Version spaces and the consistency problem

Haym Hirsh, Nina Mishra, Leonard Pitt

Research output: Contribution to journalArticlepeer-review


A version space is a collection of concepts consistent with a given set of positive and negative examples. Mitchell [Artificial Intelligence 18 (1982) 203-226] proposed representing a version space by its boundary sets: the maximally general (G) and maximally specific consistent concepts (S). For many simple concept classes, the size of G and S is known to grow exponentially in the number of positive and negative examples. This paper argues that previous work on alternative representations of version spaces has disguised the real question underlying version space reasoning. We instead show that tractable reasoning with version spaces turns out to depend on the consistency problem, i.e., determining if there is any concept consistent with a set of positive and negative examples. Indeed, we show that tractable version space reasoning is possible if and only if there is an efficient algorithm for the consistency problem. Our observations give rise to new concept classes for which tractable version space reasoning is now possible, e.g., 1-decision lists, monotone depth two formulas, and halfspaces.

Original languageEnglish (US)
Pages (from-to)115-138
Number of pages24
JournalArtificial Intelligence
Issue number2
StatePublished - Jul 2004


  • Boundary sets
  • Consistency problem
  • Inductive learning
  • Version spaces

ASJC Scopus subject areas

  • Language and Linguistics
  • Linguistics and Language
  • Artificial Intelligence


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