Formal Verification of the Correspondence Between Call-by-Need and Call-by-Name

  • Masayuki Mizuno
  • Eijiro Sumii
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10818)


We formalize the call-by-need evaluation of \(\lambda \)-calculus (with no recursive bindings) and prove its correspondence with call-by-name, using the Coq proof assistant.

It has been long argued that there is a gap between the high-level abstraction of non-strict languages—namely, call-by-name evaluation—and their actual call-by-need implementations. Although a number of proofs have been given to bridge this gap, they are not necessarily suitable for stringent, mechanized verification because of the use of a global heap, “graph-based” techniques, or “marked reduction”. Our technical contributions are twofold: (1) we give a simpler proof based on two forms of standardization, adopting de Bruijn indices for representation of (non-recursive) variable bindings along with Ariola and Felleisen’s small-step semantics, and (2) we devise a technique to significantly simplify the formalization by eliminating the notion of evaluation contexts—which have been considered essential for the call-by-need calculus—from the definitions.



We thank the anonymous reviewers for valuable comments and suggestions. This work was partially supported by JSPS KAKENHI Grant Number 15H02681 and 16K12409.


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Tohoku UniversitySendaiJapan

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