Compressibility of Closed Sets - Heidelberg University · Douglas Cenzer, Chris Porter, Ferit Toska...

Compressibility of Closed Sets


Douglas Cenzer, Chris Porter, Ferit ToskaCCR 2015, Heidelberg, Germany

June 26, 2015

Douglas Cenzer, Chris Porter, Ferit Toska CCR 2015, Heidelberg, GermanyCompressibility of Closed Sets


Table of contents

1 Introduction

2 SPM Complexity of Subsets of 2N

3 Compressibility and Homeomorphism of Subsets of 2N



Introduction

Background

Kolmogorov complexity M(x) = y where M is a universalprefix free machine.

For the optimal compression, with K (y) = |x |, the compressedstrings may be taken to be random, that is, incompressible.

The compression of an infinite sequence Y ∈ 2N to X ∈ 2N

via a Turiing machine M has long been an interesting topic

Kucera-Gacs showed that any infinite sequence Y is Turingreducible to a ML-random sequence R.



Introduction

Background








Introduction

Background








Introduction

Background








Introduction

Use

ρX (n) is the use of oracle X to compute output Y � n

Ryabkov showed that the optimal compression of Y will haverelative use ρ−(X ,Y ) = lim infn ρ

X (n)/n equal to theconstructive Hausdorff dimension dim(Y ).

Moreover,this compression can be done uniformly for allelements of a subset S of 2N, with optimal compressiondim(S).

In both cases, compression was obtained arbitrarily close tooptimal.



Introduction

Use








Introduction

Use








Introduction

Use








Introduction

Optimal Compression

Doty gave a similar result for the constructive packingdimension Dim(Y ) = ρ+(X ,Y ) = lim supn ρ

X (n)/n.

Moreover, optimal compression was achieved.

Doty also considered computable and sub-computabledimension and compression.



Introduction

Optimal Compression


X (n)/n.





Introduction

Optimal Compression


X (n)/n.





Introduction

Constructive Dimension

Constructive dimension for X ∈ 2N is characterized as follows:

Theorem (Mayordomo)

dim(X ) = lim infn

K (X � n)

n

Theorem (Athreya)

Dim(X ) = lim supn

K (X � n)

n



Introduction



Theorem (Mayordomo)

dim(X ) = lim infn

K (X � n)

n

Theorem (Athreya)

Dim(X ) = lim supn

K (X � n)

n



Introduction



Theorem (Mayordomo)

dim(X ) = lim infn

K (X � n)

n

Theorem (Athreya)

Dim(X ) = lim supn

K (X � n)

n



SPM Complexity of Subsets of 2N

Strict Process Machines

M : {0, 1}∗ → {0, 1}∗ such that dom(M) is a tree and ifv � u, then M(v) � M(u).

Compare to Levin’s monotone machines, where if v ≺ u andboth M(u) and M(v) are defined, then M(v) � M(u).

For total M, this defines a computable function F : 2N → 2N.M total on a tree T defines F on the closed set [T ].

There is a universal strict process machine.




























Description

Let M be a strict process machine. Define FM : 2N → 2N byX ∈ dom(FM) ⇐⇒ {|M(X � n)| : n ∈ N} is unbounded, andFM(X ) = Y where X is the M-description of Y

X describes Y = FM(X ) with rate c if

(∃∞n) (c · n ≤ |M(X � n)|)

We say that Y is c-compressible by M

M complexity of Y :sM(y) = inf{ 1c : y is c-compressible by M}.




Description



(∃∞n) (c · n ≤ |M(X � n)|)






Description



(∃∞n) (c · n ≤ |M(X � n)|)






Complexity

P is an M-description of Q if Q = {FM(X ) : X ∈ P}P describes Q with a rate c > 0 if each X ∈ P describesY = FM(X ) with rate c .

The SPM complexity of Q:

spM(P,Q) = inf{ 1c : P describes Q with a rate c}spM(Q) = inf{sM(P,Q) : P is an M-description of Q}sp(Q) = spM(Q) for a universal strict process machine M.




Complexity







Complexity







Complexity







Complexity







Complexity







Results of Toska

Theorem

For any X ∈ 2N, dim(X ) = sp(X ).

Theorem

For any Q ⊂ 2N, sp(Q) = sup{sp(Y ) : Y ∈ Q}

Corollary

For any Q ⊂ 2N, dim(Q) = sp(Q).




Results of Toska

Theorem


Theorem


Corollary





Results of Toska

Theorem


Theorem


Corollary





For Closed Sets

We can now improve the corollary as follows:

If Q is closed, then Q can be optimally compressed by aclosed set P with dim(P) = 1.

Question

Can we make P a random closed set

If Q is effectively closed, then Q may be compressedarbitrarily close to optimal by a Π0

1 class P.

If dim(Q) is computable, then there is optimal compression.




For Closed Sets



Question



1 class P.





For Closed Sets



Question



1 class P.





For Closed Sets



Question



1 class P.




Compressibility and Homeomorphism of Subsets of 2N

Decidability

Definition

For a closed set P, let TP = {w ∈ {0, 1}∗ : (∃Z ∈ P)w ≺ Z}P is effectively closed (a Π0

1 class) if TP is a co-c.e. set

P is decidable if TP is a computable set

P is 1-decidable if there is a computable functionf : {0, 1}∗ → (N ∪ {∞}) such thatf (w) = card{Z ∈ P : w ≺ Z}.

