The Convergence of Functions to Fixedpoints of Recursive Definitions

The classical method for constructing the least fixedpoint of a recursive definition is to generate a sequence of functions whose initial element is the totally undefined function and which converges to the desired least fixedpoint. This method, due to Kleene, cannot be generalized to allow the construction of other fixedpoints. In this paper we present an alternate definition of convergence and a new fixe&poinc access method of generating sequences of functions for a given recursive definition. The initial function of the sequence can be an arbitrary function, and the sequence will always converge to a fixedpoint that is “close” to the initial function. This defines a monotonic mapping from the set of partial functions onto the set of all fixedpoints of the given recursive definition. This research was supported by the Advanced Research Projects Agency of the Department of, Defense under Contract MDA 903-76-C-0206. The authors were also affiliated with the Departmivt of Applied Mathematics of the Weizmann Institute of Science during the period of this research. Manna & Shamir