TY - JOUR

T1 - Version spaces and the consistency problem

AU - Hirsh, Haym

AU - Mishra, Nina

AU - Pitt, Leonard

N1 - Funding Information:
✩ This paper expands and updates the results presented by Hirsh, Mishra, and Pitt (1997). * Corresponding author. E-mail addresses: hirsh@cs.rutgers.edu (H. Hirsh), nmishra@theory.stanford.edu (N. Mishra), pitt@uiuc.edu (L. Pitt). 1 Supported in part by NSF Grant IRI-9209795. 2 Supported in part by NSF Grant EIA-0137761. 3 Supported in part by NSF Grant IIS-9907483.

PY - 2004/7

Y1 - 2004/7

N2 - 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.

AB - 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.

KW - Boundary sets

KW - Consistency problem

KW - Inductive learning

KW - Version spaces

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U2 - 10.1016/j.artint.2003.04.003

DO - 10.1016/j.artint.2003.04.003

M3 - Article

AN - SCOPUS:2442676919

SN - 0004-3702

VL - 156

SP - 115

EP - 138

JO - Artificial Intelligence

JF - Artificial Intelligence

IS - 2

ER -