RELATIVIZATION IN RANDOMNESS
JOHANNA N.Y. FRANKLIN
Abstract. We examine the role of relativization in randomness through the lenses of van Lambalgen’s Theorem and lowness for randomness. Van Lambalgen’s Theorem states that the join of two
sequences is random if and only if each of the original sequences is random relative to the other; a
sequence is said to be low for randomness if it is not computationally strong enough to derandomize
any random sequence. The way in which one relativizes a randomness notion can affect the truth
of van Lambalgen’s Theorem or the class of sequences that are low for that randomness notion,
making these topics ideal approaches for examining different relativizations and thus for gaining
insight into the randomness notions we relativize.
1. Introduction
Relativization is one of the cornerstones of computability theory: it allows us to think about
sets of natural numbers not only as individual objects but in relation to each other. In this paper,
I will explore the ways in which we make use of the concept of relativization in the study of
algorithmic randomness. We begin with a basic introduction to relativization as it appears in
classical computability theory. In Section 2, we consider van Lambalgen’s Theorem, and in Section
3, we consider lowness for randomness. Finally, in Section 4, we summarize these results and
suggest some further areas for exploration.
1.1. Notation. Our notation will generally follow [13] and [45]. We will use lowercase Roman
letters such as n and m to represent natural numbers, lowercase Greek letters such as σ and τ
to represent binary strings of finite length, and uppercase Roman letters such as A and B to
represent subsets of the natural numbers. Often, we will associate a subset of the natural numbers
A with its characteristic function χA , which can be represented as the infinite binary sequence
χA (0)χA (1)χA (2) . . .. This means that we can discuss subsets of ω and infinite binary sequences
interchangeably. If we wish to consider the initial segment of length n of a binary sequence A, we
will write An, and if we want to discuss the length of a finite binary string σ, we will write |σ|.
When written without qualifiers, a “sequence” may be taken to be infinite and a “string” may be
taken to be finite.
We will work in the Cantor space, which is the probability space of infinite binary sequences. We
say that the basic open set defined by a finite binary string σ is [σ]: the set of all infinite binary
sequences extending σ. When we discuss measure in the Cantor space, we will always use Lebesgue
measure, or the standard “coin-flip” measure, and symbolize this with µ: the Lebesgue measure of
1
a basic open set [σ] is 2|σ|
, or the fraction of infinite binary sequences that begin with σ.
Date: May 31, 2015.
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1.2. Relativization in computability theory. Before we consider relativization in the context of
algorithmic randomness, let us discuss it first in its most basic form: relativization of sets of natural
numbers. (For a basic reference, see [45] or Chapter 2 of [13].) The first functions we consider in
computability theory are the primitive recursive functions: those formed by the closure of the
constant function 0, the successor function, and the projection functions under the operations of
composition and primitive recursion. However, this does not allow us to consider partial functions in
general or functions associated with algorithms that involve unbounded searches. In order to do so,
we close the class of primitive recursive functions under unbounded search using the minimization
operator µ to get the class of partial recursive functions. This class can also be defined via the
λ-calculus, Turing machines, or register machines and is generally called the partial computable
functions [45]. Since all of the different approaches to computability that people have developed
thus far have resulted in the same class of functions, we adopt the Church-Turing thesis, which
states that if a function is partial recursive if and only if an informal description of an algorithm
that will compute this function can be given [6, 49].
This universality makes it reasonable to use these functions as the basic tools of computability
theory. If such a function ϕ is total, we call it simply a computable function, and a set that is the
domain of a partial computable function (or, alternately, the range of a computable function or
the empty set) is called a computably enumerable (c.e.) set. We note that since there is a natural
computable bijection between the natural numbers and the set of finite binary strings, we may
sometimes talk about a c.e. set of finite binary strings without loss of generality.
When we discuss relativized computation, though, we need one more thing: the use of an oracle.
Here, we allow the partial computable function to use an initial segment of a given set of natural
numbers A in its computation.1 Instead of saying, for instance, that ϕ(n) = m, we say that
Φ(n, σ) = m, meaning that given an input n and a finite binary string σ, Φ outputs m. Since only
a finite initial segment of a sequence A can be used in any particular computation, two infinite
sequences A and B may have the same initial segment σ and thus result in the same answer for
the functional Φ with input n but not necessarily result in the same answer for the functional Φ for
other inputs because longer initial segments may be required for those computations on which they
may not agree. Since we generally use an infinite sequence A as an oracle, we will write ΦA (n) = m
to indicate that there is an initial segment σ of A such that Φ(n, σ) = m.
This allows us to consider the relationship between two sets A and B. We say that A is Turing
reducible to B if there is a functional Φ such that ΦB (n) = χA (n) for all natural numbers n; that
is, if some algorithm can, given a natural number n and an appropriately long initial segment of
B, determine whether n is in A. We express this relationship as A ≤T B. For instance, any set
A is Turing reducible to its complement Ac : If we wish to find out whether n is in A, we look at
Ac (n + 1), which is Ac (0) . . . Ac (n). If Ac (n) = 0, n must be in A; if Ac (n) = 1, n cannot be
in A. In fact, for any set A, A ≤T Ac and Ac ≤T A. Two mutually reducible sets like these are
Turing equivalent and are said to be in the same Turing degree. Two sets in the same Turing degree
have the same computational strength since given knowledge of the first, we can know everything
about the second and vice versa. We can also compare the computational strength of two sets in
different Turing degrees: we can say that B is stronger computationally than A if A <T B (that
is, if knowledge of B allows us to compute A but not vice versa), or we can say that B and A are
incomparable if neither can compute the other (B|T A).
1We call a partial computable function that is permitted to use an oracle like this a functional.
RELATIVIZATION IN RANDOMNESS
3
Turing reducibility is actually quite weak. For instance, there is no limitation on the length of
the initial segment of B required to compute χA (n). We only know that such an initial segment
must exist. Furthermore, we do not know whether, for any given functional Φ, there is an oracle
A and an input n for which ΦA (n) does not converge. We now define two stronger reducibilities
to address these concerns, beginning with the first. If there is a functional Φ and a computable
function f such that Φ(n, B f (n)) = χA (n) for every natural number n, we say that A is weak
truth-table reducible to B and write A ≤wtt B. In this case, the only restriction on the computation
is that an initial segment of B with a length bounded computably in n must be enough to compute
χA (n). For instance, we can say that A is weak truth-table reducible to Ac for any A: we can see
from the example above that f (n) = n + 1 is an appropriate computable function.
Note that for Turing reductions and weak truth-table reductions, ΦX (n) may diverge for any X.
This leads us to an even stronger reducibility: truth-table reduction. We say that A is truth-table
reducible to B if it is weakly truth-table reducible to B via a functional Φ that converges not only
for B on every input, but for every possible oracle on every input.
We can talk about the weak truth-table degrees and the truth-table degrees just as we talked
about the Turing degrees before. It is clear that
A ≤tt B =⇒ A ≤wtt B =⇒ A ≤T B.
These different reducibilities will come into play when we discuss van Lambalgen’s Theorem in
Section 2. However, first we must discuss relativization specifically in the context of randomness.
