Bottomed Stream
Representations of spaces
induced by dynamical systems
Hideki Tsuiki
Kyoto University
!
Representing Streams II
２０１４, Jan. 20 ー31, Leiden
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Computation over infinite sequences
0
1
0
•
1
Stream Programming (Type2
machine [Weihrauch, et.al])
0
1
0
0
1 0
1
Input
Type2 Machine
(Stream Programming)
Output
%
%
1
•
Real number computation via
binary expansion into infinite
sequences.
L
F
J
9
0
0
Unnatural computational notion on R.
%\%H%‘F~$jJ8;zNs$rMQ$$$?!"
0LAj6u4V>e$N7W;;
Admissible Representation
5
4
3
5
4
3
2
2
0
1
0.0
1
0.5
1
1.0
0.0
0
1
0.5
1.0
•
One cannot determine the first digit of 1/2 from any approximation.
Intervals of 0 and 1 are overlapping only at one point.
•
Giving more redundancy so that one can output 0 or 1 when
enough good approximation is given.
•
Admissible representation. (Signed digit representation, etc.)
[Weihrauch, Schroder,…].
Type2-machie + Adm. rep
[0,1]
admissible representa.on (Signed-­‐digit, Cauchy sequence, …)
{0,1}ω
•
•
Type2-machine
•
This idea dates back to [Turing 36].
(One of the) standard ways of defining
the notion of computation over the
reals.
I have a different approach.
Gray Representation
5
4
3
5
4
3
2
2
1
1
0.0
•
•
0.5
1.0
0.0
Taking the “exclusive or” of a digit and the
next digit.
Expansion based on Gray-code.
0.5
1.0
n
0
1
2
3
4
5
6
7
8
…
Binary
0
1
10
11
100
101
110
111
1000
Gray
0
1
11
10
110
111
101
100
1100
Gray Representation
5
4
3
1 0
1 0
1 0
5
4
3
0
2
1 0
2
1
1
01
1
0 1
0.0
•
•
•
•
•
•
0.5
1.0
0.0
0
0
0.5
1.0
For T = {0,1,⊥}, unique representation of [0,1] with T- sequences.
⊥ means undefinedness or non-terminating computation.
Each sequence contains at most one ⊥.
Define Tωn as the set of T-sequences with at most n copies of ⊥.
Tωn has the subspace topology of the Scott topology of Tω .
This rep. is a topological embedding φ of [0,1] into Tω1 .
How to Output Gray-code
5
4
3
2
0
1
1
0.0
0.5
1.0
1
How to Output Gray-code
5
4
3
⊥
1
⊥ ⊥ ⊥ ⊥
...
2
0
1
⊥ ⊥ ⊥ ⊥
...
1
1
⊥ ⊥ ⊥ ⊥
...
⊥
1
0 ⊥
0 ⊥
0 ⊥
0 ...
1
0.0
0.5
1.0
⊥：undefined currently.
IM2-machine
0
1
•
•
•
Two-head access to input/
output tapes.
0
1
Indeterministic behavior
depending on how to
input when both input
heads have values.
More generally, defined as
an n-head machine.
0
1
0
Input
IM2-Machine
(two-heads I/O)
Output
1
0
0
1
0
1 0
IM2-machine + Gray-code
IM21-machine + Gray-rep. and Type2-machine + Signed-digit rep.
induce the same computability notion on [0,1]
{0,1, }ω
[0,1]
[0,1]
admissible
representation
IM2 machine
{0,1}ω
•
A machine is operating directly on points.
•
Computation which reflect the structure
Type2-machine
of the space.
Bottomed Stream
and
related
works
Relation between
regularities of them
Combinatorial Dynamical System
(joint work with Izumi
property
FullTakeuti and Shuji
folding maps
Yamada)
Generalization
Gray
Independent
[Ohta,T,Y
amada, 2010]
Subbase
5
4
3
Dimension
of bottomed
seq.
2
Gray
1
0.0
0.5
1.0
[T, Yamada, 2009]
0
1
0
1
0
1
0
1 0
IM2 machine
Input
(two-heads I/O)
Output
1
0
[T, 2000]
0
IM2-Machine
[T, 2002]
1
0
Programming Language
Computation
Implementation
similar to REAL
as an extension
PCF
of Haskell
[Terayama,T, 2013]
[T, 2004]
[T, 2012]
of Conversion
from Binary to
topology
[T, Sugihara 2005]
Domain
Theory
Domain
Representation
Implementation in
Concurrent Logic
Languages
[T, 2004]
[T, Tsukamoto, 2013]
[T, 2005]
n
BD :
•
•
•
•
•
domain of n⊥-sequences
0̅, 1̅ : output of 0 or 1 from the blue head.
Finite/infinite-time output-tape states of an IM2-machine:
∞
̅
̅
quotient of {0, 1, 0, 1} with a ̅ b = b a for a, b ∈ {0, 1}.
—> BD1 : the domain of n⊥-sequences.
