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Algorithms for ordinal arithmetic
 In 19th International Conference on Automated Deduction (CADE
, 2003
"... 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 A ..."
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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
Rewrite Proofs and Computations
 Proof and Computation
, 1995
"... . Rewriting is a general paradigm for expressing computations in various logics, and we focus here on rewriting techniques in equational logic. When used at the proof level, rewriting provides with a very powerful methodology for proving completeness results, a technique that is illustrated here. We ..."
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. Rewriting is a general paradigm for expressing computations in various logics, and we focus here on rewriting techniques in equational logic. When used at the proof level, rewriting provides with a very powerful methodology for proving completeness results, a technique that is illustrated here. We also consider whether important properties of rewrite systems such as confluence and termination can be proved in a modular way. Finally, we stress the links between rewriting and tree automata. Previous surveys include [21; 18; 37; 12; 45; 46]. The present one owes much to [21]. Keywords. completion, confluence, critical pair, ground reducibility, inductive completion, local confluence, modularity, narrowing, ordersorted algebras, rewrite rule, rewriting, term algebra, termination, tree automata. 1 Introduction The use of equations is traditional in mathematics. Its use in computer science has culminated with the success of algebraic specifications, a method of specifying software by enc...
Halfway Up To the Mathematical Infinity I: On the Ontological & Epistemic Sustainability of Georg Cantor’s Transfinite Design
, 2009
"... Georg Cantor was the genuine discoverer of the Mathematical Infinity, and whatever he claimed, suggested, or even surmised should be taken seriously – albeit not necessary at its face value. Because alongside his exquisite in beauty ordinal construction and his fundamental powerset description of ..."
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Georg Cantor was the genuine discoverer of the Mathematical Infinity, and whatever he claimed, suggested, or even surmised should be taken seriously – albeit not necessary at its face value. Because alongside his exquisite in beauty ordinal construction and his fundamental powerset description of the continuum, Cantor has also left to us his obsessive presumption that the universe of sets should be subjected to laws similar to those governing the set of natural numbers, including the universal principles of cardinal comparability and wellordering – and implying an ordinal recreation of the continuum. During the last hundred years, the mainstream settheoretical research – all insights and adjustments due to Kurt Gödel’s revolutionary insights and discoveries notwithstanding – has compliantly centered its efforts on ad hoc axiomatizations of Cantor’s intuitive transfinite design. We demonstrate here that the ontological and epistemic sus
Algorithms for Ordinal Arithmetic
"... 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 A ..."
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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 ffl0. We use a compact notation for the ordinals up to ffl0 (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 Introduction Termination proofs are of critical importance for the mechanical verification of computing systems. This is the case even with reactive systems, nonterminating systems that are engaged in ongoing interaction with an environment (e.g., network protocols, operating systems, and distributed databases), as termination proofs are used to show that some desirable behavior is not postponed forever, i.e., to establish liveness properties. Proving termination amounts to showing that a relation is wellfounded [1]. Since every wellfounded relation can be extended to a total order that is orderisomorphic to an ordinal, it makes sense to base termination proofs on the ordinals. The theory of the ordinal numbers has been studied extensively for over 100 years and is at the core of Cantor's set theory [46]. The ordinals also play a crucial role in logic, e.g., Gentzen proved the consistency of Peano arithmetic using induction up to ffl0 [13]. Since Genten's work, proof theorists routinely use ordinals to establish the consistency of various theories. To obtain constructive proofs, constructive ordinals notations are employed [29, 34]. The general theory of ordinal notations was initiated by Church and Kleene [7] and is recounted in Chapter 11 of Roger's book on computability [26].
L R I MECHANICALLY PROVING TERMINATION USING POLYNOMIAL INTERPRETATIONS
, 2004
"... Mechanically proving termination using polynomial interpretations ..."
Order Theory of (ω ωi,+) and the Monadic Second Order Theory of
, 2006
"... Abstract. We give a new simple proof of the decidability of the First ..."
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Retrieving the Mathematical Mission of the Continuum Concept from the Transfinitely Reductionist Debris of Cantor’s Paradise Extended Abstract
, 2011
"... To the memory of my spouse Eugenia. À mes trois bienfaiteurs québécois: ..."