1.3. Basic randomness concepts. Just as sets that are not themselves computable can be computed by stronger sets, random sequences can be “derandomized” if certain other sequences are
used as oracles. For instance, no sequence is random relative to itself. Furthermore, if A ≤T B,
then A is not random relative to B: B can be used to compute A and thus to predict it perfectly.
To treat this topic in proper detail, we begin by recalling the definitions of Martin-Löf randomness,
computable randomness, and Schnorr randomness, the three main types of randomness we will use
as case studies in this paper. We will present two types of definitions, one based on the idea that a
random sequence should not be predictable and the other based on the idea that a random sequence
should have no rare measure-theoretic properties. We first turn our attention to the predictability
characterization and define the type of function we will use to formalize this concept.
Definition 1.1. A martingale m is a function m : 2<ω → R≥0 such that for every finite binary
string σ,
m(σ a 0) + m(σ a 1)
m(σ) =
.
2
In other words, a martingale is a fair betting function: the amount you lose if you are wrong is
equal to the amount you win if you are right. We say that a martingale is c.e. if the values m(σ) are
uniformly left-c.e. real numbers (that is, computably approximable from below) and computable if
the values m(σ) are uniformly computable real numbers (that is, computably approximable from
both above and below).
Now we can discuss the “winning conditions” for a martingale m on a sequence: we say that m
succeeds on a sequence A if
lim sup m(An) = ∞.
n
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This tells us that for any real number r, we can find an n such that m(An) ≥ r, so m succeeds on
A if the amount m can win by betting on A is unbounded.
Now we can define the three randomness notions mentioned above. Recall that an order function
is computable, nondecreasing, and unbounded.
Definition 1.2. Let A be an infinite binary sequence.
(1) A is Martin-Löf random if no c.e. martingale succeeds on it.
(2) A is computably random if no computable martingale succeeds on it.
(3) A is Schnorr random if there is no computable martingale m such that for some order
function h, m(An) ≥ h(n) for infinitely many n.
In short, Martin-Löf and computable randomness are distinguished by the type of martingale that
will not succeed on such a sequence, and a Schnorr random sequence is one on which a computable
martingale may succeed but not very quickly. We call the classes of Martin-Löf, computable, and
Schnorr random sequences ML, Comp, and Schnorr, respectively. We note that Schnorr randomness
is a strictly weaker notion than computable randomness and that computable randomness is a
strictly weaker notion than Martin-Löf randomness (so Schnorr ⊃ Comp ⊃ ML).
The other characterization that we will mention at this point is measure-theoretic.
Definition 1.3. A Martin-Löf test is a sequence of uniformly c.e. sets hVi i where each Vi is a set
of finite binary strings and µ([Vi ]) ≤ 21i for each i. An infinite binary sequence A is Martin-Löf
random if and only if for every Martin-Löf test hVi i, A 6∈ ∩i [Vi ].
In other words, a sequence is Martin-Löf random if it has no properties that can be “captured”
by a Martin-Löf test.
While the test definition of computable randomness is too complicated to give here [9], Schnorr
randomness can be defined similarly. A Schnorr test is a Martin-Löf test with an additional
requirement: instead of saying that µ([Vi ]) ≤ 21i for each i, we require that µ([Vi ]) = 21i for each
i. This leads to a more effective notion, since this means that we can tell whether σ is in Vi for a
1
of 21i . If σ has been enumerated
given i by enumerating Vi until the measure of [Vi ] is within 2|σ|
into Vi at this point, clearly it is in Vi ; otherwise, there is not enough space for it to go in later and
σ will never enter Vi . As for Martin-Löf randomness, a sequence A is Schnorr random if and only
if for every Schnorr test hVi i, A 6∈ ∩i [Vi ].
One difference between Martin-Löf randomness and computable and Schnorr randomness is the
presence or absence of a universal martingale or test. We note without ceremony that there is a
universal Martin-Löf test: a test hUi i such that for every Martin-Löf test hVi i, ∩i [Vi ] ⊆ ∩i [Ui ].
In short, if we are trying to determine whether a sequence is Martin-Löf random, we need only
check to see whether it is contained in the intersection of the universal test. However, there is no
universal test for Schnorr randomness. In fact, given any Schnorr test, we can find a computable
sequence that it does not capture. Since no computable sequence is Schnorr random, universality
fails in a very strong way indeed. The same situation holds for martingales: there is a universal
martingale for Martin-Löf randomness but not for computable or Schnorr randomness.
When we relativize a martingale to a sequence B, we permit the use of B as an oracle in
the calculation of the martingale m. Instead of m being c.e. or computable, it would be c.e. or
computable in B. For instance, we could build a martingale computable from B that succeeds on
B by betting its entire capital on the correct bit of B at each place.
RELATIVIZATION IN RANDOMNESS
5
When we relativize a test for randomness to a sequence B, we simply permit the use of B as
an oracle in the enumeration of the test components Vi . For instance, if we want to build a test
that captures B itself, we can use B as an oracle and define ViB to be {B(0) . . . B(i − 1)}. This
means that each ViB will consist of an initial segment of B of length i, and clearly we will have
B ∈ ∩i [ViB ]. We could also build a test that captures the sequences that were computable from B,
for instance. We can write the class of sequences that are Martin-Löf (or Schnorr) random relative
to B as MLB (or SchnorrB ). Note that for Martin-Löf randomness, defining MLB is as simple as
calculating hUiB i, but it is more complicated for Schnorr randomness because we must consider
each Schnorr test individually.
This gives us yet another way of comparing sequences. Turing reducibility allows us to compare
their computational strength; relative randomness allows us to compare their randomness content.
In fact, several reducibilities have been defined based on the idea of relative randomness (see Chapter
9 of [13]). However, for the moment, we will confine ourselves to a discussion of the consequences
of a sequence being Martin-Löf random, computably random, or Schnorr relative to another. Our
main topics of discussion will involve the following points:
• If two sequences are combined in a computable way, when does this result in a random
sequence?
• Which sequences are so information-poor that they are incapable of derandomizing any
random sequence?
The first of these questions will be addressed in Section 2 and the second in Section 3. At some
points, I may simply write “random.” If I do so, this means that what I am writing is true regardless
of which of these three randomness notions is under consideration.
2. van Lambalgen’s Theorem
We begin by considering what it means to combine two sequences. Our plan is to do so in such
a way that the resulting sequence will computably encode both of the original sequences. We say
that the join of two sets A and B is the set
A ⊕ B = {2n | n ∈ A} ∪ {2n + 1 | n ∈ B}.
This set codes A in its even bits and B in its odd bits. However, there is no reason we have to
partition the bits of the join based on their parity: we could choose any computable subset F of ω
and let the nth F -place be a 1 if n ∈ A and 0 otherwise and the nth F c -place be a 1 if n ∈ B and
0 otherwise. In cases where we do not use the traditional odd-even join and instead decompose ω
into F and F c for some other F , we will write A ⊕F B. By the Church-Turing thesis, we can see
that A ≤T A ⊕ B, B ≤T A ⊕ B, and A ⊕ B ≡T A ⊕F B for every computable set F .
