## Algorithms for ordinal arithmetic (2003)

Venue: | In 19th International Conference on Automated Deduction (CADE |

Citations: | 11 - 5 self |

### BibTeX

@INPROCEEDINGS{Manolios03algorithmsfor,

author = {Panagiotis Manolios and Daron Vroon},

title = {Algorithms for ordinal arithmetic},

booktitle = {In 19th International Conference on Automated Deduction (CADE},

year = {2003},

pages = {243--257},

publisher = {Springer–Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Proofs of termination are essential for establishing the correct behavior of computing systems. There are various ways of establishing termination, but the most general involves the use of ordinals. An example of a theorem proving system in which ordinals are used to prove termination is ACL2. In ACL2, every function defined must be shown to terminate using the ordinals up to ɛ0. We use a compact notation for the ordinals up to ɛ0 (exponentially more succinct than the one used by ACL2) and define efficient algorithms for ordinal addition, subtraction, multiplication, and exponentiation. In this paper we describe our notation and algorithms, prove their correctness, and analyze their complexity. 1

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Citation Context ...versions of our algorithms in the ACL2 system, mechanically verified their correctness, and developed a library of theorems that can be used to significantly automate reasoning involving the ordinals =-=[22]-=-. Our library substantially increases the extent to which ACL2 can automatically reason about the ordinals, as previously none of the ordinal arithmetic operations were defined and proving termination... |

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and termination
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Citation Context ...ystem in question. We present a solution to the ordinal arithmetic problem for a notational system denoting the ordinals up to ɛ0. Partial solutions to this problem appear in various books and papers =-=[29, 10, 8, 12, 24, 30, 34]-=-, e.g., it is easy to find a definition of < for various ordinal notations, but we have not found any statement of the problem nor any comprehensive solution in previous work. In this paper, we provid... |

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Citation Context ...ω + α < [ω + w · 3] m ·2+ω 5 ·5) ω + β to α < β. Our library is now distributed with the latest version of ACL2 (version 2.7) and has already been used to give a constructive proof of Dickson’s lemma =-=[33]-=-. Knowledge of ACL2 is not required to read this paper, as only thesalgorithms and their complexity are discussed; a discussion of the ACL2 library will appear elsewhere [22]. The paper is organized a... |

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Citation Context ...nd any statement of the problem nor any comprehensive solution in previous work. In this paper, we provide an efficient solution to the ordinal arithmetic problem (for ɛ0). We use a notational system =-=[2]-=- that is exponentially more succinct than the one used in ACL2 and we give efficient algorithms, whose complexity we analyze. The importance of efficient algorithms becomes apparent when one considers... |

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Citation Context ...stems that has been used to mechanically verify protocols, pipelined machines, and distributed systems [21, 19, 32]. Ordinals have also played a key role in projects to implement polynomial orderings =-=[23]-=- and multiset relations [27]. The relationship between proof theoretic ordinals and term rewriting is explored in [9, 12]. Even though ordinal notations have been studied and used extensively by vario... |