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Reasoning about partial functions with the aid of a computer

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Abstract

Partial functions are ubiquitous in both mathematics and computer science. Therefore, it is imperative that the underlying logical formalism for a general-purpose mechanized mathematics system provide strong support for reasoning about partial functions. Unfortunately, the common logical formalisms — first-order logic, type theory, and set theory — are usually only adequate for reasoning about partial functionsin theory. However, the approach to partial functions traditionally employed by mathematicians is quite adequatein practice. This paper shows how the traditional approach to partial functions can be formalized in a range of formalisms that includes first-order logic, simple type theory, and Von-Neumann—Bernays—Gödel set theory. It argues that these new formalisms allow one to directly reason about partial functions; are based on natural, well-understood, familiar principles; and can be effectively implemented in mechanized mathematics systems.

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Supported by the MITRE-Sponsored Research program. This paper is a written version (with references) of an address given at the Partial Functions and Programming: Foundational Questions conference held 17 February 1995 at the University ol California at Irvine.

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Farmer, W.M. Reasoning about partial functions with the aid of a computer. Erkenntnis 43, 279–294 (1995). https://doi.org/10.1007/BF01135375

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Keywords

  • Computer Science
  • Logical Formalism
  • Strong Support
  • Traditional Approach
  • Type Theory