Now we recall the first question presented above: If two sequences are combined in a computable
way, when does this result in a random sequence? We can see immediately that it need not always
do so. It is clearly necessary that both sequences be random themselves. However, that alone is
not enough to guarantee the randomness of their combination. Even if A is a random sequence,
A⊕A won’t be random because half of the bits will be predictable from the bits immediately before
them. What if we take A ⊕ B for two different random sequences A and B? This may still not
be random. For instance, suppose that A ≤T B. Since B can compute A, it may be possible to
predict the bits of A in A ⊕ B from the bits of B that have already appeared, which will result in
the same problem as before.
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FRANKLIN
This question was answered by van Lambalgen for Martin-Löf randomness in 1990 [50]. Intuitively, the answer to this question for Martin-Löf randomness is precisely what it should be. For
A ⊕ B to be Martin-Löf random, not only do both A and B have to be Martin-Löf random, they
must be random relative to each other. We state this result formally here:
van Lambalgen’s Theorem. [50] The following statements are equivalent.
(1) A is Martin-Löf random relative to B and B is Martin-Löf random relative to A.
(2) A ⊕ B is Martin-Löf random.
The claim that (1) implies (2) is known as the “hard” direction and the claim that (2) implies
(1) is known as the “easy” direction.
To prove the “easy” direction, we simply assume that B is not Martin-Löf random relative to
A and that thus B ∈ ∩i UiA and use this to produce a Solovay test2 that contains A ⊕ B. This
contradicts the assumption that A ⊕ B is Martin-Löf random.
One of the standard proofs of the “hard direction” is by Nies [10]. We assume for a contradiction
that A ⊕ B is not Martin-Löf random and, from a Martin-Löf test containing A ⊕ B, produce a
sequence of sets hSi i. We then construct a Martin-Löf test based on these Si s. If this test contains
A, then we have a contradiction because A cannot even be Martin-Löf random unrelativized; if it
does not, we can use this information about A to construct an A-Martin-Löf test that contains B.
van Lambalgen’s Theorem has a very interesting porism. The proof shows us that A ⊕ B is
Martin-Löf random precisely when A is Martin-Löf random and B is Martin-Löf random relative
to A. However, the roles of A and B in this proof are interchangeable, so we can draw the following
conclusion:
Porism 2.1. A is Martin-Löf random relative to B if and only if B is Martin-Löf random relative
to A.
The argument is rather straightforward. If A is Martin-Löf random and B is Martin-Löf random
relative to A, then A ⊕ B is Martin-Löf random. Since A ⊕ B is Martin-Löf random, B ⊕ A must
be Martin-Löf random, and therefore B must be Martin-Löf random and A must be Martin-Löf
random relative to B.
In short, if two sequences are Martin-Löf random and the first is random relative to the second,
the second must also be random relative to the first. This sort of symmetry is quite surprising and
very pleasing.
However, we should consider van Lambalgen’s Theorem in the context of not only Martin-Löf
randomness but also computable and Schnorr randomness. For a long time, it was assumed that the
“hard” direction of van Lambalgen’s Theorem was also true for computable and Schnorr randomness
“with essentially the same proof” (p. 276 of [13]). However, no formal proof of either statement was
given until 2011, when Franklin and Stephan proved that the “hard” direction holds for Schnorr
randomness.
Theorem 2.2. [20] If A is Schnorr random and B is Schnorr random relative to A, then A ⊕ B
must be Schnorr random.
Their proof, like Nies’s for the Martin-Löf case, involves supposing that A is Schnorr random, B
is A-Schnorr random, and A ⊕ B is not Schnorr random for a contradiction and then considering
2A Solovay test gives us another measure-theoretic characterization of Martin-Löf randomness [46].
moment, simply note that anything contained in a Solovay test is not Martin-Löf random.
For the
RELATIVIZATION IN RANDOMNESS
7
two separate cases. However, unlike Nies’s proof, they use martingales instead of the test characterization. They use a computable martingale that succeeds on A ⊕ B in the sense of Schnorr to
construct a set S of lengths of strings on which this martingale has values above a certain level. If
there are infinitely many lengths in S with a certain technical property, then A can be shown to be
not Schnorr random; if there are finitely many, then B can be shown to be not A-Schnorr random.
Miyabe and Rute have an alternate proof of this result in [39] that uses integral tests.3 They
begin with a Schnorr integral test witnessing that A ⊕ B is not Schnorr random and use this to
construct a Schnorr integral test relative to A witnessing that B is not Schnorr random relative to
A.
No such proof is known for computable randomness, so the following question is still open:
Question 2.3. If A is computably random and B is computably random relative to A, must A ⊕ B
be computably random?
On the other hand, it has long been known that the “easy” direction of van Lambalgen’s Theorem
does not hold for Schnorr or computable randomness; namely, that there is a Schnorr (computably)
random sequence A ⊕ B such that at least one of A and B is not Schnorr (computably) random
relative to the other. While the first result to this effect appears in [35], the following argument
appearing in [41], due to Kjos-Hanssen, is simpler.
A high Turing degree is a Turing degree with an element that wtt-computes a function that
dominates every computable function, and a minimal Turing degree is one with no degree strictly
between itself and 0. The Cooper Jump Inversion Theorem states that a minimal high Turing
degree exists [7], and every high Turing degree contains a computably random sequence A ⊕ B [42].
Since A and B are Turing computable from A ⊕ B and A ⊕ B belongs to a minimal degree, A and
B must either be in the same Turing degree as A ⊕ B or computable. Since no random sequence
is computable, they must both be in the same Turing degree as A ⊕ B. However, this means that
A ≡T B, so A and B are mutually computable and therefore neither of them is computably random
relative to the other.
Since every high degree contains a Schnorr random sequence that is not computably random
[42], the above argument holds for Schnorr randomness as well.
Additional results have been obtained in this area. Yu proved that this direction of van Lambalgen’s Theorem fails for computably random sequences bounded below 00 by a c.e. set [51], and
Franklin and Stephan proved that every high degree contains a computably random sequence for
which van Lambalgen’s Theorem fails for both computable and Schnorr randomness [20].
Franklin and Stephan’s proof in particular sheds light on the reason that a Schnorr or computably
random sequence does not necessarily decompose into two mutually Schnorr or computably random
sequences. They begin with an arbitrary high degree and a set A in it that wtt-computes a function
that dominates all computable functions. As a preliminary, they use this dominating function
mentioned above to construct a martingale m that subsumes all computable martingales; that is,
if any computable martingale succeeds on a sequence, m will too. They then construct a sequence
B that is contained in that high degree on which m does not succeed. This sequence will therefore
be computably random.
3An integral test is a lower semicomputable function on 2ω whose integral is ≤ 1 (in the case of an Martin-Löf
integral test) or equal to 1 (in the case of a Schnorr integral test) [33, 37].
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FRANKLIN
To ensure that van Lambalgen’s Theorem fails for B, they construct it in finite initial segments,
alternating two kinds of steps in the construction. In the first kind of step, they extend the current
finite segment of B by another that has two properties:
(1) m does not increase very much on this new part of B and
(2) this new part of B codes an initial segment of the set A.
The first condition ensures that B is not computably or Schnorr random itself; the second
ensures that A ≤T B. The rest of the construction makes it clear that B ≤T A: a martingale
that is computable from A determines B, and all other aspects of the construction are themselves
computable.
