## Ordinal arithmetic: Algorithms and mechanization (2006)

Venue: | Journal of Automated Reasoning |

Citations: | 5 - 3 self |

### BibTeX

@ARTICLE{Manolios06ordinalarithmetic:,

author = {Panagiotis Manolios and Daron Vroon},

title = {Ordinal arithmetic: Algorithms and mechanization},

journal = {Journal of Automated Reasoning},

year = {2006},

pages = {1--37}

}

### OpenURL

### Abstract

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 which 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. 1.

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Citation Context ...nal numbers provide a general setting for establishing termination proofs. 1.1. Previous Work The ordinal numbers were introduced by Cantor over 100 years ago and are at the core of modern set theory =-=[10, 11, 12]-=-. They are an extension of the natural numbers into the transfinite and are an important tool in logic, e.g., after Gentzen’s proof of the consistency of Peano arithmetic using the ordinal number ε0 [... |

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Citation Context ... to prove that the floating-point operations performed by AMD microprocessors are IEEE-754 compliant [41, 54], to analyze bit and cycle accurate models of the Motorola CAP, a digital signal processor =-=[9]-=-, and to analyze a model of the JEM1, the world’s first silicon JVM (Java Virtual Machine) [24]. Termination proofs have played a key role in various projects that use ACL2 to verify reactive systems.... |

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Citation Context ...nal numbers provide a general setting for establishing termination proofs. 1.1. Previous Work The ordinal numbers were introduced by Cantor over 100 years ago and are at the core of modern set theory =-=[10, 11, 12]-=-. They are an extension of the natural numbers into the transfinite and are an important tool in logic, e.g., after Gentzen’s proof of the consistency of Peano arithmetic using the ordinal number ε0 [... |

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Citation Context ...tions, the measure function is written ω 2 (n − r) + ω(r − s) + k. ordinals.tex; 16/03/2005; 12:41; p.2sOrdinal Arithmetic: Algorithms and Mechanization 3 were carried out with the Isabelle/ZF system =-=[44, 46]-=-. A version of the reflection theorem was also proved by Bancerek, using Mizar [3]. Another line of work is by Belinfante, who has used Otter to prove elementary theorems of ordinal number theory [4, ... |

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Citation Context ...rdinals have also played a key role in projects to implement polynomial orderings [39] and multiset relations [52]. The relationship between proof theoretic ordinals and term rewriting is explored in =-=[15, 20]-=-. 1.2. The Ordinal Arithmetic Problem Despite the fact that ordinals have been studied and used extensively by various communities for over 100 years, we have not been able to find a comprehensive tre... |

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Citation Context ...a model of the JEM1, the world’s first silicon JVM (Java Virtual Machine) [24]. Termination proofs have played a key role in various projects that use ACL2 to verify reactive systems. For example, in =-=[32]-=-, we develop a theory of refinement for reactive systems that has been used to mechanically verify protocols, pipelined machines, and distributed systems [35, 31, 60]. Ordinals have also played a key ... |

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Checking a large routine
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Citation Context ...leene [13] and is recounted in Chapter 11 of Roger’s book on computability [50]. An early use of the ordinals for proving program termination is due to Alan M. Turing, who in 1949 wrote the following =-=[64, 42]-=-. The checker has to verify that the process comes to an end. Here again he should be assisted by the programmer giving a further definite assertion to be verified. This may take the form of a quantit... |

31 | Partial functions in ACL2
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Citation Context ...ry recursive call decrease according to this measure. Only terminating recursive definitions can be so admitted under the definitional principle. (However, “partial functions” can be axiomatized; see =-=[33, 34]-=-.) As a theorem prover, ACL2 is an industrial-strength version of the Boyer-Moore theorem prover [8]. Of special note is its “industrialstrength,” e.g., it has been used to prove some of the largest a... |

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Citation Context ...sion 2.8 to allow it to use our new representation and library. We conclude with Section 9, where we also outline future work. 2.1. Set Theoretic Ordinals 2. Ordinals We review the theory of ordinals =-=[17, 30, 57]-=-. A relation ≺ is wellfounded if every decreasing sequence is finite. A woset is a pair 〈X,≺〉, where X is a set, and ≺ is a well-ordering, a total, well-founded relation, over X. Given a woset, 〈X,≺〉,... |

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Citation Context ...ustrial-scale problems by companies such as AMD, Rockwell Collins, Motorola, and IBM. ACL2 was used to prove that the floating-point operations performed by AMD microprocessors are IEEE-754 compliant =-=[41, 54]-=-, to analyze bit and cycle accurate models of the Motorola CAP, a digital signal processor [9], and to analyze a model of the JEM1, the world’s first silicon JVM (Java Virtual Machine) [24]. Terminati... |

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Citation Context ... verify reactive systems. For example, in [32], we develop a theory of refinement for reactive systems that has been used to mechanically verify protocols, pipelined machines, and distributed systems =-=[35, 31, 60]-=-. Ordinals have also played a key role in projects to implement polynomial orderings [39] and multiset relations [52]. The relationship between proof theoretic ordinals and term rewriting is explored ... |

