Normality and Differentiability
Pablo Ariel Heiber
Universidad de Buenos Aires
joint work with Verónica Becher
What is normality?
simply normal
r ∈ [0, 1] is simply normal to base b if and only if every b-digit
has the same frequency in the expansion of r in base b
normal
r ∈ [0, 1] is normal to base b if and only if it is simply normal in
bases b, b2 , b3 , ...
What is differentiability?
differentiable
f : [0, 1] → [0, 1] non-decreasing is differentiable at x if and only
if
µ(f ([r, r + h]))
f (r + h) − f (r)
Df (r) = lim
= lim
h→0 µ([r, r + h])
h→0
h
converges to a finite value.
differentiable
f : bω → bω non-decreasing is differentiable at x if and only if
µ(f ((x n)bω ))
n→∞ µ((x n)bω )
Df (x) = lim
converges to a finite value.
What links differentiability and normality?
finite-state computable
f : bω → bω is finite-state computable if there is a finite-state
transducer F = hb, Q, q0 , δ, oi without λ output cycles such that
for every n, o∗ (q0 , x n) is a prefix of f (x).
finite-state computable
f : [0, 1] → [0, 1] is finite-state computable in base b if and only if
there is a finite-state computable g : bω → bω such that for
every x ∈ bω , v(g(x)) = f (v(x)).
v : bω → [0, 1]
X
v(x) =
x[i]b−i
i
Examples {0, 1}ω → {0, 1}ω
double bits
division by 2
f
0 →
7
00
1 →
7
11
0/00
1/11
q0
1n 0
1ω
f
7
→
01n
7
→
01ω
1/0
q0
0/0
q1
0/1
compress zeroes
00
01
10
11
f
7
→
7
→
7
→
7
→
0
10
110
111
0/0
1/10
q1
0/0
1/1
q0
0/λ
q2
1/11
1/1
Differentials and finite-state computation
F
= hb, Q, q0 , δ, oi computes f
Fq = hb, Q, q , δ, oi computes fq
Df (x) =
µ(f ((x n)bω ))
n→∞ µ((x n)bω )
lim
µ(o∗ (q0 , x n)fδ∗ (q0 ,xn) (bω ))
n→∞
µ((x n)bω )
lim
lim
b−|o
∗ (q ,xn)|
0
n→∞
lim bn−|o
n→∞
µ(fδ∗ (q0 ,xn) (bω ))
b−|xn| µ(bω )
∗ (q ,xn)|
0
µF,δ∗ (q0 ,xn)
Differentials and finite-state computation
Df (x) = lim bn−|o
n→∞
∗ (q ,xn)|
0
µF,δ∗ (q0 ,xn)
Difficulties
Definition has lim instead of lim inf: cannot just work on k
digits at a time for large k
Replace with cycle analysis: input strings form a maximal
prefix free set, but output strings do not
division by 2
1n 0
1ω
f
7
→
01n
7
→
01ω
1n 0 is maximal prefix free,
but 01n is not
compress zeroes
f
00 7→ 0
01 7→ 10
10 →
7
110
11 →
7
111
Df ((0010)ω ) oscilates
Meaning of differential
logb Df (x) = lim n − |o∗ (q0 , x n)| + logb µF,δ∗ (q0 ,xn)
n→∞
− logb µF,q measures the information stored at q
Proposition
Df (x) = k > 0 means f “growth rate” is k around x
f (x) (n − logb k) can be calculated from x n
f ((x n)bω ) ⊆ (f (x) (n − logb k))bω
Df (x) = 0 means f “growth rate” is asymptotically 0
around x
∀c, f (x) (n + c) can be calculated from x n
∀c, f ((x n)bω ) ⊆ (f (x) (n + c))bω
What links differentiability and normality?
Main Theorem
For a sequence x ∈ bω , the following are equivalent:
1
2
3
v(x) is normal to base b.
Every non-decreasing function f : bω → bω finite-state
computable is differentiable at x.
Every non-decreasing function f : [0, 1] → [0, 1] finite-state
computable in base b is differentiable at v(x).