In the second kind of step, the current finite initial segment σ of B is extended by a single bit
that can be determined from σ. The locations of these bits can be calculated computably, which
determines the way the sequence is determined for the join: the locations of these bits form the
computable set F . If B is separated into two pieces, one comprised of the bits at locations in F
and one comprised of the bits at locations in F c , we get two infinite binary sequences. We call the
sequence formed by the F -bits X and the sequence formed by the F c -bits Y . Clearly, X ⊕F Y = B.
However, X ≤T Y , so X is not computably random relative to Y . In fact, a slight additional
argument can be used to show that X is not even Schnorr random relative to Y (this argument is
necessary because X can still be Schnorr random relative to Y if X is computable from Y as long
as the computation of X from Y is very slow).
Now we can develop our intuition for the ways in which this direction of van Lambalgen’s
Theorem can fail for computable or Schnorr randomness. Suppose that we have a computably
random sequence B that can be written as X ⊕ Y , where X is not computably random relative to
Y . This means that there must be a computable martingale m0 relative to Y that succeeds on X.
How, then, could that martingale not be converted to a computable martingale that would succeed
on X ⊕ Y ? There are two possible ways. The first is the simplest: it may not be possible to define
such a martingale on X ⊕ Z for every sequence Z. If that happens, then the result will not be a
computable martingale. The second is inherent in the argument above: perhaps m0 would have to
see too many bits of Y before betting on a bit of X. In Franklin and Stephan’s proof, their choice
of F guarantees that there is a computable bound on the amount of Y that must be seen before a
given bit of X is determined. If no such bound exists, a computable martingale cannot be found.
This situation is reminiscent of the difference between Turing reducibility and truth-table reducibility: the conditions for success mirror the additional requirements for a Turing reduction to
be a truth-table reduction. This leads us to consider a new question. If we consider a different
reducibility, can we reformulate van Lambalgen’s Theorem in such a way that this direction would
hold for computable randomness or for Schnorr randomness? It turns out that the answer is yes,
that van Lambalgen’s Theorem is indeed sensitive to the reducibility used in the relativization of
these randomness notions.
It had been clear for some time that computable and Schnorr randomness behaved much more like
each other than like Martin-Löf randomness. As previously mentioned, there is a universal MartinLöf test, but no universal test for computable or Schnorr randomness. Martin-Löf randomness
can be characterized simply in terms of predictability, measure, or Kolmogorov complexity (to be
discussed in Section 3); the definitions are more complicated for the other two. The ways these
randomness notions behave with respect to van Lambalgen’s Theorem only confirm this difference.
RELATIVIZATION IN RANDOMNESS
9
It is intuitive to see why this should be so. Martin-Löf randomness is based in c.e. notions.
The martingales are c.e. in the predictability formulation, and the test components need only be
approximable from below. Computable randomness and Schnorr randomness, on the other hand,
are based in computable notions. The martingales involved must be computable, and the test
components must be approximable from both above and below.
This leads us to draw an analogy between these randomness notions and the different reducibilities discussed in Section 1. Martin-Löf randomness seems to be analogous to Turing reducibility:
partial computable functions need not be defined on all inputs. Computable and Schnorr randomness, on the other hand, seem to be analogous to tt-reducibility, in which every reduction is total for
every oracle. These analogies have led several people to consider the relationship between Schnorr
randomness and tt-reducibility.
Franklin and Stephan were the first to do this. In a paper that will be discussed further in
Section 3, they showed that two ways of being far from Schnorr random that seemed very different
when Turing reducibility was considered actually coincide when tt-reducibility is used instead [19].
To relativize Schnorr randomness to a sequence A, instead of requiring that the martingale m be
Turing computable from A, they used the stronger condition that m ≤tt A. While they could have
chosen to relativize the bound function as well, they argued that the end result is the same and
opted for the simpler definition.
Miyabe argued in [36] that the fact that van Lambalgen’s Theorem does not hold for Schnorr
randomness suggests that the standard relativization for Schnorr randomness is not the most appropriate one. In [36] and [39], Miyabe (and then Miyabe and Rute) introduced variants on Schnorr
randomness and computable randomness in which the relativizations were based on truth-table
reductions rather than Turing reductions. Miyabe called the Schnorr randomness version “truthtable Schnorr randomness” in [36], and he and Rute changed the terminology to “uniformly relative
Schnorr randomness” in [39], where they prove that this is equivalent to Franklin and Stephan’s
notion in [19].
In Section 4 of [36], Miyabe claimed a proof that van Lambalgen’s Theorem holds for uniformly
relative Schnorr randomness that closely parallels the standard proof for Martin-Löf randomness
mentioned previously. However, an unwarranted assumption was made in this proof: it had been
assumed that, given a Schnorr test relativized to A, hViA i, the measure of ViA could be computed
uniformly from A, a code for Vi , and the measure of Vi . This is corrected in [39] with a new proof
using integral tests. Furthermore, the particular assumption that had been made sheds light on
the reason that van Lambalgen’s Theorem does not hold for Schnorr randomness with the standard
relativization: this type of computation may not allow us to distinguish between two sets with very
different measures provided that the difference in measure results from a small “gap” [39].
In addition, as mentioned above, they provide a remarkably short and elegant proof of the
“hard” direction of van Lambalgen’s Theorem for standard Schnorr randomness and then show the
following result for uniformly relative Schnorr randomness.
Theorem 2.4 (Theorem 4.7, [39]). A ⊕ B is Schnorr random if and only if A is Schnorr random
and B is Schnorr random uniformly relative to A.
The proof of the “easy” direction is given in [36]; the proof of the “hard” direction relies on
a technical lemma. We assume that A is Schnorr random and A ⊕ B is not and take a Schnorr
integral test t witnessing that A⊕B is not Schnorr random. We use this test to construct a uniform
10
FRANKLIN
Schnorr integral test tA witnessing that B is not Schnorr random uniformly relative to A. This test
is striking in its simplicity: for each X, we set tX (Y ) = t(X ⊕ Y ).
Miyabe and Rute also address the situation for computable randomness. While they do not
prove a theorem precisely parallel to Theorem 2.4, they are able to show a slightly weaker theorem
using a relatively traditional martingale approach:
Theorem 2.5 (Theorem 5.3, [39]). A ⊕ B is computably random if and only if A is computably
random uniformly relative to B and B is computably random uniformly relative to A.
To show the “hard” direction, we take a martingale m witnessing that A ⊕ B is not computably
A
random and use it as in the proof of Theorem 2.4 to construct two martingales mB
1 and m2 that
mimic the behavior of m on X ⊕ B and A ⊕ X, respectively. Then we show that either mB
1
succeeds on A or mA
succeeds
on
B.
The
“easy”
direction
is
proven
in
[36]
using
the
test
definition
2
of computable randomness and showing that if A ⊕ B is computably random, then B must be
computably random uniformly relative to A.
The authors noted that this theorem has an interesting corollary that shows immediately that
computable randomness differs from uniformly relative computable randomness. As previously
mentioned, several people have shown that there are two computably random sequences whose join
is computably random that are not computably random relative to the other. However, Theorem
2.5 shows that the two computably random sequences in question must still be computably random
uniformly relative to each other.
