Convergent approximate solving of first-order constraints by approximate quantifiers

  • Authors:

  • Stefan Ratschan

  • Affiliations:

  • Max-Planck-Institut für Informatik, Saarbruecken, Germany

  • Venue:
  • ACM Transactions on Computational Logic (TOCL)
  • Year:
  • 2004

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Abstract

Exactly solving first-order constraints (i.e., first-order formulas over a certain predefined structure) can be a very hard, or even undecidable problem. In continuous structures like the real numbers it is promising to compute approximate solutions instead of exact ones. However, the quantifiers of the first-order predicate language are an obstacle to allowing approximations to arbitrary small error bounds. In this article, we remove this obstacle by modifying the first-order language and replacing the classical quantifiers with approximate quantifiers. These also have two additional advantages: First, they are tunable, in the sense that they allow the user to decide on the trade-off between precision and efficiency. Second, they introduce additional expressivity into the first-order language by allowing reasoning over the size of solution sets.