Formal Verification of the Correspondence Between Call-by-Need and Call-by-Name
We formalize the call-by-need evaluation of λ $$\lambda $$ -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...
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| Published in | Functional and Logic Programming Vol. 10818; pp. 1 - 16 |
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| Main Authors | , |
| Format | Book Chapter |
| Language | English |
| Published |
Switzerland
Springer International Publishing AG
01.01.2018
Springer International Publishing |
| Series | Lecture Notes in Computer Science |
| Subjects | |
| Online Access | Get full text |
| ISBN | 3319906852 9783319906850 |
| ISSN | 0302-9743 1611-3349 |
| DOI | 10.1007/978-3-319-90686-7_1 |
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| Summary: | We formalize the call-by-need evaluation of λ $$\lambda $$ -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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| Bibliography: | Original Abstract: We formalize the call-by-need evaluation of λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}-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. |
| ISBN: | 3319906852 9783319906850 |
| ISSN: | 0302-9743 1611-3349 |
| DOI: | 10.1007/978-3-319-90686-7_1 |