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Ordinal Arithmetic: Algorithms and Mechanization

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Abstract

Termination proofs are of critical importance for establishing the correct behavior of both transformational and reactive computing systems. A general setting for establishing termination proofs involves the use of the ordinal numbers, an extension of the natural numbers into the transfinite that were introduced by Cantor in the nineteenth century and are at the core of modern set theory. We present the first comprehensive treatment of ordinal arithmetic on compact ordinal notations and give efficient algorithms for various operations, including addition, subtraction, multiplication, and exponentiation. Using the ACL2 theorem proving system, we implemented our ordinal arithmetic algorithms, mechanically verified their correctness, and developed a library of theorems that can be used to significantly automate reasoning involving the ordinals. To enable users of the ACL2 system to fully utilize our work required that we modify ACL2, e.g., we replaced the underlying representation of the ordinals and added a large library of definitions and theorems. Our modifications are available starting with ACL2 version 2.8.

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  1. College of Computing, CERCS Lab, Georgia Institute of Technology, 801 Atlantic Drive, Atlanta, GA, 30332-0280, USA

    Panagiotis Manolios & Daron Vroon

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  1. Panagiotis Manolios
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  2. Daron Vroon
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Correspondence to Panagiotis Manolios.

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Manolios, P., Vroon, D. Ordinal Arithmetic: Algorithms and Mechanization. J Autom Reasoning 34, 387–423 (2005). https://doi.org/10.1007/s10817-005-9023-9

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