Abstract
An axiomatic theory represents mathematical knowledge declaratively as a set of axioms. An algorithmic theory represents mathematical knowledge procedurally as a set of algorithms. A biform theory is simultaneously an axiomatic theory and an algorithmic theory. It represents mathematical knowledge both declaratively and procedurally. Since the algorithms of algorithmic theories manipulate the syntax of expressions, biform theories—as well as algorithmic theories—are difficult to formalize in a traditional logic without the means to reason about syntax. Chiron is a derivative of von-Neumann-Bernays-Gödel (nbg) set theory that is intended to be a practical, general-purpose logic for mechanizing mathematics. It includes elements of type theory, a scheme for handling undefinedness, and a facility for reasoning about the syntax of expressions. It is an exceptionally well-suited logic for formalizing biform theories. This paper defines the notion of a biform theory, gives an overview of Chiron, and illustrates how biform theories can be formalized in Chiron.
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Farmer, W.M. (2007). Biform Theories in Chiron. In: Kauers, M., Kerber, M., Miner, R., Windsteiger, W. (eds) Towards Mechanized Mathematical Assistants. MKM Calculemus 2007 2007. Lecture Notes in Computer Science(), vol 4573. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73086-6_6
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