The New Normal: We Cannot Eliminate Cuts in Coinductive Calculi, But We Can Explore Them

TL;DR

This paper demonstrates that cut elimination is impossible in certain coinductive sequent calculi but introduces practical methods for working with cuts, including a new proof system and heuristics for cut discovery.

Contribution

It proves that cut cannot be eliminated in coinductive extensions of LJ and presents practical tools for coinductive proof exploration and cut management.

Findings

Cut is not eliminable in coinductive LJ (CLJ).

CoLP yields cut-free proofs with fixpoint terms in CLJ.

A new heuristic method for discovering cut formulae in CLJ.

Abstract

In sequent calculi, cut elimination is a property that guarantees that any provable formula can be proven analytically. For example, Gentzen's classical and intuitionistic calculi LK and LJ enjoy cut elimination. The property is less studied in coinductive extensions of sequent calculi. In this paper, we use coinductive Horn clause theories to show that cut is not eliminable in a coinductive extension of LJ, a system we call CLJ. We derive two further practical results from this study. We show that CoLP by Gupta et al. gives rise to cut-free proofs in CLJ with fixpoint terms, and we formulate and implement a novel method of coinductive theory exploration that provides several heuristics for discovery of cut formulae in CLJ.