Similarly, we define the domain BDn of n⊥-sequences as the
quotient of {0(k), 1(k) | k < n} ∞with a(k+1) b(l) = b(l) a(k) for a, b ∈ {0, 1}.
The set of limit elements of BDn is Tωn .
010ω 110ω
Tω1
0⊥0
ω
０１ ⊥ ⊥
⊥10ω
1
...........................................................
0⊥1 00
̅
1
01
0⊥0
0
⊥10
0
̅
0
1
0
0
̅
0
1
0
̅
1
０⊥ ⊥ ⊥
⊥00
̅
0
…
0
̅
0
1
…
…
0
⊥ １⊥ ⊥
̅
0 ⊥⊥⊥⊥ 1
1
…
…
Finite/infinte-time state of a tape for the
Gray-embedding
•
•
•
We need ⊥1, but not ⊥0.
There is a point x such that φ(x) = ⊥1…., but not ⊥0….
Finite-tme states : {φ(x)|n : x ∈ X, n ∈ N}.
(here, p|n is the prefix of a with length n.)
•
•
Infinite-time states: ideal completion.
c.f. signed digit representation.
The space of minimal limit elements homeomorphic to I
010ω 110ω
ω
⊥10
..............................................................................
010 100 110
00
01
̅ 1 0
1
0
0
0⊥1
0
11 1⊥1
⊥10
̅ 1 1
0
̅
1
1
1
1
̅
1
10
0
For every n-­‐dimensional separable metrizable space X, there exists an embedding φ of X in BDn such that X is homeomorphic to the minimal-­‐limit set of the domain constructed in this way.
Regular n -domain
……
̅
0
0⊥1
00
̅
1
0
⊥10
01
1
0
0
0
̅ 1
0
⊥1
̅
1
1⊥1
11
1
1
̅
1
10
0
1
B
̅
1
0,1
•
A language A ⊆ {0, 1, 0̅, 1}̅ * recognizable by a finite
state automaton F which satisfies 𝛿(s, ab̅ ) = 𝛿(s,
ba) for every state s of F.
Take a quotient of A with ab̅ =ba .
Can be generalized to a subdomain of BDn.
We call such a domain a regular n⊥-domain.
•
a n⊥-domain is full if {0,1}* ⊆ A.
•
•
•
A
0,1
Regular n -domain
•
Definition. For n ∈ N, consider the alphabet {0(k), 1(k) | k
< n} . Let A be a language recognizable by a finite-state
automaton F such that 𝛿(s, a(k+1) b(l)) = 𝛿(s, b(l) a(k)) for every
state s of F. A regular n⊥-domain D is the ideal
completion of a poset K obtained as the quotient of such a
language A with a(k+1) b(l) = b(l) a(k) for a, b ∈ {0, 1}.
!
•
A n⊥-domain is a subdomain of BDn.
Gray-code: itinerary of the tent function.
•
Tent function:
⇢
2x
f (x) =
1 2x
(0  x  1/2)
(1/2 < x  1)
•
Dynamical system f: I → I.
•
Itinerary of the tent function:
Infinite sequence of {0,1,⊥} which codes whether
x is mapped to [0,1/2), (1/2, 1], or {1/2} by its repetition.
•
8
< 0
1
'f (x)(n) =
:
?
(0  f n (x) < 1/2)
n
(1/2 < f (x)  1)
n
n
(1/2
<
f
(x)  1)
(f (x) = 1/2)
1
0
0
1/2
S0,0 S0,1
S1,0 S1,1
S2,0 S2,1
S3,0 S3,1
Gray-expansion is nothing but itinerary of the tent function.
1
Generalization to other
dynamical systems.
•
A dynamical system with the notion of full-itinerary.
!
•
Each point x is identified by its itinerary φ(x) .
!
•
Full-folding map
Exact full-folding map
{φ(x)|k : x ∈ X, k ∈ N} generates a regular n⊥-domain for some n.
Exact full-folding map with this
property.
Full-folding map
•
X
X: compact metrizable space.
f
flip
X
•
Definition: A continuous map f: X →X is a full-folding map if,
for a regular open set X0 and X1 = X \ X0,
f |X̄i : X̄i ⇠
= X (homeo) for i = 0, 1.
U is regular open if it is the interior of its closure.
•
•
•
We define C = X \ (X0 ∪ X1 ) .
f is 2-1 on X0 ∪ X1 and 1-1 on C.
We define homeomorphisms go : X → Xi (i = 0, 1) as gi = f |X̄i
1
.
