Fixed points and closure operators: Programmological aspects
The present article continues the research of others into the declarative nature of specification languages that describe the properties of objects in the form of solutions of equations with the left-hand side solved for the unknown, i.e., equations of the form x = f(x), x {element_of} D, where D is the data universe and f is an operator on D. The main purpose of this study is to elucidate the essence of the iterative processes that can be used to find the solutions of this equation, i.e., the fixed points of the operator f. In the programmological context, the data universe should be viewed as a data type, i.e., the set of data should be endowed with an appropriate structure. Operating on the lowest level of abstraction, we regard the data universe as a partially ordered set (poset), and thus speak of the first approximation, the next approximation, and so on. The relevant programmological applications also impose certain restrictions on the operators. Specifically, we consider operators that preserve the initial relation on data. Indeed, if x{prime} is an approximation to x, then we naturally stipulate that f(x{prime}) be an approximation to f(x). Moreover, the operators in applications are always effective; and as we know, effectiveness implies monotonicity. Thus, we consider equations of the form x = f(x), x {element_of} D, where D is a poset and f a monotone operator on D. Contrary to other references, where we used direct methods traceable to the work of Tarski, the main instrument in this study are the so-called a-chains of the element a of a poset relative to an operator. This enables us, first, to describe the fixed points of a monotone operator on any poset and, second, to construct the closure operator on the set of fixed points of the initial monotone operator.
- Sponsoring Organization:
- USDOE
- OSTI ID:
- 457576
- Journal Information:
- Cybernetics and Systems Analysis, Vol. 31, Issue 1; Other Information: PBD: Sep 1995; TN: Translated from Kibernetika i Sistemnyi Analiz; No. 1, 113-121(Jan-Feb 1995)
- Country of Publication:
- United States
- Language:
- English
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