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Taming the wild ant-lion; a counterexample to a conjecture of Böhm

Published online by Cambridge University Press:  07 August 2015

RICK STATMAN
RICK STATMAN*
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A Email: statman@cs.cmu.edu
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Extract

In Barendregt (1984), Corrado Böhm conjectured that every adequate (Barendregt 1984, 6.4.2 (ii)) numeral system of normal combinators has normal successor, predecessor and zero test. In this note, we give a counterexample to this conjecture. Our example is shown to have no normal zero test. Böhm has informed us that Intrigila (1994) has given an example with no normal successor. Our strategy, in terms of the ant-lion paradigm, is to pry open the trap so wide that it enters its active state before its jaws are shut.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 27 , Special Issue 5: Computing with Lambda-terms. A Special Issue Dedicated to Corrado Böhm for his 90th Birthday , June 2017 , pp. 734 - 737
Copyright
Copyright © Cambridge University Press 2015 

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References

Barendregt, H.P. (1984). The Lambda Calculus. Its Syntax and Semantics, Studies in logic and the foundations of mathematics, vol. 103, North-Holland, Amsterdam.Google Scholar
Böhm, C. (1989). Subduing self-application. In: 16th International Colloquium on Automata, Languages and Programming, ICALP89, Stresa, Italy, July 11–15. Springer Lecture Notes in Computer Science 372 108–122.CrossRefGoogle Scholar
Böhm, C. and Intrigila, B. (1994). The ant-lion paradigm for strong normalization. Information and Computation 114 (1) 30–49.CrossRefGoogle Scholar
Intrigila, B. (1994). Some results on numerical systems in lambda-calculus. Notre Dame Journal of Formal Logic 35 (4) 523–541.Google Scholar