Abstract
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.
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Notes
- 1.
We believe that our approach can be adopted in other proof assistants as well.
- 2.
Strictly speaking, the reduction rules shown here are called standard reduction rules in their paper, as opposed to non-deterministic reduction. Note that the let-binding \(\mathbf {let}~x=M~\mathbf {in}~N\) is non-recursive.
- 3.
Although this argument seems to be a proof by contradiction, our actual Coq proof is constructive, using an induction on the finite reduction sequence of \({\xrightarrow {\mathrm {name}}}\circ {\xrightarrow {\beta }_*}\) from \(M^\pitchfork \) as we shall see in Sect. 5.
- 4.
Another drawback is that evaluation contexts may introduce an arbitrary number of bindings and therefore need to be indexed by that number to coexist with de Bruijn indices, requiring heavy natural number calculations—like the Omega [9] library for Presburger arithmetic—in the mechanized proofs. Our approach will also obviate the need for such calculations.
- 5.
Indeed, we also formalized the original semantics and proved its correspondence to call-by-name. See: https://github.com/fetburner/call-by-need.
- 6.
This definition is adopted from the accessibility predicate Acc in Coq.
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Acknowledgments
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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Mizuno, M., Sumii, E. (2018). Formal Verification of the Correspondence Between Call-by-Need and Call-by-Name. In: Gallagher, J., Sulzmann, M. (eds) Functional and Logic Programming. FLOPS 2018. Lecture Notes in Computer Science(), vol 10818. Springer, Cham. https://doi.org/10.1007/978-3-319-90686-7_1
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