On the Power of Discontinuous Approximate Computations

Comparision operations are used in algebraic computations to avoid degeneracies but are also used in numerical computations to avoid huge roundo errors On the other hand the classes of algorithms using only arithmetic operations are the most studied in complexity theory and are used e g to obtain fast parallel algorithms for numerical problems In this paper we study by using a simulation argument the relative power of di erent sets of operations for computing with approximations We prove that comparisions can be simulated e ciently and with the same error bounds for most inputs by arithmetic operations when divisions are present To develop our simulation strategy we combine notions imported from approximation theory and topology with complexity and error bounds