Why Computational Complexity Requires Stricter Martingales

Abstract

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 ξ₀,ξ₁,ξ₂,... of random variables such that E(ξ_n+1ξ₀,..., ξ_n) = ξ_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, because under this definition every computable martingale can be simulated by a polynomial-time computable martingale.

This research was supported in part by National Science Foundation Grant 9988483.