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Citation Context ...ls against industrial design simulation tools, before subjecting the ACL2 models to proof [56]. ACL2 models of microprocessors have been executed at 90% of the speed of comparable C simulation models =-=[23]-=-. As a mathematical logic, ACL2 may be thought of as first-order predicate calculus with equality, recursive function definitions, and mathematical induction. The primitives of applicative Common Lisp... |

24 |
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Citation Context ...h the consistency of logical theories [57, 62]. To obtain constructive proofs, constructive ordinals notations are employed. The general theory of ordinal notations was initiated by Church and Kleene =-=[13]-=- and is recounted in Chapter 11 of Roger’s book on computability [50]. An early use of the ordinals for proving program termination is due to Alan M. Turing, who in 1949 wrote the following [64, 42]. ... |

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Citation Context ...pliant [41, 54], to analyze bit and cycle accurate models of the Motorola CAP, a digital signal processor [9], and to analyze a model of the JEM1, the world’s first silicon JVM (Java Virtual Machine) =-=[24]-=-. Termination proofs have played a key role in various projects that use ACL2 to verify reactive systems. For example, in [32], we develop a theory of refinement for reactive systems that has been use... |

18 |
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Citation Context ...e ordinals, but there are well known notations that can represent ordinals up to Γ0 (which is needed to show termination of some term rewrite systems [15, 20]) and further into the Veblen hierarchies =-=[65]-=- and further still [40, 58, 59]. Another possible extension is to define additional operations on ordinals, e.g., division, taking logs, etc. Finally, a promising direction for future work is to use o... |

16 | abczewski. Mechanizing set theory. Cardinal arithmetic and the axiom of choice
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Citation Context ...the proof that for any infinite cardinal, κ, we have κ ⊗ κ = κ, most of the first chapter of Kunen’s excellent book on set theory [30], and the equivalence of eight forms of the well-ordering theorem =-=[49]-=-. More recently, Paulson has mechanized the proof of the relative consistency of the axiom of choice and has proved the reflection theorem [48, 47]. Paulson and Grabczewski’s efforts required reasonin... |

15 |
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Citation Context ...leene [13] and is recounted in Chapter 11 of Roger’s book on computability [50]. An early use of the ordinals for proving program termination is due to Alan M. Turing, who in 1949 wrote the following =-=[64, 42]-=-. The checker has to verify that the process comes to an end. Here again he should be assisted by the programmer giving a further definite assertion to be verified. This may take the form of a quantit... |

15 |
A mechanically checked proof of correctness of the AMD K5 floating point square root microcode
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Citation Context ...em prover [8]. Of special note is its “industrialstrength,” e.g., it has been used to prove some of the largest and most complicated theorems ever proved about commercially designed digital artifacts =-=[41, 54, 53, 55, 56, 9, 24]-=-. The theorem prover is an integrated system of ad hoc proof techniques that include simplification, generalization, induction, and many other techniques. Simplification is the main technique and incl... |

13 |
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Citation Context ... 46]. A version of the reflection theorem was also proved by Bancerek, using Mizar [3]. Another line of work is by Belinfante, who has used Otter to prove elementary theorems of ordinal number theory =-=[4, 5, 6]-=-. There is much more work that can be mentioned, but we end by listing some of the theorem proving systems for which there exists support for the ordinals: Nqthm [8], ACL2 [26], Coq [7], PVS [43], HOL... |

13 |
On computer-assisted proofs in ordinal number theory
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Citation Context ... 46]. A version of the reflection theorem was also proved by Bancerek, using Mizar [3]. Another line of work is by Belinfante, who has used Otter to prove elementary theorems of ordinal number theory =-=[4, 5, 6]-=-. There is much more work that can be mentioned, but we end by listing some of the theorem proving systems for which there exists support for the ordinals: Nqthm [8], ACL2 [26], Coq [7], PVS [43], HOL... |

13 | Linking theorem proving and model-checking with well-founded bisimulation
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(Show Context)
Citation Context ... verify reactive systems. For example, in [32], we develop a theory of refinement for reactive systems that has been used to mechanically verify protocols, pipelined machines, and distributed systems =-=[35, 31, 60]-=-. Ordinals have also played a key role in projects to implement polynomial orderings [39] and multiset relations [52]. The relationship between proof theoretic ordinals and term rewriting is explored ... |

13 |
RTL verification: A floating-point multiplier
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(Show Context)
Citation Context ...odels of commercial floating-point designs have been executed on millions of test cases to “validate” the models against industrial design simulation tools, before subjecting the ACL2 models to proof =-=[56]-=-. ACL2 models of microprocessors have been executed at 90% of the speed of comparable C simulation models [23]. As a mathematical logic, ACL2 may be thought of as first-order predicate calculus with e... |

12 | Proof-theoretic techniques for term rewriting theory
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- 1988
(Show Context)
Citation Context ...rdinals have also played a key role in projects to implement polynomial orderings [39] and multiset relations [52]. The relationship between proof theoretic ordinals and term rewriting is explored in =-=[15, 20]-=-. 1.2. The Ordinal Arithmetic Problem Despite the fact that ordinals have been studied and used extensively by various communities for over 100 years, we have not been able to find a comprehensive tre... |

