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Cornucopia of Randomness
Verónica Becher
Departamento de Computación
Universidad de Buenos Aires
Joos Heintz’s 60th birthday – October 2005
Tossing a fair coin?
11111111111111111111111111111111111
0101010101010101010101010101010101
0010100010010101000001011111001001
Towards a definition of randomness
A random sequence…
should be unpredictable,
should be lawless,
should lack structure,
should avoid distinguishing properties,
should pass all conceivable statistical randomness tests.
Chaitin’s notion of randomness
“lack of structure”
Any structure or regularity can be used to compress the sequence, so,
the initial segments of a random infinite sequence are incompressible.
Chaitin’s definition (1975)
x{0,1} is random if its initial segments are algorithmically incompressible.
x is random if c n H(x | n ) > n-c
H is prefix- program size-complexity (variant of Kolmogorov complexity)
Proposition. Random sequences are Borel normal and not computable.
G.Chaitin “A theory of program size formally identical to Information theory”, J ACM 1975
Another definition of randomness?
random sequences should have no distinguishing properties…
A naive idea:
a sequence x{0,1} is random if it is in no set of Lebesgue measure 0.
Of course, since singletons have measure 0, there is no such sequence.
Martin Löf’s definition (1966)
Random sequences are those that avoid every effectively presented
measure 0 set of a certain kind (effective G of measure 0).
Formalizes the idea that a random sequence should pass every
conceivable statistical test.
Corollary. The set of Martin Löf random reals has measure 1.
Martin Löf “The definition of random sequences” Information and Control 9, 1966.
The two definitions coincide
Theorem (Schnorr): Chaitin random  Martin Löf random.
As with Church’s thesis for the definition of an algorithm,
this equivalence can be regarded as supporting the definition
of randomness.
The definition extends immediately to R, identifying R with {0,1}
Dyadic rationals may have two representations, but this is not
important for the measure-theoretic considerations made in this
work.
Almost all real numbers are random.
How can we come up with specific examples?
R
?
Examples of not random reals
rational numbers
computable numbers (e.g., , e, )
Liouville numbers
characteristic functions of r.e. sets
Computable reals are not random
They can be dramatically compressed: there is an algorithm
(encoded with a fixed number of bits) for their whole fractional
expansion.
e.g. Rationals, , 0.010101...
Pseudo-random numbers
"Anyone who considers arithmetical
methods of producing random digits is,
of course, in a state of sin.”
John von Neumann
Chaitin (1975) gave the first example of a random real
U
=
2
-|p|
= (domain(U){0,1} )
U(p) halts
where U:{0,1}*-> {0,1}* is any optimal (hence universal) function
with prefix-free domain.
Also known as a self-delimiting optimal Turing machine
Proposition. 0 < U < 1
U is the probability that the machine U with an arbitrary input halts.
R

U
-numbers
There are countably many self-delimiting optimal Turing machines
U1, U2 , U3 ...
Thus, there are countably many halting probabilities
1, 2 , 3 ...
R

1 2 3
Computably enumerable and random reals
Definition. A real is left c.e. if its left cut is recursively enumerable.
Proposition. -numbers are left c.e.
Theorem (Calude et al. and Kučera-Slaman 2001)
-number  left computably enumerable and random.
R


?
random
computable
c.e
Degrees of randomness
Turing machines provide the primary notion of effective computability.
When machines are equipped with an oracle we have relative computability.
This induces a definition of randomness relative to some oracle.
Definition. A real is random in B iff its initial segments are algorithmically
incompressible even with the help of oracle B.
random is 1-random
random in ’ is 2-random
random in (n-1) is n-random
and the obvious examples are
 = the halting probability of U, is 1-random’
’ = the halting probability of U’, is 2-random
’’ = the halting probability of U’’ , is 3-random
.
.
.
R


’
’’
’’’
Except for , all these examples are defined using oracles.
Cornucopia of randomness
Joint work with Serge Grigorieff (Université Paris 7)
Grigorieff’s conjecture
Given a non-empty set, the probability that an optimal Turing machine
produces an output in this set is random.
Moreover, “the harder” the set, “the more random” the probability.
( If the set is 0n (0n)-complete, then such probability is n-random )
Grigorieff’s conjecture
Given a non-empty set, the probability that an optimal Turing machine
produces an output in this set is random.
Moreover, “the harder” the set, “the more random” the probability.
( If the set is 0n (0n) complete, then such probability is n-random)
Positive instances (1-random)
If the set is {0,1}*, the above probability is .
For infinite r.e. sets, the above probability is random (Chaitin, 1988).
Finite sets lead to randomness for some class of optimal machines (B-G)
0n complete sets have random probability (Miller 2005).
Grigorieff’s conjecture
Given a non-empty set, the probability that an optimal Turing machine
produces an output in this set is random.
Moreover, “the harder” the set, “the more random” the probability.
( If the set 0n (0n) complete then such probability is n-random)
Negative instances
The conjecture fails for some 0n sets with rational probability (Miller 2004)
0n complete sets do not give n-randomness.
The 01 case is still open.
R


