## Ordinal arithmetic in ACL2 (2003)

Venue: | In ACL2 Workshop 2003 |

Citations: | 9 - 6 self |

### BibTeX

@INPROCEEDINGS{Manolios03ordinalarithmetic,

author = {Panagiotis Manolios and Daron Vroon},

title = {Ordinal arithmetic in ACL2},

booktitle = {In ACL2 Workshop 2003},

year = {2003},

pages = {2--2003}

}

### OpenURL

### Abstract

Abstract. Ordinals form the basis for termination proofs in ACL2. Currently, ACL2 uses a rather inefficient representation for the ordinals up to ɛ0 and provides limited support for reasoning about them. We present algorithms for ordinal arithmetic on an exponentially more compact representation than the one used by ACL2. The algorithms have been implemented and numerous properties of the arithmetic operators have been mechanically verified, thereby greatly extending ACL2’s ability to reason about the ordinals. We describe how to use the libraries containing these results, which are currently distributed with ACL2 version 2.7. 1

### Citations

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Citation Context ...e proofs, constructive ordinal notations are used [16, 20]. The general theory of these notations was initiated by Church and Kleene [4] and is reviewed in Chapter 11 of Roger’s book on computability =-=[15]-=-. Although ordinal notations have been studied extensively by various communities for over a century, we have been unable to find a comprehensive treatment of arithmetic for ordinal notations. We defi... |

294 |
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Citation Context ...n ACL2. In Section 5, we consider complexity issues. Section 6 contains our conclusions and outlines future work. 2 Ordinals 2.1 Set-Theoretic Ordinals We briefly review the theory of ordinal numbers =-=[7, 12, 16]-=-. A relation, ≺ is said to be well-founded if every decreasing sequence is finite. A well-ordering is a total, well-founded relation. A woset is a pair 〈X,≺〉, such that ≺ is a well-ordered relation ov... |

270 |
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Citation Context ... p) otherwise For example, f(5) = 5, f(ω 4 +ω 4 +ω 2 +ω+3) = (4 4 2 1 . 3), and f(ω ω +ω 2 +ω 2 ) = ((1 . 0) 2 2 . 0). A further explanation of the ordinal representation used in ACL2 can be found in =-=[10]-=-. Our representation of the ordinals is similar to ACL2’s, but uses the left distributive property of ordinal multiplication over addition to make it more compact. It is a well known result of the the... |

162 | Basic Proof Theory
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Citation Context ...is practice was introduced by Gentzen, when he proved the consistency of Peano arithmetic using induction up to ɛ0 [9]. In order to obtain constructive proofs, constructive ordinal notations are used =-=[16, 20]-=-. The general theory of these notations was initiated by Church and Kleene [4] and is reviewed in Chapter 11 of Roger’s book on computability [15]. Although ordinal notations have been studied extensi... |

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Citation Context ...is practice was introduced by Gentzen, when he proved the consistency of Peano arithmetic using induction up to ɛ0 [9]. In order to obtain constructive proofs, constructive ordinal notations are used =-=[16, 20]-=-. The general theory of these notations was initiated by Church and Kleene [4] and is reviewed in Chapter 11 of Roger’s book on computability [15]. Although ordinal notations have been studied extensi... |

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Citation Context ...that systems such as ACL2 use ordinals as the basis for termination proofs. The theory of ordinals has been studied for over 100 years, since it was introduced by Cantor as the core of his set theory =-=[1, 2]-=- (see also the English translation [3]). Ordinals have subsequently played an important role in logic, e.g., they are routinely used to prove the consistency of logical systems. This practice was intr... |

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Citation Context ... the basis for termination proofs. The theory of ordinals has been studied for over 100 years, since it was introduced by Cantor as the core of his set theory [1, 2] (see also the English translation =-=[3]-=-). Ordinals have subsequently played an important role in logic, e.g., they are routinely used to prove the consistency of logical systems. This practice was introduced by Gentzen, when he proved the ... |

49 |
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Citation Context ... logic, e.g., they are routinely used to prove the consistency of logical systems. This practice was introduced by Gentzen, when he proved the consistency of Peano arithmetic using induction up to ɛ0 =-=[9]-=-. In order to obtain constructive proofs, constructive ordinal notations are used [16, 20]. The general theory of these notations was initiated by Church and Kleene [4] and is reviewed in Chapter 11 o... |

