Ordinal Arithmetic: Algorithms and Mechanization

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. (2000): 68W30, 03F15. Mathematics Subject Classifications

Three equivalent ordinal notation systems in cubical Agda

“…Formalisations Several formalisations of ordinals and ordinal notation systems exist in the literature. Manolios and Vroon [21] represents ordinals below ε 0 in the ACL2 theorem prover, based on a variation of Cantor normal form with ω β 1 c 1 + . .…”

“…Hence the program must terminate by the well-foundedness of the order on ordinals. At first, such proofs were carried out using pen and paper [11,14], but with advances in proof assistants, also machine-checked proofs can be produced [21,24]. As a first step, one must then represent ordinals inside a theorem prover.…”

The Ideal Approach to Computing Closed Subsets in Well-Quasi-orderings

“…While the above applies to any recursive ordinal, the applications that we are aware of usually only need ordinals below ǫ 0 , for which the Cantor Normal Form is well known and understood, and leads to natural data structures [47]. One can push this at least to all ordinals below the larger ordinal Γ 0 [21].…”

Efficient execution in an automated reasoning environment

“…Up through Version 2.7, the ACL2 system provided a notation for representing these ordinals, a function to check if an object represents an ordinal in this notation, and a function for comparing the magnitude of two ordinals, but had only very limited support for reasoning about and constructing ordinals. In fact, while the set theoretic definitions of arithmetic operations were given by Cantor in the 1800's, algorithms for arithmetic operations on ordinal notations were not studied in any comprehensive way until recently, when Manolios and Vroon provided efficient algorithms, with complexity analyses, for ordinal arithmetic on the ordinals up to ǫ 0 , using a notational system that is exponentially more succinct than the one used in ACL2 Version 2.7 [36,39]. The above notations and algorithms were implemented in the ACL2 system, their correctness was mechanically verified, and a library of theorems developed that can be used to significantly automate reasoning involving the ordinals [37].…”