Kleene chain completeness and fixedpoint properties

Abstract

Fixedpoint theorems play a fundamental role in denotational semantics of programming languages. In 1955, Tarski [6] and Davis [l] showed that a lattice L is complete if and only if every monotonic function f: L + L has a fixedpoint. Since then, other fixedpoint properties are considered, by imposing either the function to be o-continuous or the fixedpoint to be the least fixedpoint. The key question is whether such fixedpoint properties can as well be characterized by some completeness properties of the given partially ordered set. In this direction, Markowsky [2] showed that a partially ordered set is chain-complete if and only if every monotonic function f: D + D has a least fixedpoint. Suppose we replace monotonic functions by o-continuous functions which play a prominent role in Scott’s theory of computation [4,5]. A slight modification of Tarski’s proof shows that if a partially ordered set D with a least element 1 is o-chain complete, then every w-continuous function f: D + D has a least fixedpoint given by UnEWfn (I). The latter result is the wellknown Tarski-Kleene-Knaster theorem. In 1978, Plotkin asked the validity of the converse of Tarski-Kleene-Knaster theorem beta use an affirmative answer would give us a characterization of w-chain complete pin-tially ordered sets in terms of the least fixedpoint property for o-continuous functions. Throughout the paper, D stands for a partially .3rdered set with a least element 1. Let us consider Plotkin’s puzzle in the followirlg version: Given a D, if every o-continuous function f: D + D has a least fixedpoint given by UIltWf’*(l), is D o-chain complete? In this paper, we answer the problem negatively (Mashburn obtained the same result independently in [3j). Despite this negative answer, we show a rather astonishing result, namely: If D is either countable or countably algebraic, then the converse of Tarski-Kleene-Knaster theorem is true. This positive answer is rather pleasing because most of the D’s used in denotational semantics are either countable or countably algebraic.