Adleman's theorem

This article gives the statement, and proof, of one complexity class being contained in another one. The smaller complexity class is BPP and the larger complexity class is P/poly.

Statement

In terms of complexity classes

The complexity class BPP (bounded error probabilistic polynomial time -- polynomial time where the probability of error is bounded below $1/2$) is contained in the complexity class P/poly (languages where, for every length of input string, there is an advice string using which the algorithm runs in polynomial time).

Loose philosophical formulation

The statement is that:

Probabilistic algorithm that is independent of length $\to$ Deterministic algorithm that depends on length

In other words, we can shed the probabilistic nature of the algorithm but must instead accept length-specific advice that shapes the algorithm.