12 |
A Mechanically Checked Proof of the AMD5K86 Floating-Point Division Program
- Moore, Lynch, et al.
- 1998
(Show Context)
Citation Context ...ustrial-scale problems by companies such as AMD, Rockwell Collins, Motorola, and IBM. ACL2 was used to prove that the floating-point operations performed by AMD microprocessors are IEEE-754 compliant =-=[41, 54]-=-, to analyze bit and cycle accurate models of the Motorola CAP, a digital signal processor [9], and to analyze a model of the JEM1, the world’s first silicon JVM (Java Virtual Machine) [24]. Terminati... |

12 |
An Incremental Stuttering Refinement Proof of a Concurrent Program in ACL2
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(Show Context)
Citation Context ... verify reactive systems. For example, in [32], we develop a theory of refinement for reactive systems that has been used to mechanically verify protocols, pipelined machines, and distributed systems =-=[35, 31, 60]-=-. Ordinals have also played a key role in projects to implement polynomial orderings [39] and multiset relations [52]. The relationship between proof theoretic ordinals and term rewriting is explored ... |

11 | Algorithms for Ordinal Arithmetic
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- 2003
(Show Context)
Citation Context ... is exponentially more succinct than the one used in ACL2 before version 2.8 and we give efficient algorithms, whose complexity we analyze. A preliminary conference version of the results appeared in =-=[36]-=-. Here, we give a more comprehensive treatment and provide full complexity and correctness proofs. Using the ACL2 theorem proving system, we implemented our ordinal arithmetic operations and mechanica... |

9 | Ordinal arithmetic in acl2
- Manolios, Vroon
- 2003
(Show Context)
Citation Context ...onstruct ordinals. With our library, users can ignore representational issues and can work in an algebraic setting. An example application is due to Sustik, who used a previous version of our library =-=[37]-=- to give a constructive proof of Dickson’s lemma [61]. This is a key lemma in the proof of the termination of Buchberger’s algorith for finding Gröbner bases, and Sustik made essential use of the ordi... |

9 | Multiset Relations: a Tool for Proving Termination
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- 2000
(Show Context)
Citation Context ...mechanically verify protocols, pipelined machines, and distributed systems [35, 31, 60]. Ordinals have also played a key role in projects to implement polynomial orderings [39] and multiset relations =-=[52]-=-. The relationship between proof theoretic ordinals and term rewriting is explored in [15, 20]. 1.2. The Ordinal Arithmetic Problem Despite the fact that ordinals have been studied and used extensivel... |

8 |
Normal functions and constructive ordinal notations
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- 1976
(Show Context)
Citation Context ...3s4 In this paper, we present a solution to the ordinal arithmetic problem for a notational system denoting the ordinals up to ε0. Partial solutions to this problem appear in various books and papers =-=[57, 16, 20, 40, 58, 62]-=-, 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. One notable exception is... |

8 |
A Mechanically Checked Proof of Correctness of the AMD5K 86 Floating-Point Square Root Microcode
- Russinoff
- 1997
(Show Context)
Citation Context ...em prover [8]. Of special note is its “industrialstrength,” e.g., it has been used to prove some of the largest and most complicated theorems ever proved about commercially designed digital artifacts =-=[41, 54, 53, 55, 56, 9, 24]-=-. The theorem prover is an integrated system of ad hoc proof techniques that include simplification, generalization, induction, and many other techniques. Simplification is the main technique and incl... |

7 | Integrating reasoning about ordinal arithmetic into ACL2
- Manolios, Vroon
- 2004
(Show Context)
Citation Context ... implementations. In addition, we modified ACL2 by replacing its then current ordinal representation (in ACL2, version 2.7) with the exponentially more succinct representation we present in Section 2 =-=[38]-=-. (Both representations are based on the Cantor normal form.) We show that our changes do not affect the soundness of the ACL2 logic by exhibiting a bijection between our ordinal representation and th... |

6 |
An extended arithmetic of ordinal numbers
- Doner, Tarski
- 1969
(Show Context)
Citation Context ...on of < for various ordinal notations, but we have not found any statement of the problem nor any comprehensive solution in previous work. One notable exception is the dissertation work of John Doner =-=[19, 18]-=-. Doner and Tarski (his adviser) study hierarchies of ordinal arithmetic operations. They give a transfinite recursive definition for binary operations Oγ for any ordinal γ. The operation O0 correspon... |

6 | Ordinal systems
- Setzer
(Show Context)
Citation Context ...re well known notations that can represent ordinals up to Γ0 (which is needed to show termination of some term rewrite systems [15, 20]) and further into the Veblen hierarchies [65] and further still =-=[40, 58, 59]-=-. Another possible extension is to define additional operations on ordinals, e.g., division, taking logs, etc. Finally, a promising direction for future work is to use our library and ACL2 as a proof ... |