Miyabe and Rute also note that uniform relativization and the standard relativization coincide
for Kolmogorov-Loveland randomness and Martin-Löf randomness [39]. This, when combined with
the facts presented above, suggests to them that uniform relativization is the proper relativization
to consider for all randomness notions: for some randomness notions, uniform relativization does
not affect the results obtained, and for Schnorr randomness and computable randomness, theorems
that ought to be true become true in the context of uniform relativization. Later, Kihara and
Miyabe would show that van Lambalgen’s Theorem holds for Kurtz randomness when uniformly
relativized but not under the standard relativization, lending more support to this statement [26].4
3. Lowness for randomness
Now we consider the second question asked at the end of Section 1: Which sequences are so
information-poor that they are incapable of derandomizing any random sequence? Our instinct is
to say that that this will depend on the type of randomness we are considering. We will begin by
discussing this type of computational weakness, called lowness, within the context of other areas
of computability theory and then show how it developed within randomness. In addition to the
references to the original papers given below, a more thorough survey of lowness for randomness
notions can be found in [16].
Lowness is one of the simplest contexts in which we can discuss relativization. We begin with a
relativizable class C and say that a set is low for C if the class generated using the set as an oracle
is precisely the same as the class generated using no oracle at all, in other words, if C A = C, and we
denote the sets that are low for C by Low(C). The term “lowness” was first introduced in 1972 [44]
and applied to the class of Turing functionals. In other words, a set A is low if the Turing jump
of A was equivalent to the Turing jump of 0. Over the past four decades, this notion has been
4A sequence A is Kurtz random if it is contained in every conull Σ0 subclass of 2ω [31].
1
RELATIVIZATION IN RANDOMNESS
11
generalized to other areas such as computational learning theory [43] and computable structure
theory [18] but has been most thoroughly developed in the context of algorithmic randomness. For
instance, randomness is the only context in which lowness for pairs of relativizable classes has ever
been discussed.
The concept of lowness for pairs, first introduced in [29], is perhaps best illustrated by an
example. We already know that every Martin-Löf random sequence is Schnorr random, that is,
that ML ⊂ Schnorr. If a sequence A is low for Schnorr, we know that SchnorrA = Schnorr and
thus that ML ⊂ SchnorrA . However, it may be possible to find a sequence B that is not low for
Schnorr such that ML ⊆ SchnorrB . Any sequence B for which ML ⊆ SchnorrB is said to be in
Low(ML, Schnorr).
We generalize this example to the more general concept of lowness for pairs: we say that A is in
Low(C, D) if DA still contains C. We note that Low(C, C) is the same as Low(C) and that it makes
no sense to consider Low(C, D) unless C ⊆ D: if C is not a subset of D, then it can certainly never
b ⊆ D, we will have
be a subset of DA . We can also see that, given a chain of classes C ⊆ Cb ⊆ D
b
b
Low(C, D) ⊆ Low(C, D), a fact that will often be useful in this section.
Which sequences, then, can belong to Low(C) when C is a class of random sequences? We begin
by observing that for any A, C A will necessarily be a subclass of C: if A is computationally strong
enough to remove even one element of C, then we will have C A ⊂ C. For instance, if we take A to
be a Martin-Löf random sequence, it cannot be low for the class ML because A belongs to ML but
not to MLA .
We will begin by discussing the class Low(ML). Thereafter, we will discuss lowness for pairs
of notions grouped by the last entry in the pair Low(−, C), as the proofs and results tend to be
similar when the last entry in the pair is held constant. Finally, we will discuss some recent work
by Kihara and Miyabe in which they investigate lowness for pairs of randomness notions under
uniform relativization.
Just as Martin-Löf randomness is generally considered to be the gold standard for randomness
notions, Low(ML) is the most robust lowness class. When a sequence is low for a class C, it is, in
the sense of relativization, very far from being in C. However, there are other ways of being “far
from random,” and in the context of Martin-Löf randomness, all of these ways coincide with being
low. Several of them are based on a third characterization of randomness based in the idea that a
random sequence should be difficult to describe. Kolmogorov complexity provides us with a way
to calculate this difficulty of description for a finite binary string.
Definition 3.1. A prefix-free machine is a function M : 2<ω → 2<ω such that the domain of M is
prefix free. We say that the prefix-free Kolmogorov complexity of a string σ relative to a prefix-free
machine M is
KM (σ) = min{|τ | | M (τ ) = σ}.
A computable measure machine is a prefix-free machine whose domain has a computable measure
[12].
We note without ceremony that there is a universal prefix-free machine U ; if we use such a
machine in a Kolmogorov complexity calculation, we simply write K instead of KU . Levin [32]
and Chaitin [5] defined a sequence A to be random if there is a constant c such that for every n,
K(An) ≥ n − c. This is equivalent to Martin-Löf randomness [5], and it is this notion in which
the following properties are based. (There is a similar characterization of Schnorr randomness
using prefix-free Kolmogorov complexity that requires the use of computable measure machines.)
12
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The properties in the first pair are truly lowness properties; the properties in the second pair are
based on the notion that a sequence is far from random if it is no more difficult to describe than a
computable sequence.
Definition 3.2. Let A be a sequence.
(1) A is low for K if there is some c such that for all σ, K(σ) ≤ K A (σ) + c (unpublished, from
Muchnik).
(2) A is low for computable measure machines if for every A-computable measure machine M ,
there is a computable measure machine M 0 and a constant csuch that for all σ, KM 0 (σ) ≤
KM (σ) + c [8].
Definition 3.3. Let A be a sequence.
(1) A is K-trivial if there is some c such that for all n, K(An) ≤ K(0n ) + c [4].
(2) A is Schnorr trivial if for every computable measure machine M , there is a computable
measure machine M 0 and a constant c such that for all n, KM 0 (An) ≤ KM (0n ) + c [9].
In each of these definitions, the first item corresponds to a notion for Martin-Löf randomness
and the second to a notion for Schnorr randomness. Neither lowness for a type of machine nor
triviality has a natural analogue for computable randomness since computable randomness has no
known natural definition in terms of Kolmogorov complexity. The following two properties also
indicate being “far from random.” In each definition, we can substitute “Schnorr” or “computable”
for “Martin-Löf” to obtain the corresponding definitions for the other two types of randomness.
Definition 3.4. Let A be a sequence.
• A is low for Martin-Löf tests if for every Martin-Löf test relative to A hViA i, there is an
unrelativized Martin-Löf test hTi i such that ∩i ViA ⊆ ∩i Ti .
• A is a base for Martin-Löf randomness if there is a sequence B ≥T A such that B is
Martin-Löf random relative to A.
The question is now how these ways of being far from random relate to lowness for randomness.
If they coincide for a given relativization, it suggests that the chosen relativization is a sensible one.
It is straightforward to see that lowness for Martin-Löf tests is equivalent to lowness for MartinLöf randomness since there is a universal Martin-Löf test. Being low for K, being K-trivial, and
being a base for Martin-Löf randomness are also equivalent to lowness for Martin-Löf randomness,
but more work is required to show these facts. Nies proved the equivalence of lowness for K and
lowness for Martin-Löf randomness in [40]. This paper also contains a proof by Nies and Hirschfeldt
that K-triviality and lowness for K are equivalent. We will begin by discussing these two results;
afterwards, we will discuss Hirschfeldt, Nies, and Stephan’s proof of the equivalence of K-triviality
and being a base for Martin-Löf randomness from [24].
