uniform test of algorithmic randomness over a general ... - CiteSeerX

10 PETER GÁCS
From now on, when referring to randomness tests, we will always assume that our space
X has recognizable Boolean inclusions and hence has a universal test. We fix a universal
test t µ (x), and call the function
d µ (x) = log t µ (x).
the deficiency of randomness of x with respect to µ. We call an element x ∈ X random with
respect to µ if d µ (x) < ∞.
Remark 3.2. Tests can be generalized to include an arbitrary parameter y: we can talk about
the universal test
t µ (x | y),
where y comes from some constructive topological space Y. This is a maximal (within
a multiplicative constant) lower semicomputable function (x, y, µ) ↦→ f(x, y, µ) with the
property µ x f(x, y, µ) 1.
♦
3.2. Conservation of randomness. For i = 1, 0, let X i = (X i , d i , D i , α i ) be computable
metric spaces, and let M i = (M(X i ), σ i , ν i ) be the effective topological space of probability
measures over X i . Let Λ be a computable probability kernel from X 1 to X 0 as defined
in Subsection 2.2. In the following theorem, the same notation d µ (x) will refer to the
deficiency of randomness with respect to two different spaces, X 1 and X 0 , but this should
not cause confusion. Let us first spell out the conservation theorem before interpreting it.
Theorem 2. For a computable probability kernel Λ from X 1 to X 0 , we have
λ y xt Λ ∗ µ(y) ∗ < t µ (x). (3.3)
Proof. Let t ν (x) be the universal test over X 0 . The left-hand side of (3.3) can be written as
u µ = Λt Λ ∗ µ.
According to (B.4), we have µu µ = (Λ ∗ µ)t Λ ∗ µ which is 1 since t is a test. If we show that
∗
(µ, x) ↦→ u µ (x) is lower semicomputable then the universality of t µ will imply u µ < t µ .
According to Proposition C.7, as a lower semicomputable function, t ν (y) can be written
as sup n g n (ν, y), where (g n (ν, y)) is a computable sequence of computable functions. We
pointed out in Subsection 2.2 that the function µ ↦→ Λ ∗ µ is computable. Therefore the
function (n, µ, x) ↦→ g n (Λ ∗ µ, f(x)) is also a computable. So, u µ (x) is the supremum of a
computable sequence of computable functions and as such, lower semicomputable. □
It is easier to interpret the theorem first in the special case when Λ = Λ h for a computable
function h : X 1 → X 0 , as in Example 2.7. Then the theorem simplifies to the following.
Corollary 3.3. For a computable function h : X 1 → X 0 , we have d h ∗ µ(h(x)) + < d µ (x).
Informally, this says that if x is random with respect to µ in X 1 then h(x) is essentially at
least as random with respect to the output distribution h ∗ µ in X 0 . Decrease in randomness
can only be caused by complexity in the definition of the function h. It is even easier to
interpret the theorem when µ is defined over a product space X 1 × X 2 , and h(x 1 , x 2 ) = x 1
is the projection. The theorem then says, informally, that if the pair (x 1 , x 2 ) is random with
respect to µ then x 1 is random with respect to the marginal µ 1 = h ∗ µ of µ. This is a very
natural requirement: why would the throwing-away of the information about x 2 affect the
plausibility of the hypothesis that the outcome x 1 arose from the distribution µ 1 ?
□