Proof sketch
Prove
1
⇒
2
2
⇒
3
3
⇒
1
via ¬ 1 ⇒ ¬ 3
What links differentiability and normality?
Main Theorem
For a sequence x ∈ bω , the following are equivalent:
1
2
3
1
v(x) is normal to base b.
Every non-decreasing function f : bω → bω finite-state
computable is differentiable at x.
Every non-decreasing function f : [0, 1] → [0, 1] finite-state
computable in base b is differentiable at v(x).
⇒ 2
differential cannot diverge (via incompressibility)
if n − |o∗ (q0 , x n)| diverges, differential is 0
b|s|−|o
∗ (q,s)|
b|s|−|o
∗ (q,s)|
µF,q = µF,q if δ ∗ (q, s) = q
µF,q = µF,δ∗ (q,s) for some visited q
eventually all terms in the sequence Df have the same
value
What links differentiability and normality?
Main Theorem
For a sequence x ∈ bω , the following are equivalent:
1
2
3
2
v(x) is normal to base b.
Every non-decreasing function f : bω → bω finite-state
computable is differentiable at x.
Every non-decreasing function f : [0, 1] → [0, 1] finite-state
computable in base b is differentiable at v(x).
⇒ 3
show that
µ(f ([r, r + h]))
µ(g((x n)bω ))
= lim
n→∞ µ((x n)bω )
h→0 µ([r, r + h])
lim
take [r, r + h] as the disjoint union of numerable cones sbω
all sbω have equal µ(f (sbω ))/µ(sbω ), so do their union
What links differentiability and normality?
Main Theorem
For a sequence x ∈ bω , the following are equivalent:
1
2
3
v(x) is normal to base b.
Every non-decreasing function f : bω → bω finite-state
computable is differentiable at x.
Every non-decreasing function f : [0, 1] → [0, 1] finite-state
computable in base b is differentiable at v(x).
¬1 ⇒ ¬3
build a compressor of x, replacing Huffman by Hu-Tucker
take long blocks to “fade” excess and retain compression
Hu-Tucker ensures the function is increasing
compression makes function diverge at x
In perspective
Lebesgue Theorem
Every non-decreasing function f : [0, 1] → [0, 1] is differentiable
at almost every real r.
Brattka-Miller-Nies computable version
Every non-decreasing computable function f : [0, 1] → [0, 1] is
differentiable at computably random reals r.
Our FS computable versions
Every non-decreasing finite-state computable function
reals r.
f : [0, 1] → [0, 1]
is differentiable at normal
ω
ω
sequences x.
f :b →b
Main references
É. Borel.
Les probabilités dénombrables et leurs
applications arithmétiques.
Rendiconti del Circolo Matematico di
Palermo, 27:247–271, 1909.
A. Broglio and P. Liardet.
Predictions with automata. symbolic
dynamics and its applications.
Contemporary Mathematics, (135):111–124,
1992.
D. Huffman.
A method for the construction of
minimum-redundancy codes.
Proceedings of the IRE, 40(9):1098–1101,
1952.
J. Dai, J. Lathrop, J. Lutz, and
E. Mayordomo.
Finite-state dimension.
Theoretical Computer Science, 310:1–33,
2004.
V. N. Agafonov.
Normal sequences and finite automata.
Soviet Mathematics Doklady, 9:324–325,
1968.
C. Bourke, J. Hitchcock, and N. Vinodch.
Entropy rates and finite-state dimension.
Theoretical Computer Science, 349:392–406,
2005.
T. C. Hu and A. C. Tucker.
Optimal computer search trees and
variable-length alphabetical codes.
SIAM Journal on Applied Mathematics,
21:514–532, 1971.
V. Brattka, J. Miller, and A. Nies.
Randomness and differentiability.
Technical Report arXiv:1104.4465, 2011.
C. P. Schnorr and H. Stimm.
Endliche automaten und zufallsfolgen.
Acta Informatica, 1:345–359, 1972.
Becher V. and Heiber P.
Normality and compressibility.
submitted, 2012.