Perspex Machine X: software development

Noble, Sam; Thomas, Benjamin A.; Anderson, James A.

doi:10.1117/12.698166

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Axiomatisation1

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2007

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“…The definition used here is different from the programmatic definition of sgn given in. 4 It is a shorthand that is used to state axiom [A31] concisely. It is not essential to the axiomatisation.…”

Section: Axiomatisationmentioning

confidence: 99%

Perspex Machine VIII: axioms of transreal arithmetic

2007

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Transreal arithmetic is a total arithmetic that contains real arithmetic, but which has no arithmetical exceptions. It allows the specification of the Universal Perspex Machine which unifies geometry with the Turing Machine. Here we axiomatise the algebraic structure of transreal arithmetic so that it provides a total arithmetic on any appropriate set of numbers. This opens up the possibility of specifying a version of floating-point arithmetic that does not have any arithmetical exceptions and in which every number is a first-class citizen.We find that literal numbers in the axioms are distinct. In other words, the axiomatisation does not require special axioms to force non-triviality. It follows that transreal arithmetic must be defined on a set of numbers that contains as a proper subset. We note that the axioms have been shown to be consistent by machine proof.

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Section: Axiomatisationmentioning

confidence: 99%