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Given a binary data stream and a filter whose output at time is for some complex , there are at most distinct values of . These values are the sums of the subsets of . It is shown that all sums are distinct unless is a unit in the ring of algebraic integers that satisfies a polynomial equation with coefficients restricted to +1, -1, and 0. Thus the size of the state space is if is transcendental, if is rational, and if is irrational algebraic but not a unit of the type mentioned. For the exceptional values of , it appears that the size of the state space grows only as a polynomial in if , but as an exponential with if .