Computing with Compact Sets: The Gray Code Case Dieter Spreen - - PowerPoint PPT Presentation

Computing with Compact Sets: The Gray Code Case

Dieter Spreen University of Siegen

joint work with

Hideki Tsuiki Kyoto University

Continuity, Computability, Constructivity: From Logic to Algorithms Kochel am See, Bavaria, 14-18 September 2015

• 1. Signed digit repesentation

Let x = 0.a1a2a3 . . . be a real number in I = [−1, +1]. Then writing a digit a ∈ SD := {−1, 0, 1} in front of a1a2a3 . . . corresponds to applying the contraction da defined by da(x) = x + a 2 to x. In this way we can think of 0.a1a2a3 . . . as an iterated application of operations da ∈ D := { da | a ∈ SD } da1 ◦ da2 ◦ da3 ◦ · · ·

Lemma

I is a complete bounded metric space and D a finite set of contractions that covers I: I =

• { da[I] | a ∈ SD }.

In particular I is compact. Consequently, for α = α0α1 . . . ∈ SDω

• n≥0

dα0 ◦ · · · ◦ dαn−1[I] contains exactly one element. Let [ [α] ] be the unique x ∈

• n≥0

dα0 ◦ · · · ◦ dαn−1[I]

Lemma

SDω is a bounded complete and hence compact metric space with metric δ(α, β) :=

• if α = β,

2− min{ n|αn=βn }

• therwise.

Proposition

◮ [

[·] ]: SDω → I is onto and uniformly continuous.

◮ The Euclidean topology on I is equivalent to the quotient

topology induced by [ [·] ]. In particular we have that each α ∈ SDω denotes a real number in I, and vice versa.

• 2. Pre-Gray code representation

Let pG := {U, D, L, R, FinL, FinR} G := {U, L, R} H := {D, FinL, FinR} Now, not every sequence in pGω is meaningful! The set coG of meaningful sequences is defined together with a set

coH by mutual coninduction: (coG, coH) is the largest subset of

pGω × pGω such that α ∈ coG → (∃β ∈ coG) α ∈ {L, R}β ∨ (∃β ∈ coH) α = Uβ α ∈ coH → (∃β ∈ coG) α ∈ {FinL, FinR}β ∨ (∃β ∈ coH) α = Dβ

Proposition

• 1. (coG, δ) is complete.
• 2. (coH, δ) is complete.

Recall that δ is bounded.

Corollary

Both, (coG, δ) and (coH, δ) are compact.

Let [ [L] ] := LR−1 [ [R] ] := LR1 [ [U] ] := U [ [FinL] ] := Fin−1 [ [FinR] ] := Fin1 [ [D] ] := D where for a ∈ {−1, 1} LRa(x) := −ax − 1 2 U(x) := x 2 Fina(x) := ax + 1 2 D(x) := x 2

Lemma

◮ d1(x) = Fin1(x) = LR1(−x) ◮ d−1(x) = Fin−1(−x) = LR−1(x) ◮ d0(x) = U(x) = D(x)

Corollary

◮ range(d1) = range(Fin1) = −

− → [0, 1], range(LR1) = ← − − [0, 1]

◮ range(d−1) = range(LR−1) = −

− − − → [−1, 0], range(Fin−1) = ← − − − − [−1, 0]

◮ range(d0) = range(U) = range(D) = [−1/2, +1/2]

Consequently, both { U, LRa | a ∈ {−1, 1} } and { D, Fina | a ∈ {−1, 1} } are finite covering sets of contractions on I. Hence, for α = α0α1 . . . ∈ coG and β = β0β1 . . . ∈ coH

• n≥0

[ [α0] ] ◦ · · · ◦ [ [αn] ][I] and

• n≥0

[ [β0] ] ◦ · · · ◦ [ [βn] ][I] contain exactly one element. Let {[ [α] ]G} :=

• n≥0

[ [α0] ]◦· · ·◦[ [αn] ][I] and {[ [β] ]H} :=

• n≥0

[ [β0] ]◦· · ·◦[ [βn] ][I]

Proposition

◮ [

[·] ]G : coG → I, [ [·] ]H : coH → I are onto and uniformly continuous.

◮ The Euclidean topology on I is equivalent to the quotient

topology induced by [ [·] ]G ([ [·] ]H).