This can be generalized to n + 1-decidability by havingf (w) = card{Z ∈ Dn(P) : w ≺ Z}




Decidability

Definition

For a closed set P, let TP = {w ∈ {0, 1}∗ : (∃Z ∈ P)w ≺ Z}P is effectively closed (a Π0

1 class) if TP is a co-c.e. set

P is decidable if TP is a computable set

P is 1-decidable if there is a computable functionf : {0, 1}∗ → (N ∪ {∞}) such thatf (w) = card{Z ∈ P : w ≺ Z}.

This can be generalized to n + 1-decidability by havingf (w) = card{Z ∈ Dn(P) : w ≺ Z}




Rank One Classes

Definition

Let P be a rank one Π01 class with unique limit path A.

Rn(P) = {Z ∈ P : n = least k(A � (k + 1) 6= Z � (k + 1))}Let `(P) = limn card(Rn) (if this exists)

Let av(P) = lim∑

k≤n cardRn

n+1 if it exists

X ∈ Rn is an isolated path which branches off from A at A(n).av(P) is the average population of these branches.




Rank One Classes

Definition

Let P be a rank one Π01 class with unique limit path A.

Rn(P) = {Z ∈ P : n = least k(A � (k + 1) 6= Z � (k + 1))}Let `(P) = limn card(Rn) (if this exists)

Let av(P) = lim∑

k≤n cardRn

n+1 if it exists

X ∈ Rn is an isolated path which branches off from A at A(n).av(P) is the average population of these branches.




Subshifts

Definition

The shift operator σ : 2N → 2N is defined byσ(X ) = (X (1),X (2), . . . ).

A closed set P is a subshift if it is closed under σ.

Suppose that P is a subshift of rank one with unique limitpath A. Then A is periodic.

Furthermore, P is 1-decidable.




Subshifts

Definition








Subshifts

Definition








Population of Subshifts

Let P be a subshift with unique limit path A = 0ω.

Then for each n, Rn+1 ⊆ Rn.

Since each Rn is finite, limn Rn = R exists, and`(P) = card(R) = av(P) is an integer.


















Examples

P = {0ω} ∪ {0n1ω : n ∈ ω}`(P) = av(P) = 1.

Q = {0ω} ∪ {0n10ω : n ∈ ω} ∪ {0n110ω : n ∈ ω}`(Q) = av(Q) = 2.




Examples

P = {0ω} ∪ {0n1ω : n ∈ ω}`(P) = av(P) = 1.

Q = {0ω} ∪ {0n10ω : n ∈ ω} ∪ {0n110ω : n ∈ ω}`(Q) = av(Q) = 2.




Homeomorphisms of Rank One Classes

The problem of determining when two Π01 classes of rank one

are computably homeomorphic, and hence automorphic, ornot, has been studied by Cenzer, Cholak+Downey,Montalban, and others.

It follows from this previous work that any two 1-decidableclosed sets of rank one are computably homeomorphic.

We will consider a stronger notion of homeomorphism, wherethe population of P serves as a characteristic.




















Compression of Rank One Classes

Theorem

Suppose that P and Q are rank one classes such that

av(P) = p, av(Q) = q exist.

P c-compresses Q.

Then, q · c ≤ p.




Main Result

Theorem

Let P and Q be 1-decidable closed sets with unique limit paths,such that av(P) = p and av(Q) = q exist.

Then there exists a bijection F : P → Q represented by acomputable strict process machine M such that P is anM-description of Q and lim |M(X �n)|

n = pq .




Main Result

Theorem

Let P and Q be 1-decidable closed sets with unique limit paths,such that av(P) = p and av(Q) = q exist.

Then there exists a bijection F : P → Q represented by acomputable strict process machine M such that P is anM-description of Q and lim |M(X �n)|

n = pq .




Corollaries

Definition

F : P → Q is a strong homeomorphism if there are computablerepresentations M,M‘ of F ,F−1 such that, for X ∈ P, Y ∈ Q,

lim|M(X ) � n)|

n= lim

|M‘(Y � n)|n

= 1

Corollary

Let P and Q be 1-decidable closed sets with unique limit pathssuch that av(P) = p and av(Q) = q exist. Then

P strongly c-compresses Q if and only if c · q ≤ p.

P is strongly homeomorphic to Q if and only if p = q.




Corollaries

Definition

F : P → Q is a strong homeomorphism if there are computablerepresentations M,M‘ of F ,F−1 such that, for X ∈ P, Y ∈ Q,

lim|M(X ) � n)|

n= lim

|M‘(Y � n)|n

= 1

Corollary

Let P and Q be 1-decidable closed sets with unique limit pathssuch that av(P) = p and av(Q) = q exist. Then

P strongly c-compresses Q if and only if c · q ≤ p.

P is strongly homeomorphic to Q if and only if p = q.




Subshifts and strong homeomorphisms

Corollary

Let P be a subshift with a unique limit point and letp = `(P) = av(P).Then P is strongly homeomorphic to{0ω} ∪ {0n1k0ω : k = 1, . . . p}.




Current and Future Work

n + 1 -decidable closed sets

Compressibility and homeomorphisms of rank n + 1 closed sets

Strong homeomorphisms of arbitrary closed sets


















Thank You


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