Convexity and logical analysis of data

نویسندگان

  • Oya Ekin
  • Peter L. Hammer
  • Alexander Kogan
چکیده

A Boolean function is called k-convex if for any pair x; y of its true points at Hamming distance at most k, every point \between" x and y is also true. Given a set of true points and a set of false points, the central question of Logical Analysis of Data is the study of the Boolean functions whose values agree with those of the given points. In this paper we examine such data sets which admit k-convex Boolean extensions. We provide polynomial algorithms for nding a k-convex extension, if any, and for nding the maximum k for which a k-convex extension exists. We study the problem of uniqueness, and provide a polynomial algorithm for checking whether all k-convex extensions agree in a point outside the given data set. We estimate the number of k-convex Boolean functions, and show that for small k this number is double-exponential. On the other hand, we also show that for very large k the class of k-convex Boolean functions is PAC-learnable.

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عنوان ژورنال:
  • Theor. Comput. Sci.

دوره 244  شماره 

صفحات  -

تاریخ انتشار 2000