not enough. This paper shows that the cut-elimination property still fails in a simple cyclic proof system even if we restrict languages to unary inductive predicates and unary functions, aiming to clarify why the cut-elimination property fails in the cyclic proof systems. The result in this paper is a sharper one than that of the first authors' previous result, which gave a counterexample using two ternary inductive predicates and a unary function symbol to show the failure of the cut-elimination property in the cyclic proof system of the first-order logic.