It is relatively straightforward to see that lowness for K implies lowness for Martin-Löf randomness: since ML can be defined in terms of Kolmogorov complexity K, MLA can be defined in
terms of K A , so every sequence that is low for K is low for Martin-Löf randomness. The proof that
lowness for Martin-Löf randomness is equivalent to lowness for K is most easily done by considering
two reducibilities: ≤LK and ≤LR .
Definition 3.5. Let A and B be sequences. We say that A ≤LK B if for all σ and some c,
K B (σ) ≤ K A (σ) + c and that A ≤LR B if MLB ⊆ MLA .
RELATIVIZATION IN RANDOMNESS
13
Both of these reducibilities let us compare the derandomization abilities of two sequences: in
each case, A is reducible to B if B can derandomize at least as much as A in the relevant context.
Since Martin-Löf randomness can be defined in terms of Kolmogorov complexity, we can see that
A ≤LK B implies A ≤LR B. Kjos-Hanssen, J. Miller, and Solomon showed that LR-reducibility
implies LK-reducibility, so we have the reverse direction as well [28].
We now turn our attention to K-triviality. Showing that lowness for K implies K-triviality is
relatively straightforward [40]: given a constant c0 witnessing A’s lowness for K and a universal
prefix-free machine U , we simply build a universal prefix-free machine M such that for every X
and σ, M X (σ) = X|U (σ)| whenever this is defined. If a sequence A is low for K, this allows us
to first pass from information about the complexity the unrelativized U gives an initial segment of
A to the complexity the A-relativized M gives it to the complexity of a string of 0s of the same
length for only the cost of a constant.
Showing that K-triviality implies lowness for K, though, is far more difficult and makes use of
the “golden run” method.
Theorem 3.6. [40] Every sequence that is K-trivial is low for K.
The proof of this result is rather complicated, so we will simply give an idea of the ingredients
of the proof and how they fit together.
We begin with a computable approximation to our K-trivial A. Our primary tools are two
Kraft-Chaitin sets L and W .5 L will allow us to use A’s K-triviality in our proof, and W will
witness A’s lowness for K.
The construction involves a tree of “runs” of procedures. At each branching point, we try to
ensure that W generates a prefix-free machine that will witness A’s lowness for K. The complications arise when A’s approximation changes. We cannot simply compensate for this change every
time; if we do, L will be too large. Therefore, we refuse to allow an axiom to enter L until it is
“approved” at a certain level of the tree.
Our tree will have 2k − 2 levels at each stage for a k calculated in advance from the index for
the machine used and a constant witnessing A’s K-triviality. The root procedure is Pk , which
calls several procedures of type Qk−1 , each of which calls Pk−1 , and so on. Each procedure has a
certain measure as its goal, and the run of each procedure enumerates a set. The only way this
enumeration will halt is if it reaches its goal or if it is cancelled by a higher-level run. Pk ’s goal is
the enumeration of a measure 1 set Ck . If this is never achieved, there is a run of another procedure,
called the “golden run,” which never returns, although all its subprocedures either return or are
cancelled. In general, if the relevant approximation to A changes quickly, Pi can add an axiom to
its Kraft-Chaitin set immediately and cancel one of the Qi s; if that Qi returns a set D, Pi will wait
for the appropriate approximation to A to change and, if it does, add D to its own Kraft-Chaitin
set.
We establish a “garbage quota” for L to account for changes in the approximation to A that
occur after we have committed some measure to an earlier version. When runs are cancelled, they
contribute to this garbage quota for L. By careful balancing of the measure goals and the garbage
quotas, we ensure that L will be a Kraft-Chaitin set. Furthermore, we can see that there is a golden
run of some procedure by assuming that every run of every procedure is either returned or cancelled
and obtaining a contradiction. We then use this golden run to enumerate the Kraft-Chaitin set W
5A Kraft-Chaitin set is a subset of ω × 2<ω with bounded measure that allows us to build a prefix-free machine
with a certain targeted property. The elements of a Kraft-Chaitin set are called “axioms.”
14
FRANKLIN
witnessing that A is low for K. Our previous balancing of measure goals and garbage quotas will
ensure that the measure of W is small enough, and the way in which we choose axioms to include
will ensure that K(τ ) is no bigger than K A (τ ) plus some constant.
At this point, we have shown that lowness for Martin-Löf randomness, lowness for Martin-Löf
tests, lowness for K, and K-triviality are all equivalent. We now turn our attention to the last
property: being a base. Kučera observed that the Kučera-Gács Theorem [22, 30] can be applied
to show that if A is low for Martin-Löf randomness, then it is a base for Martin-Löf randomness.
Hirschfeldt, Nies, and Stephan completed the proof of this equivalence:
Theorem 3.7. [24] If a sequence is a base for Martin-Löf randomness, then it is K-trivial.
Here, we fix a sequence B ≥T A that is A-Martin-Löf random and enumerate a Kraft-Chaitin set
Ln for each n ∈ ω in such a way that one of them will witness A’s K-triviality. An A-Martin-Löf test
is built from uniformly Σ01 classes with certain technical properties, and the Ln are defined based
on these Σ01 classes. Therefore, all the mentioned characterizations of being far from Martin-Löf
random coincide using the standard relativization.
We now turn our attention to classes of the form Low(−, Comp). Nies proved the following
results in [40]. Interestingly, the characterization of Low(ML, Comp) is necessary for his proof of
the characterization of Low(Comp).
Theorem 3.8. [40] A sequence A is in Low(ML, Comp) if and only if A is low for K.
We note that since ML ⊆ ML ⊆ ML ⊆ Comp, Low(ML) ⊆ Low(ML, ML) ⊆ Low(ML, Comp).
Therefore, we simply need to show that any element of Low(ML, Comp) is low for K.
This proof uses the martingale characterization of computable randomness: we construct a
martingale functional L such that A is low for K if LA only succeeds on sequences that are not
Martin-Löf random.6 Rather than build L directly, we set L to be the sum of the martingale
functionals Ln that we actually construct.
As before, we build a Kraft-Chaitin set W witnessing that such an A must be low for K. We use
an auxiliary set R, a c.e. open subset of the Cantor space with measure less than one containing
all sequences that are not Martin-Löf random. This set R is used to ensure that W is truly a
Kraft-Chaitin set and has measure no greater than 1: we wait to see if a certain set enters R before
committing a certain axiom to W . As we carry out this procedure to construct our W , we build
Ln by choosing certain strings that may be initial segments of non-Martin-Löf random sequences
and ensuring that all sequences of a certain type have a high value on those strings. If this happens
infinitely often, then L will succeed on these sequences.
We now turn our attention to Low(Comp). Since ML ⊆ Comp ⊆ Comp ⊆ Comp, Low(Comp) ⊆
Low(Comp, Comp) ⊆ Low(ML, Comp). A result of Bedregal and Nies states that every element of
Low(Comp) is hyperimmune free [2]. Since every sequence that is low for K is ∆02 , every element
of Low(Comp) must be ∆02 , and since the only hyperimmune-free ∆02 sequences are the computable
sequences, Low(Comp) is simply the computable sequences.
Although we cannot discuss any analogue to K-triviality or lowness for K for computable randomness, the bases for computable randomness have been partially characterized with the following
two results. Recall that a sequence is DNC if it computes a total function f such that f (e) is never
equal to Φe (e) for any e and that a Turing degree is PA if it can compute a complete extension of
Peano Arithmetic.