Randomness from infinite computations
Consider the space of finite and infinite sequences {0,1} 
(or more generally the set of increasing sequences of elements
from a computable partially ordered set).
We consider the “upper-cone” topology on this set (Beware!).
Theorem (B-G 2003)
Monotone Turing machines U: {0,1} -> {0,1} are exactly the
effective continuous maps for the usual topology on the Cantor
space and the upper-cone topology on the target space.
Theorem (B-G 2004, 2005)
Let O be 0n and effectively hard for the class 0n ({0,1} ). Then,
the probability that an optimal monotone machine U produces
an output in O, i.e.,
(U-1(O)),
is n-random.
If O is … then, the probability that a monotone machine
performing infinite computations gives an output in O is ….
1-random
[Space R, topology of semi-intervals (0,q) and (q,), O=R]
2-random
[Space {0,1}, O ={0,1}*]
[ Space (N), O= finite sets]
3-random
[Space (N), O = recursive sets
O = r.e. sets
O = cofinite sets]
Cornucopia of randomness
If O is 0n and effectively hard for the class 0n ({0,1}  )
we give a method to define sets that are 0n+m and 0n+m effectively
hard, for every m.
We use classes S= Rec, Fin, Inf, Cof, Coinf, Exists, All
rules for combining them given by certain operators .
Theorem (B-G 2005)
Let O be 0n and effectively hard for the class 0n ({0,1}).
Then (U-1( O )) is n+m -random.
R


Happy birthday dear Joos!
Émile Borel
B= b1 b2 b3 b4...
bi= 0 if the i-th string is not a question in French.
1 if it is a question with a positive answer
2 if it is a question with a negative answer
“La connaissance de ce nombre donnerait donc la solution
à toutes les énigmes, en infinité dénombrable, qui peuvent
être posées dans le domaine de la science, de la curiosité,
de l´histoire et de la metaphysique.”
Émile Borel, La Définition en Mathematiques, 1948
Émile Borel
B= b1 b 2 b3 b4...
“Tout cela est pure fantaisie”
É. Borel, La Définition en Mathematiques, 1948
We can not prove the properties of this number.
Chaitin’s definition of randomness
Program size complexity
H(s) = min{ |p| : U(p) = s}
x is random iff c  n H(x n) > n - c
The characteristic number of
Turing’s Halting Problem
A = a1 a2 a3 ....
ai = 1 iff U(pi) halts
is not random
Indicate how many programs halt among the first n,
and there is an algorithm that tells you which ones!
Turing’s application to the Entscheidungsproblem
U(p1)= b11 b12 b13 ....
U(p2)= b21 b22 b23 ....
U(p3)= b31 b32 b33 ....
.....
: b11 b22 b33 ....
 is not in the table!
Why not?
It is undecidable whether the i-th program prints an i-th digit
Turing’s application to the Entscheidungsproblem
It is undecidable whether the i-th program prints an i-th digit
It is undecidable whether the i-th program will ever print 0.
It is undecidable whether the i-th program halts.
 encodes the Halting problem
very compactly!
Knowing the first n digits of  we can algorithmically
determine all halting programs up to length n.
Enumerate the halting programs until their contribution
reaches the first n digits of . Any halting program p not
yet enumerated contributes to  with 2-|p|. Thus, |p| > n.
The characteristic number
of the Halting problem is not random
A = a1 a2 a3 ....
where ai = 1 iff i(i) halts
(i)i is a recursive enumeration of the partial recursive functions
Proposition: A is not computable, not random
Algorithm: tell me how many among the first n halt and I will find them.
The first n digits can be compressed to log n digits plus a constant.
Chaitin’s example of random real (1975)
U =
2
-|p|
U(p) halts
where U:{0,1}*-> {0,1}* partial recursive, prefix-free domain, optimal.
(U is a self-delimiting universal Turing machine)
Proposition: 0 < U < 1
Proposition: U = ( domain(U){0,1} )= P(U halts)
Theorem : U is not computable and random.