The complexity of Tarski's fixed point theorem

Tarski's fixed point theorem guarantees the existence of a fixed point of an order-preserving function f:L->L defined on a nonempty complete lattice (L,@?) [B. Knaster, Un theoreme sur les fonctions d'ensembles, Annales de la Societe Polonaise de Mathematique 6 (1928) 133-134; A. Tarski, A lattice theoretical fixpoint theorem and its applications, Pacific Journal of Mathematics 5 (1955) 285-309]. In this paper, we investigate several algorithmic and complexity-theoretic topics regarding Tarski's fixed point theorem. In particular, we design an algorithm that finds a fixed point of f when it is given (L,@?) as input and f as an oracle. Our algorithm makes O(log|L|) queries to f when @? is a total order on L. We also prove that when both f and (L,@?) are given as oracles, any deterministic or randomized algorithm for finding a fixed point of f makes an expected @W(|L|) queries for some (L,@?) and f.

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