In this paper we study fixpoints of operators on lattices and bilattices in a systematic and principled way. The key concept is that of an approximating operator, a monotone operator on the product bilattice, which gives approximate information on the original operator in an intuitive and well-defined way. With any given approximating operator our theory associates several different types of fixpoints, including the Kripke-Kleene fixpoint, stable fixpoints, and the well-founded fixpoint, and relates them to fixpoints of operators being approximated. Compared to our earlier work on approximation theory, the contribution of this paper is that we provide an alternative, more intuitive, and better motivated construction of the well-founded and stable fixpoints. In addition, we study the space of approximating operators by means of a precision ordering and show that each lattice operator O has a unique most precise - we call it ultimate - approximation. We demonstrate that fixpoints of this ultimate approximation provide useful insights into fixpoints of the operator O. We then discuss applications of these results in logic programming.
|Number of pages||38|
|Journal||Information and Computation|
|State||Published - Jul 1 2004|
Bibliographical noteFunding Information:
This paper is based upon work supported by the National Science Foundation under Grants Nos. 9874764, 0097278, and 0325063. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors gratefully acknowledge helpful comments by anonymous referees and by Nikolai Pelov.
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ASJC Scopus subject areas
- Theoretical Computer Science
- Information Systems
- Computer Science Applications
- Computational Theory and Mathematics