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A Meta Lambda Calculus with Cross-Level Computation

Published:14 January 2015Publication History
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Abstract

We propose meta lambda calculus Lambda-* as a basic model of textual substitution via metavariables. The most important feature of the calculus is that every beta-redex can be reduced regardless of whether the beta-redex contains meta-level variables or not. Such a meta lambda calculus has never been achieved before due to difficulty to manage binding structure consistently with alpha-renaming in the presence of meta-level variables. We overcome the difficulty by introducing a new mechanism to deal with substitution and binding structure in a systematic way without the notion of free variables and alpha-renaming.

Calculus Lambda-* enables us to investigate cross-level terms that include a certain type of level mismatch. Cross-level terms have been regarded as meaningless terms and left out of consideration thus far. We find that some cross-level terms behave as quotes and `eval' command in programming languages. With these terms, we show a procedural language as an application of the calculus, which sheds new light on the notions of stores and recursion via meta-level variables.

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      • Published in

        cover image ACM SIGPLAN Notices
        ACM SIGPLAN Notices  Volume 50, Issue 1
        POPL '15
        January 2015
        682 pages
        ISSN:0362-1340
        EISSN:1558-1160
        DOI:10.1145/2775051
        • Editor:
        • Andy Gill
        Issue’s Table of Contents
        • cover image ACM Conferences
          POPL '15: Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages
          January 2015
          716 pages
          ISBN:9781450333009
          DOI:10.1145/2676726

        Copyright © 2015 ACM

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        Association for Computing Machinery

        New York, NY, United States

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        • Published: 14 January 2015

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