There is a simple technique to translate between truth-table functionals and quick process machines that allows further characterisations of computable randomness, Schnorr randomness and weak randomness in terms of truth-table reducibility. However, for these characterisations to hold, we need to be careful about how we define the use of a truth-table reduction. Nerode observed that a truth-table reduction can be regarded as a Turing reduction that is total on all oracles [68]. We will take this as our definition of a truth-table reduction. Given a truth-truth-table reduction Φ we now define φX(n) to be the largest query made of the oracle X during the computation of ΦX(m)for any m ≤ n.
Theorem 6.3.1. (i). A sequence X ∈ 2ω is not computably random if and only if there exist a truth-table functional Φ, and a sequence Y ∈ 2ω, such that ΦY = X and ∀c ∃n φY(n) ≤ n − c.
(ii). A sequence X ∈ 2ωis not Schnorr random if and only if there exist a truth-table functional Φ, a sequence Y ∈ 2ω, and a strictly increasing computable function msuch that ΦY = X and ∃∞n φY(m(n)) ≤ m(n) − n.
6.3. QUICK PROCESS MACHINES AND TRUTH-TABLE FUNCTIONALS 107
(iii). A sequence X ∈ 2ωis not weakly random if and only if there exist a truth-table functional Φ, a sequence Y ∈ 2ω, and a strictly increasing computable function msuch that ΦY = X and ∀n φY(m(n)) ≤ m(n) − n.
Proof. The right to left direction of the above statements can be established by constructing a martingale d from a truth-table functional Φ as follows. Let d(σ) = µ{X : ΦX σ} · 2|σ|. The fact that Φ is total on all oracles makes d computable. Now if ΦY = X, then we have that d(X n) ≥ 2n−φY(n). The right to left direction for (i) and (ii) follow immediately. For (iii) an application of Lemma 6.2.15 is needed.
To establish the left to right direction, given a quick process machine P , we define a truth-table functional Φ that computes ΦX(n) by finding the shortest initial segment of τ of X such that |P (τ )| > n and setting ΦX(n)to be the nth bit of this output.
Now if X is not computably random, then by Corollary 6.2.14, there is some quick process machine P such that ∀c ∃n CP(X n) = n − c. Further by applying Lemma 6.2.10, for all c, we can take some τcwith P (τc) = X (|τc|+c) and such that {τc : c ∈ ω} is a chain. Let Y = S
cτc. Thus ΦY = X and φY(|P (τc)| − 1) ≤ |P (τc)| − c. If X is Schnorr random or weakly random then the proof proceeds similarly.
The author would like to note that this characterisation of computable ran-domness in terms of truth-table reducibility has been independently arrived at by Laurent Bienvenu and Chris Porter.
Chapter 7
Non-Computable Measures
This chapter is joint work with Joseph Miller of the University of Wisconsin-Madison.
It is based on research undertaken with Miller at the University of Heidelberg in July 2009. The work in this chapter has been accepted for publication in the Transactions of the American Mathematical Society.
7.1 Defining randomness
Let X be an element of Cantor space and µ a Borel probability measure on Cantor space. What should it mean for X to be random with respect to µ? If µ is a computable measure, then early work of Levin showed that µ-randomness can be seen as essentially a variant on randomness for Lebesgue measure [94].
This leaves the question of how to define randomness if µ is non-computable.
We will show that the two approaches that have previously been used to de-fine µ-randomness for non-computable measures µ, are equivalent. Later, in Theorem 7.4.12, we will provide another characterisation of µ-randomness us-ing the enumeration degrees.
We would like to find a natural generalisation of Martin-L ¨of randomness to non-computable probability measures. One approach is to generalise Martin-L ¨of tests. This approach immediately runs into the difficult question of what sort of oracle access a test should have. It is reasonable to expect that a test for a measure µ should be able to compute the µ-measure of any basic clopen set. However, there are continuum many Borel probability measures on Can-tor space, so in order to make these measures accessible to the techniques of computability theory, we will make use of some basic concepts of computable analysis. We will define all the concepts we need. For further background on computable analysis, the reader is referred to Weihrauch [92], who gives a modern development of the subject. Classical computability theory studies
Cantor space (2ω) and Baire space (ωω). The main idea behind computable analysis is to transfer the notions of computability theory to other structures via representations of those structures. If S is a set, a representation of S is just a surjective function (possibly partial) ρ : 2ω → S. The representation induces a computability-theoretic structure on S. We will also use the word “represen-tation” in another, less standard, sense. If R ∈ 2ω and ρ(R) = x, we call R a representation of x.