Real Kneading
[Wikipedia]
Itinerary of a full-folding map
If 𝑓:𝑋→𝑋 is a full-­‐folding map, we define its i6nerary 𝜑𝑓:𝑋→{0,1,⊥}𝜔 as •
Exact full-folding map
flip
•
•
•
•
•
flip
flip
X is divided into 2𝑛 regions 𝑅(𝑝) (𝑝∈{0,1}𝑛) by taking n inverses. De[ine 𝑔𝑝=𝑔𝑝(0)∘𝑔𝑝(1)∘…∘𝑔𝑝(𝑛−1) . cl 𝑅(𝑝) =𝑔𝑝(𝑋) and cl 𝑅(𝑝) is homeomorphic to 𝑋. We say that a f-­‐f. map 𝑓 is exact if max{diam cl 𝑅(𝑝) | 𝑝∈{0,1}𝑛 }
shrinks to 0 as 𝑛→∞ We need exactness for coding!
•
It is easy to show that the tent map is the only exact f.f. map on I=[0,1] modulo conjugacy. !
•
Characteriza6on of exact full-­‐folding 2 maps on D=𝐼 which induce regular n⊥-­‐domains.
f : X ! X is conjugate to
g : Y ! Y if there is a homeomorphism h : X ! Y such that
E f =g h
X
h
Y
f
g
X
h: homeo.
Y
First, consider the case X is homeomorphic to the unit disk D and g0 and g1 are similarity maps. • Recutangular full-­‐folding map. • Triangular full-­‐folding map.
Rectangular Full-­‐folding map.
1 011
001
0 0 010
000
1
110
100
111
101
!
!
1 0 0 1
Itinerary of (x, y): interleaving inverted itineraries of x and y with tent map
flip
: ⊥⊥00111…
Triangular Full-­‐folding map
011
010
001
100
101
110
f(C)
000
111
g0
11
X0
f
X
X1
g1
flip
: ⊥0⊥⊥0111…
Full-­‐Folding maps of 𝐷
flip
11 Categories of Full-­‐folding maps on D
A(, A)
B(, B)
C(, C)
E(, E)
G
J
I
H
F
Acute f.f. map
D(, D)
K
Only A, B, J contain exact full-­‐folding maps.
Unimodal maps
•
•
•
1
X1
h：I → I is a unimodal map if h(0) = h(1) and h is increasing on [0,1/2] and decreasing on [1/2, 1]. I6neray is defined and plays an important role in Kneading Theory [Milnor-­‐Thurston, …] 𝜑(1/2) determine approx. behavior of S
the dynamical system. S
I6nerary can be exact, but not full S
except for the tent func6on and its conjugates.
X0
•
0
0,0
S0,1
1,0
S1,1
2,0
S2,1
1/2
000… does not exist.
1
Boundary restriction of a f.f. map.
• f
•
Conversely, we can form a f.f. map from a unimodal map, if
it is not conjugate to the tent map.
Itineraries of f.f. map and unimodal map.
• X1
011
010
X0
001
011
010
110
111
000, 100… do not exist.
101
000, 100… exist inside the disk.
000
001
100
101
110
111
Itineraries of a and b determine exact f.f. map.
• A
B
J
Cutting/Co-­‐cutting time
• (Hofbauer, Keller, Bruin, …)
There a
re f
our c
ases.
For a f.f. map of category A, !
an = a for some n. The number of boaoms in the i6nerary of a is infinite. Do not induce a n⊥-­‐domain for any n. • Both cubng-­‐6me and co-­‐cubng 6mes are finite. Kneading sequence pω for an odd p. No exact f. f. map. • Cubng 6me is infinite, co-­‐cubng 6me is finite. Kneading sequence pω for an even p. No exact f. f. map. • Both cubng and co-­‐cubng 6mes are infinite. • Post cri6cally finite. ( ∪fn(a) is a finite set) Kneading sequence qpω for nonempty p, q (ultimately periodic). Exact f.f. maps exist, construc6on via complex dynamical system. • Otherwise… f.f. map with dense Inverse image of the boundary. Similar result for categories B, taking care of both cri6cal points.
•
Theorems
• A conformal full-folding map (i.e., both g0 and g1 (z̄) are conformal maps from the upper half disk) on I 2 is semi-conjugate to a
complex quadratic rational map with real coefficients. Therefore,
modulo conjugacy, it has the form
az 2 + b
f (z) = 2
.
cz + d
with a, b, c, d 2 R. The conjugacy map identifies complex conjugacy to map to the closure of the upper half plane and make a
Mobius transformation to the unit disk.
• It is exact and induces a full rational n?-domain if f is postcritically finite and has no super attractive periodic orbit.
Correspondence between regularity of tape-states of
a machine and regularity of the dynamical system
which determine the code.
34
Conclusion
Bottomed sequences enable symbolic manipulation of
continuous objects.
•
•
•
The space of bottomed sequences is continuous and, at the
same time, can be manipulated symbolically.
•
Separable metrizable spaces can be embedded into the space of
bottomed sequences.
Studied how properties of continuous objects (with respect to
coding) are reflected to the structure of an IM2-machine operating
on it.
•
Dimension as the minimal number of heads.
•
Regularity of the dynamical system which induces the coding as
regularity of the machine behavior with respect to the next input.
We are interested in logical/coalgebraic treatment of bottomed
sequences for programming over them.