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Citation Context ...iation? Solving this problem amounts to defining algorithms for arithmetic operations in the given notational system. While partial solutions to this problem are presented in various texts and papers =-=[16, 6, 8, 14, 17, 20]-=-, (for example, definitions for < appear in many of the above sources), we have not found any similar statement of this problem nor any comprehensive solution in previous work. In a companion paper, w... |

29 |
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Citation Context ...n ACL2. In Section 5, we consider complexity issues. Section 6 contains our conclusions and outlines future work. 2 Ordinals 2.1 Set-Theoretic Ordinals We briefly review the theory of ordinal numbers =-=[7, 12, 16]-=-. A relation, ≺ is said to be well-founded if every decreasing sequence is finite. A well-ordering is a total, well-founded relation. A woset is a pair 〈X,≺〉, such that ≺ is a well-ordered relation ov... |

24 |
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Citation Context ...hmetic using induction up to ɛ0 [9]. In order to obtain constructive proofs, constructive ordinal notations are used [16, 20]. The general theory of these notations was initiated by Church and Kleene =-=[4]-=- and is reviewed in Chapter 11 of Roger’s book on computability [15]. Although ordinal notations have been studied extensively by various communities for over a century, we have been unable to find a ... |

12 | Proof-theoretic techniques for term rewriting theory
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Citation Context ... our representation to larger countable ordinals. Of particular interest is the countable ordinal Γ0, which is much larger than ɛ0 and is needed to show the termination of some term rewriting systems =-=[5, 8]-=-. There are notational systems for ordinals up to Γ0 and even for much larger countable ordinals [14, 17, 18], and it would be interesting to find algorithms for manipulating these notations. A final ... |

11 | Algorithms for Ordinal Arithmetic
- Manolios, Vroon
- 2003
(Show Context)
Citation Context ...paper, we present efficient algorithms for ordinal addition, subtraction, multiplication, and exponentiation on succinct ordinal representations, prove their correctness, and analyze their complexity =-=[13]-=-. In this paper, we discuss the ACL2 mechanization, as we use ACL2 to define our ordinal representation and arithmetic algorithms. Our representation is exponentially more compact than the current rep... |

8 |
Normal functions and constructive ordinal notations
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(Show Context)
Citation Context ...iation? Solving this problem amounts to defining algorithms for arithmetic operations in the given notational system. While partial solutions to this problem are presented in various texts and papers =-=[16, 6, 8, 14, 17, 20]-=-, (for example, definitions for < appear in many of the above sources), we have not found any similar statement of this problem nor any comprehensive solution in previous work. In a companion paper, w... |

6 | Ordinal systems
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(Show Context)
Citation Context ...iation? Solving this problem amounts to defining algorithms for arithmetic operations in the given notational system. While partial solutions to this problem are presented in various texts and papers =-=[16, 6, 8, 14, 17, 20]-=-, (for example, definitions for < appear in many of the above sources), we have not found any similar statement of this problem nor any comprehensive solution in previous work. In a companion paper, w... |

5 | Ordinal arithmetic with list structures
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(Show Context)
Citation Context |

4 |
Beiträge zur Bgründung der transfiniten Mengenlehre
- Cantor
(Show Context)
Citation Context ...that systems such as ACL2 use ordinals as the basis for termination proofs. The theory of ordinals has been studied for over 100 years, since it was introduced by Cantor as the core of his set theory =-=[1, 2]-=- (see also the English translation [3]). Ordinals have subsequently played an important role in logic, e.g., they are routinely used to prove the consistency of logical systems. This practice was intr... |

4 | Ordinal systems part 2: One inaccessible
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- 2000
(Show Context)
Citation Context ...hich is much larger than ɛ0 and is needed to show the termination of some term rewriting systems [5, 8]. There are notational systems for ordinals up to Γ0 and even for much larger countable ordinals =-=[14, 17, 18]-=-, and it would be interesting to find algorithms for manipulating these notations. A final possibility is to explore heuristics that allow ACL2 to guess measure functions based on the ordinals; the ho... |

4 |
Proof of Dixon’s lemma using the ACL2 theorem prover via an explicit ordinal mapping
- Sustik
- 2003
(Show Context)
Citation Context ...antly extends ACL2’s ability to reason about ordinals and ordinal arithmetic. This library is included with ACL2 version 2.7, and has already been used to give a constructive proof of Dickson’s Lemma =-=[19]-=-. Despite the fact that the theory of ordinals has been studied for over 100 years, we believe we are the first to give a full solution to the ordinal arithmetic problem [13] and the first to give mec... |