6A martingale functional is a Turing functional that is a martingale for every oracle.
RELATIVIZATION IN RANDOMNESS
15
Theorem 3.9. [24] Every non-DNC ∆02 sequence is a base for computable randomness, but no
sequence in a PA degree is.
Corollary 3.10. [24] An n-c.e. sequence is a base for computable randomness if and only if it is
Turing incomplete.
To show that a non-DNC ∆02 sequence is a base for computable randomness, we take such a
sequence A and construct a generic sequence G such that (A ⊕ G)0 ≡T A00 . This sequence is used
as an oracle to build a Z ≥T A that is A-computably random. The other half of this theorem is
proven by noting that a sequence A with PA degree will compute a sequence B with PA degree
such that A is PA relative to B. We then show that if there is a sequence computing A that is
A-computably random, we can conclude that B is both K-trivial and not K-trivial and derive a
contradiction.
The proof of the corollary uses a theorem in [25] that states that a Turing incomplete n-c.e.
sequence is not DNC and is thus a base for computable randomness. Furthermore, such a sequence
that is Turing complete will necessarily have PA degree and thus cannot be a base.
We now turn our attention to lowness for Schnorr randomness. While the study of van Lambalgen’s Theorem for Schnorr randomness suggested the importance of choosing a nonstandard
relativization, the study of lowness for Schnorr randomness emphasizes that Miyabe’s relativization is far more appropriate than the standard one.
We can begin by observing that lowness for Schnorr tests clearly implies lowness for Schnorr
randomness; however, the converse is not obvious since there is no universal Schnorr test. In order
to characterize the Low(−, Schnorr) classes, we need a new concept: traceability.
Definition 3.11. A sequence A is c.e. traceable if there is an order function p such that for all
f ≤T A, there is a computable function r such that the following conditions hold for all n:
• f (n) ∈ Wr(n) and
• |Wr(n) | ≤ p(n).
A sequence is computably traceable if we can replace Wr(n) with Dr(n) in the above definition, where
Dk is the k th canonical finite set.
We note that all computably traceable sequences are hyperimmune free and that we may think
of a computably traceable sequence as uniformly hyperimmune free.
This notion allowed Terwijn and Zambella to characterize the sequences that were low for Schnorr
tests.
Theorem 3.12. [48] A sequence is computably traceable if and only if it is low for Schnorr tests.
Later, Kjos-Hanssen, Nies, and Stephan showed that the computably traceable sequences were
precisely those that were low for Schnorr randomness. They began by showing that the c.e. traceable
sequences were precisely those in Low(ML, Schnorr).
Theorem 3.13. [29] A sequence is c.e. traceable if and only if it is in Low(ML, Schnorr).
To prove this, we assume that if a sequence A is c.e. traceable, we can take a Schnorr test relative
to A and use a trace for A to approximate the components of this Schnorr test in a computable
way. This approximations allow us to construct a Martin-Löf test whose intersection contains the
intersection of the relativized Schnorr test.
16
FRANKLIN
Conversely, if A is in Low(ML, Schnorr), we need to construct a c.e. trace for an arbitrary f ≤T A.
We code this f into a Schnorr test relative to A and recognize that the intersection of this A-Schnorr
test is contained in each element of the universal Martin-Löf test. We can use one element of this
unrelativized universal Martin-Löf test to build a c.e. trace for f . When we build this trace, we
must ensure that we don’t include too many options for each value f (n) while ensuring at the same
time that we include the correct value. This is possible due to our method of encoding the initial
segments of f into the A-Schnorr test.
Finally, we use this result to find the other classes Low(−, Schnorr).
Theorem 3.14. [29] Let A be a sequence. The following are equivalent:
(1) A is computably traceable.
(2) A is in Low(Schnorr).
(3) A is in Low(Comp, Schnorr).
Terwijn and Zambella had already shown that (1) implies (2), and it can be seen from the
relationship between Comp and Schnorr that (2) implies (3). Therefore, the only thing to do is to
show that (3) implies (1). This follows from the same paper of Bedregal and Nies mentioned earlier
[2]: they showed that every sequence in Low(Comp, Schnorr) must be hyperimmune free, and since
every hyperimmune free c.e. traceable sequence is actually computably traceable, we are done.
We now consider lowness for computable measure machines, which is analogous to lowness for
K. This is also equivalent to computable traceability:
Theorem 3.15. [8] A sequence A is low for computable measure machines if and only if it is
computably traceable.
Since lowness for computable measure machines implies lowness for Schnorr randomness, it is
easy to prove the forward direction. To prove the other direction, we assume that A is a computably
traceable sequence and choose a computable measure machine; we then use the computable traceability of A to construct a sequence of small finite sets with a certain technical property. We then
build a Kraft-Chaitin set from this sequence that allows us to construct a computable measure
machine witnessing A’s lowness for computable measure machines.
Bases for Schnorr randomness and Schnorr triviality have also been the subject of study. At first
glance, the Schnorr trivial sequences appear to be very different from the sequences that are low
for Schnorr randomness. Downey, Griffiths, and LaForte proved that there is a Turing complete
Schnorr trivial sequence [9], and Franklin proved that, in fact, every high Turing degree contains
a Schnorr trivial sequence [15]. The Schnorr trivial Turing degrees do not form an ideal since they
are not closed downward, as the K-trivial Turing degrees do [40]. However, Franklin showed the
following relationship between Schnorr triviality and lowness for Schnorr randommness:
Theorem 3.16. [14] A sequence is low for Schnorr randomness if and only if it is Schnorr trivial
and hyperimmune free.
To see this, we begin by assuming that A is low for Schnorr randomness and fixing an arbitrary
computable measure machine M . We use an enumeration of this machine and the computable
traceability of A to construct another computable measure machine M 0 witnessing the Schnorr
triviality of A with respect to M . To prove the other direction, we assume that A is hyperimmune
free but not Schnorr low. We then take a function g ≤T A that witnesses that A is not computably
traceable and, since A is hyperimmune free, observe that the use function for this g relative to
RELATIVIZATION IN RANDOMNESS
17
all possible A is a computable function. Now, for every computable measure machine Mi , we can
define a computable trace that contains all strings of a certain length with low complexity. Since
A is not Schnorr low, there is an n for which g A (n) is not in this trace, and thus A will not be
Schnorr trivial with respect to this Mi .
This result caused Franklin to suggest that Turing reducibility was not the right framework for
Schnorr randomness and that a more natural framework might be the truth-table degrees [14].
As mentioned in Section 2, she and Stephan later showed that Schnorr triviality and lowness for
Schnorr randomness actually coincide when one considers truth-table reducibility. They use the
relativization below that is equivalent to Miyabe’s in [19]:
Definition 3.17. A is tt-low for Schnorr randomness if for every Schnorr random set B, there is no
martingale m ≤tt A and no computable function g such that for infinitely many n, m(Bg(n)) ≥ n.
This allows them to prove the following theorem that is far more parallel to the results for
Martin-Löf randomness mentioned earlier.
Theorem 3.18. [19] Let A be a sequence. The following are equivalent:
(1) A is Schnorr trivial.
(2) A is tt-low for Schnorr randomness.
