Distance between Herbrand interpretations: A measure for approximations to a target concept

  • Shan-Hwei Nienhuys-Cheng
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1297)


We can use a metric to measure the differences between elements in a domain or subsets of that domain (i.e. concepts). Which particular metric should be chosen, depends on the kind of difference we want to measure. The well known Euclidean metric on ℜn and its generalizations are often used for this purpose, but such metrics are not always suitable for concepts where elements have some structure different from real numbers. For example, in (Inductive) Logic Programming a concept is often expressed as an Herbrand interpretation of some firstorder language. Every element in an Herbrand interpretation is a ground atom which has a tree structure. We start by defining a metric d on the set of expressions (ground atoms and ground terms), motivated by the structure and complexity of the expressions and the symbols used therein. This metric induces the Hausdorff metric h on the set of all sets of ground atoms, which allows us to measure the distance between Herbrand interpretations. We then give some necessary and some sufficient conditions for an upper bound of h between two given Herbrand interpretations, by considering the elements in their symmetric difference.


metric space inductive logic programming logic programming 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Shan-Hwei Nienhuys-Cheng
    • 1
  1. 1.Department of Computer Science,H4-19Erasmus University RotterdamRotterdamthe Netherlands

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