Strict Ideal Completions of the Lambda Calculus

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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 res- ulting 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 ex- tends the corresponding metric-based calculus. Three of our calculi are infinitarily normalising and confluent; their unique infinitary normal forms are exactly the Böhm-like trees of the cor- responding metric-based calculi. Our calculi dispense with the infinitely many ⊥-rules of the metric-based calculi. The fully non-strict calculus (called 111) consists of only β-reduction, while the other two calculi (called 001 and 101) require two additional rules that precisely state their strictness properties: λx.⊥ → ⊥ (for 001) and ⊥ M → ⊥ (for 001 and 101).
Original languageEnglish
Title of host publication3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)
Number of pages16
PublisherSchloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH
Publication dateJul 2018
Article number8
Publication statusPublished - Jul 2018
EventInternational Conference on Formal Structures for Computation and Deduction - Oxford, United Kingdom
Duration: 9 Jul 201812 Jul 2018
Conference number: 3


ConferenceInternational Conference on Formal Structures for Computation and Deduction
Country/TerritoryUnited Kingdom
Internet address


  • lambda calculus
  • infinitary rewriting
  • Böhm trees
  • meaningless terms
  • confluence


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