(3) The tt-degree of A is computably traceable (here, we trace the functions f that are tt-reducible
to A instead of all f ≤T A).
The proof of this result involves a characterization of Schnorr triviality using computable probability distributions. This characterization is used to show that any function tt-computable from
a Schnorr trivial sequence can be computably traced given an arbitrary bound function (in other
words, that (1) implies (3)). We then translate this trace into information about martingales with
the help of a technical lemma and show that A is tt-low for Schnorr randomness (so (3) implies
(2)). At this point, we can do a highly calculational proof, once again using computable probability
distributions, to show that (2) implies (1).
An additional characterization of Schnorr triviality found in [19] allows us to prove a corollary
that serves as another indication that the truth-table degrees are the natural habitat of Schnorr
randomness:
Corollary 3.19. [19] The join of any two Schnorr trivial sequences is also Schnorr trivial, so the
Schnorr trivial sequences form an ideal in the truth-table degrees.
We conclude our discussion of lowness for Schnorr randomness with a discussion of bases. The
bases for Schnorr randomness are precisely the sequences that can’t compute 00 [21], which is
clearly a different class from those that are low for Schnorr randomness. Franklin and Stephan also
considered a truth table version of bases: one in which A is a tt-base for Schnorr randomness if
and only if there is a B ≥tt A such that B is A-Schnorr random. While they showed that this is
not equivalent to Schnorr triviality, as hoped, they do show in [19] that it is sufficient for Schnorr
triviality.
Miyabe, once again, took this study a step farther by defining bases for uniform Schnorr tests
and uniformly computable martingales:
Definition 3.20. [38] Let A ∈ 2ω , and let O be the class of open sets in 2ω .
18
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ML
Comp
Schnorr
ML K-trivial
K-trivial
c.e. traceable
Comp
computable computably traceable
Schnorr
computably traceable
Table 1. Characteristics of lowness classes: standard relativization
(1) A uniform Schnorr test is a computable function f : 2ω × ω → O such that hX, ii 7→
µ(f (X, i)) is computable and µ(f (X, i)) ≤ 21i for all n. A sequence hUiA i is a Schnorr test
uniformly relative to A if UiA = f (A, i) for all i and some uniform Schnorr test f .
(2) A uniform martingale test is a computable function m : 2ω × 2<ω → R>0 such that mZ =
m(Z, −) is a martingale for every sequence Z.
Definition 3.21. [38] Let A be a sequence.
(1) A is a base for uniform Schnorr tests if for every computable martingale m uniformly
relative to A, there is a B ≥tt A that is Schnorr random uniformly relative to A for m.
(2) A is a base for uniformly computable martingales if for every computable martingale m
uniformly relative to A, there is a B ≥tt A that is computably random uniformly relative
to A for m.
Miyabe then goes on to prove that these notions are equivalent to Schnorr triviality:
Theorem 3.22. [38] Let A be a sequence. The following are equivalent:
(1) A is Schnorr trivial.
(2) A is a base for uniformly computable martingales.
(3) A is a base for uniform Schnorr tests.
It is clear that (2) implies (3), and the proof that (3) implies (1) is inherent in Franklin and
Stephan’s work [19]. To prove that (1) implies (2), we construct our set B using the Space Lemma
[34] and the fact that A must be computably tt-traceable.
A summary of these lowness classes can be found in Table 1. Lowness for other randomness notions such as Kurtz randomness, weak 2-randomness, and difference randomness have been studied
as well [3, 11, 17, 23, 28, 47], but we will not consider these in this survey since they do not cast
light on the role of relativization in the same way that a comparison of lowness for Martin-Löf
randomness and Schnorr randomness, in particular, does.
Kihara and Miyabe have also considered an idea they call LowF for pairs of randomness notions:
LowF (C, D) is the set of sequences A such that C is contained in the class of sequences that are
in D uniformly relative to A [27]. They characterize LowF (C, D) for several pairs of randomness
notions where C and D range over ML, Schnorr, and Kurtz randomness in terms of various forms
of traceability and anticomplexity [27]. Their proofs rely heavily on previous characterizations of
Low(C, D) and technical means of uniformizing the proofs of these characterizations. An example
of their results is given below:
Theorem 3.23. [27]
(1) A is in LowF (ML, Schnorr) if and only if A is c.e. tt-traceable.
(2) A is in LowF (Schnorr) if and only if A is computably tt-traceable.
RELATIVIZATION IN RANDOMNESS
19
The first of these results is proven using a covering property of Bienvenu and Miller [3] and
several technical lemmas; the second is inherent in Franklin and Stephan [19].
We can see that these results are parallel to the results with the standard relativization: the only
difference is that here, we use tt-traceability instead of Turing traceability.
All the natural ways in which a sequence can be far from Martin-Löf random coincide, and Miyabe
and Rute demonstrated that uniform relativization coincides with the standard relativization in
that setting. When we consider Schnorr randomness under the standard relativization, the lowness
notions (lowness for Schnorr randomness, lowness for Schnorr tests, and lowness for computable
measure machines) all coincide, but triviality and being a base coincide neither with the lowness
notions nor with each other. However, under uniform relativization, all these notions coincide.
This lends incredible strength to the argument that uniform relativization is the most reasonable
approach to relativizing definitions related to Schnorr randomness and possibly any randomness
notion.
4. Conclusion
As we have seen, the truth of van Lambalgen’s Theorem and the class of sequences that are low
for a certain type of randomness are highly dependent on the relativization used. Based on what
“ought” to be true (van Lambalgen’s Theorem should be true; lowness and triviality should coincide
when both notions are defined), we can determine which type of relativization is most appropriate
for which randomness notion. It is apparent that the standard Turing-based relativization is the
most appropriate for Martin-Löf randomness, while a relativization based in truth-table reducibility
is the most appropriate for Schnorr randomness.
We have said very little about the role that choosing a relativization plays with computable
randomness. This is a much more difficult notion to work with than Martin-Löf or Schnorr randomness since its characterization in terms of martingales is the only simple one it has. While
Miyabe and Rute have shown that uniformly relative computable randomness makes van Lambalgen’s Theorem hold when the standard relativization does not, it is unclear whether a somewhat
weaker reducibility would also work. Since we have so few useful characterizations of ways in which
a sequence can be far from computable randomness, it is difficult to study relativization in this
context. The difficulty of characterizing bases using the standard Turing reduction suggests, once
again, that this is not the correct reducibility to use; however, it does not help us identify a more
useful relativization.
There are other components of the study of randomness in which relativization is also important,
such as highness: given two classes C and D such that D ⊆ C, a sequence A is high for the pair
(C, D) if C A ⊆ D, that is, if A is computationally strong enough to shrink C enough to fit inside of
D. This concept was introduced in [21] and pursued in [1], but only the standard relativizations
have ever been considered, and there are still several open questions remaining. We suggest that
highness for pairs be more strongly investigated using the standard relativization and that a notion
HighF (C, D) parallel to Kihara and Miyabe’s LowF (C, D) be investigated in parallel. Considering
the differences in these classes may be very instructive and may provide another avenue to consider
the role of relativization in randomness.
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Department of Mathematics, Room 306, Roosevelt Hall, Hofstra University, Hempstead, NY 115490114, USA
E-mail address: johanna.n.franklin@hofstra.edu