We will takeP(2ω)to be the set of all Borel probability measures on Cantor space. We will let ρ : 2ω → P(2ω)be a representation of P(2ω). In Section 7.2 we will give a detailed definition of such a representation ρ but for now it is enough to specify that if ρ(R) = µ, then we can uniformly in R compute the µ measure of any basic clopen set in Cantor space.
As we access measures via representations, one approach is to define ran-domness in terms of representations. The following definitions, while not identical, are equivalent to that of Reimann [72] and Reimann and Slaman [73, 74].
Definition 7.1.1. Let µ ∈P(2ω)and let R ∈ 2ω be a representation of µ.
(i). An R-test is a uniform (in R) sequence {Vi}i∈ω of Σ01(R) sets such that µ(Vi) ≤ 2−i.
(ii). X ∈ 2ωpasses an R-test if X 6∈T
iVi. (iii). X ∈ 2ωis R-random if it passes all R-tests.
A universal R-test exists for the same reason that a universal Martin-L ¨of test exists. Given R, we would like to enumerate all R-tests by enumerating all (uniform in R) sequences of R-c.e. sets, halting any enumeration if it would exceed the measure bound. There is a small technical obstruction. Using R, we can compute a sequence, approximating from above, the µ-measure of a basic clopen set. Hence, we can pause an enumeration until a stage when our approximation from above guarantees that we can add the next element without exceeding the measure bound. Note that this could cause a problem if some test {VnR}n∈ω had ViR = 2−i for some i. The enumeration of this test could be paused forever. However, in this case, the test {Vn+1R }n∈ω defines the same null set and avoids this problem. This shows that we can (essentially) enumerate all R-tests, uniformly in R, so we can build a universal R-test. Even better, because the construction is uniform, there is a uniform sequence of c.e.
sets Un such that if UnR = {[τ ] : hτ, σi ∈ Un and σ ≺ R}, then {UnR}n∈ω is a universal R-test. Call {Un}n∈ω a universal oracle Martin-L¨of test.
7.1. DEFINING RANDOMNESS 111
As noted by Reimann, the problem with Definition 7.1.1 is that it is depen-dent on the representation. Given any measure, it is possible to encode any sequence into some representation of that measure. Hence for all µ ∈ P(2ω) and all X ∈ 2ω, there is a representation R of µ such that X is not R-random.
A natural way to overcome this problem is with the following definition.
Definition 7.1.2. A sequence X ∈ 2ωis µ-random if there exists a representation Rof µ such that X is R-random.
Our goal is to show that, at least in Cantor space, this definition gives the same class of randoms for a measure as does the concept of a uniform test.
Uniform tests are an alternative approach to randomness for non-computable measures. They were introduced by Levin and developed by G´acs, and also by Hoyrup and Rojas [38, 40, 57]. While uniform tests can be applied to general probability spaces, in this chapter, we will only be concerned with Cantor space. We take {Bi}i∈ωto be an enumeration of open balls inP(2ω). The details of this enumeration will be provided in the following section.
Definition 7.1.3. Consider a function t : P(2ω) × 2ω → R≥0∪ {∞}.
(i). The graph of t is {(µ, X, r) : t(µ, X) > r}. We say that the under-graph of t is c.e. open, if it is equal toS
hi,σ,qi∈W Bi × [σ] × [0, q) for some c.e. set W ⊆ ω × 2<ω× Q.
(ii). We call t a uniform test if its under-graph is c.e. open and for every µ ∈ P(2ω)we haveR t(µ, X) dµ ≤ 1.
(iii). X ∈ 2ωpasses a test t for a measure µ if t(µ, X) is bounded.
(iv). X ∈ 2ωis µ-random for uniform tests if it passes all tests for measure µ.
By a straightforward theorem of G´acs, later refined by Hoyrup and Rojas, it is sufficient to consider a single universal uniform test.
Definition 7.1.4. A uniform test t is universal if for all uniform tests t0 there is a constant c > 0 such that for all µ ∈ P(2ω) and X ∈ 2ω, we have t(µ, X) ≥ c · t0(µ, X).
Theorem 7.1.5(G´acs; Hoyrup and Rojas [38, 40]). There exists a universal uni-form test.
The following theorem will establish that these two approaches, one based on representations and the other on uniform tests, are equivalent.
Theorem 7.1.6. For any measure µ and X ∈ 2ωwe have that X is µ-random if and only if X is µ-random for uniform tests.
Before proving this theorem, we need to take a more detailed look atP(2ω) and at representations of probability measures.