• 3. Compact sets

Let K(I) be the collection of all nonempty compact subsets of I.

Proposition

K(I) endowed with the Hausdorff metric µH is a bounded complete, hence compact metric space. Berger/Spreen: There is no finite set of contractions on K(I) that covers K(I). So, it is impossible to represent nonempty compact subsets of I by infinite streams. As we will see, however, they can be represented by infinite trees.

Definition

A digital tree is a nonempty set T ⊆ pG∗ of finite sequences of elements of pG that is closed under initial seqments and has no maximal elements, that is, the empty sequence [] is in T and whenever R0 . . . Rn ∈ T, then R0 . . . Rn−1 ∈ T and R0 . . . RnR ∈ T, for some R ∈ pG. Note that each such tree is finitely branching. Moreover, each element R0 . . . Rn ∈ T can be continued to an infinite path α in T. In the following we write α ∈ T to mean that α is an infinite path

• f T and identify T with the set of its infinite paths.

Let T be the set of all digital trees and TG := { T ∩ coG | T ∈ T } TH := { T ∩ coH | T ∈ T }

Lemma

(TG, TH) is the largest subset of T × T such that T ∈ TG → (∃T1, . . . , Tn ∈ T)(∃R1, . . . , Rn ∈ G)(∀1 ≤ i ≤ n) [Ri ∈ {L, R} → Ti ∈ TG] ∧ [Ri = U → Ti ∈ TH] ∧ T =

n

• i=1

RiTi T ∈ TH → (∃T1, . . . , Tn ∈ T)(∃R1, . . . , Rn ∈ H)(∀1 ≤ i ≤ n) [Ri ∈ {FinL, FinR} → Ti ∈ TG] ∧ [Ri = D → Ti ∈ TH] ∧ T =

n

• i=1

RiTi.

Proposition

◮ TG endowed with the Hausdorff metric δH induced by δ is a

bounded complete, hence compact metric space.

◮ TH endowed with the Hausdorff metric δH induced by δ is a

bounded complete, hence compact metric space.

Lemma

Every tree in TG is compact and every nonempty compact subset

• f coG is the set of infinite paths of a digital tree in TG. Similarly,

for TH.

Definition

For T ∈ TG set [ [T] ]TG := { [ [α] ]G | α ∈ T }.

Since [ [·] ]G is continuous, [ [T] ]TG is compact. Conversely, for any K ∈ K(I), [ [·] ]−1

G [K] is closed and hence compact, as TD is

compact.

Lemma

The nonempty compact subsets of I are exactly the values of trees in TG.

Proposition

◮ [

[·] ]TG : TG → I is onto and uniformly continuous.

◮ The topology on K(I) induced by the Hausdorff metric is

equivalent to the quotient topology induced by [ [·] ]TG .

Let D := { da | a ∈ {−1, 0, +1} }, G := {U, LR−1, LR1}, H := {D, Fin−1, Fin1}. In Berger/Spreen a representation-free coinductive approach to computations with compact subsets of I was presented. Let CK be the largest subset of K(I) such that CK(A) → (∃B ⊆ D)(∃(Ab)b∈B ∈ K(I)B)(∀b ∈ B)CK(Ab) ∧ A =

• b∈B

b[Ab].

Lemma (Berger/Spreen)

K(I) = CK.

Let (GK, HK) be the largest subset of K(I) × K(I) such that GK(A) → (∃E ⊆ G)(∃(Ae)e∈E ∈ K(I)E) A =

• e∈E

e[Ae] ∧ (∀e ∈ E) [e ∈ {LR−1, LR1} → GK(Ae)] ∧ [e = U → HK(Ae)], HK(A) → (∃F ⊆ H)(∃(Af )f ∈F ∈ K(I)F) A =

• f ∈F

f [Af ] ∧ (∀f ∈ F) [f ∈ {Fin−1, Fin1} → GK(Af )] ∧ [f = D → HK(Af )].

Theorem

GK = CK.

◮ From the proof a computable translator between realizers for

both sides can be derived, that is, between trees in TG and trees over SD.

◮ In Berger/Spreen the equivalence between the latter

representation and the Cauchy representation for nonempty compact sets was shown.

◮ Consequently, pre-Gray computability of nonempty compact

subsets of I is equivalent to TTE computability.