Why Computational Complexity Requires Stricter Martingales

The word "martingale" has related, but different, meanings in probability theory and theoretical computer science. In computational complexity and algorithmic information theory,a martingale is typically a function d on strings such that E(d(wb)|w) = d(w) for all strings w, where the conditional expectation is computed over all possible values of the next symbol b. In modern probability theory a martingale is typically a sequence ?0, ?1, ?2, . . . of random variables such that E(?n+1|?0, . . . , ?n) = ?n for all n.This paper elucidates the relationship between these notions and proves that the latter notion is too weak for many purposes in computational complexity,b ecause under this definition every computable martingale can be simulated by a polynomial-time computable martingale.