Complexity Analysis via Approach Spaces

Complexity of a recursive algorithm typically is related to the solution to a recurrence equation based on its recursive structure. For a broad class of recursive algorithms we model their complexity in what we call the complexity approach space, the space of all functions in X = ]0,∞]Y , where Y can be a more dimensional input space. The set X, which is a dcpo for the pointwise order, moreover carries the complexity approach structure. There is an associated selfmap on the complexity approach space X such that the problem of solving the recurrence equation is reduced to finding a fixed point for . We will prove a general fixed point theorem that relies on the presence of the limit operator of the complexity approach space X and on a given well founded relation on Y. Our fixed point theorem deals with monotone selfmaps that need not be contractive. We formulate conditions describing a class of recursive algorithms that can be treated in this way.