Strict Ideal Completions of the Lambda Calculus

Author: Patrick Bahr

Paper Information

Title:Strict Ideal Completions of the Lambda Calculus
Authors:Patrick Bahr
Proceedings:FSCD Presented Papers
Editor: Helene Kirchner
Keywords:lambda calculus, infinitary rewriting, Böhm trees, meaningless terms, confluence, normalisation

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 β-reduction with infinitely many ‘⊥-rules’, which contract meaningless terms directly to ⊥. Three of the resulting Böhm reduction calculi have unique infinitary normal forms corresponding to Böhm-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 infinitary normalising and confluent; their unique infinitary normal forms are exactly the Böhm-like trees of the corresponding metric-based calculi. Our calculi dispense with the infinitely many ⊥-rules of the metric-based calculi. The fully non-strict calculus (111) consists of only β-reduction, while the other two calculi (001 and 101) require two additional rules that precisely state their strictness properties: λx.⊥ → ⊥ and ⊥ M → ⊥.

Talk:Jul 11 10:00 (Session 60C: Lambda Calculus)