Strict Ideal Completions of the Lambda Calculus

TL;DR

This paper introduces a new approach to infinitary lambda calculi using ideal completion, simplifying the existing metric-based methods while preserving key properties like normalization and confluence.

Contribution

It develops a theory of infinitary lambda calculi based on ideal completion, extending metric-based calculi and eliminating the need for infinitely many $ot$-rules.

Findings

Calculi are infinitarily normalising and confluent.

Unique infinitary normal forms correspond to B"ohm-like trees.

Simplifies the calculus by removing infinitely many rules.

Abstract

The infinitary lambda calculi pioneered by Kennaway et al. extend the basic lambda calculus by metric completion to infinite terms and reductions. Depending on the chosen metric, the resulting infinitary calculi exhibit different notions of strictness. To obtain infinitary normalisation and infinitary confluence properties for these calculi, Kennaway et al. extend $\beta$-reduction with infinitely many `$\bot$-rules', which contract meaningless terms directly to $\bot$. Three of the resulting B\"ohm reduction calculi have unique infinitary normal forms corresponding to B\"ohm-like trees. In this paper we develop a corresponding theory of infinitary lambda calculi based on ideal completion instead of metric completion. We show that each of our calculi conservatively extends the corresponding metric-based calculus. Three of our calculi are infinitarily normalising and confluent; their unique infinitary normal forms are exactly the B\"ohm-like trees of the corresponding metric-based calculi. Our calculi dispense with the infinitely many $\bot$-rules of the metric-based calculi. The fully non-strict calculus (called $111$) consists of only $\beta$-reduction, while the other two calculi (called $001$ and $101$) require two additional rules that precisely state their strictness properties: $\lambda x.\bot \to \bot$ (for $001$) and $\bot\,M \to \bot$ (for $001$ and $101$).