Formal verifications of call-by-need and call-by-name evaluations with mutual recursion

Masayuki Mizuno, Eijiro Sumii

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

2 Citations (Scopus)


We present new proofs—formalized in the Coq proof assistant—of the correspondence among call-by-need and (various definitions of) call-by-name evaluations of λ-calculus with mutually recursive bindings. For non-strict languages, the equivalence between high-level specifications (call-by-name) and typical implementations (call-by-need) is of foundational interest. A particular milestone is Launchbury’s natural semantics of call-by-need evaluation and proof of its adequacy with respect to call-by-name denotational semantics, which are recently formalized in Isabelle/HOL by Breitner (2018). Equational theory by Ariola et al. is another well-known formalization of call-by-need. Mutual recursion is especially challenging for their theory: reduction is complicated by the traversal of dependency (the “need” relation), and the correspondence of call-by-name and call-by-need reductions becomes non-trivial, requiring sophisticated structures such as graphs or infinite trees. In this paper, we give arguably simpler proofs solely based on (finite) terms and operational semantics, which are easier to handle for proof assistants (Coq in our case). Our proofs can be summarized as follows: (1) we prove the equivalence between Launchbury’s call-by-need semantics and heap-based call-by-name natural semantics, where we define a sufficiently (but not too) general correspondence between the two heaps, and (2) we also show the correspondence among three styles of call-by-name semantics: (i) the natural semantics used in (1); (ii) closure-based natural semantics that informally corresponds to Launchbury’s denotational semantics; and (iii) conventional substitution-based semantics.

Original languageEnglish
Title of host publicationProgramming Languages and Systems - 17th Asian Symposium, APLAS 2019, Proceedings
EditorsAnthony Widjaja Lin
Number of pages21
ISBN (Print)9783030341749
Publication statusPublished - 2019
Event17th Asian Symposium on Programming Languages and Systems, APLAS 2019 - Bali, Indonesia
Duration: 2019 Dec 12019 Dec 4

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume11893 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference17th Asian Symposium on Programming Languages and Systems, APLAS 2019
City Bali


Dive into the research topics of 'Formal verifications of call-by-need and call-by-name evaluations with mutual recursion'. Together they form a unique fingerprint.

Cite this