Sort it out with monotonicity: translating between many-sorted and unsorted first-order logic

  • Authors:

  • Koen Claessen;Ann Lillieström;Nicholas Smallbone

  • Affiliations:

  • Chalmers University of Technology, Gothenburg, Sweden;Chalmers University of Technology, Gothenburg, Sweden;Chalmers University of Technology, Gothenburg, Sweden

  • Venue:
  • CADE'11 Proceedings of the 23rd international conference on Automated deduction
  • Year:
  • 2011

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Abstract

We present a novel analysis for sorted logic, which determines if a given sort is monotone. The domain of a monotone sort can always be extended with an extra element. We use this analysis to significantly improve well-known translations between unsorted and many-sorted logic, making use of the fact that it is cheaper to translate monotone sorts than non-monotone sorts. Many interesting problems are more naturally expressed in many-sorted first-order logic than in unsorted logic, but most existing highly-efficient automated theorem provers solve problems only in unsorted logic. Conversely, some reasoning tools, for example model finders, can make good use of sort-information in a problem, but most problems today are formulated in unsorted logic. This situation motivates translations in both ways between many-sorted and unsorted problems. We present the monotonicity analysis and its implementation in our tool Monotonox, and also show experimental results on the TPTP benchmark library.