Abstract numeration systems

(1)

ÉmilieCharlier UniversitélibredeBruxelles

Colloquium Turku, May 7,2012

(2)

A positional numeration system(PNS) isgiven by asequenceof integers

U = (U

i

)

i≥0

suchthat

I

U

₀

= 1

I

∀i U

_i

< U

_i+1

I

(U

i+1

/U

i

)

i≥0

isbounded

→

C

U

= sup

i≥0

dU

i+1

/U

i

e

The greedy

U

-representationofa positiveinteger

n

is the unique word

rep

U

(n)

= c

`−1

· · · c

0

over

Σ

U

= {0, . . . , C

U

− 1}

satisfying

n =

`−1

X

i=0

c

i

U

i

, c

`−1

6= 0

and

∀t

t

X

i=0

c

i

U

i

< U

t+1

.

(3)

A set

X ⊆ N

is

U

-recognizableor

U

-automatic ifthe subset

rep

_U

(X) = {rep

_U

(x) : x ∈ X}

of

Σ

∗

(4)

Integer base

b

≥ 2

U = (b

i

₎

i≥0

rep

U

, Σ

U

→

rep

b

, Σ

b

Σ

b

= {0, · · · , b − 1}

L

b

= rep

b

(N) = Σ

∗

b

\ 0 Σ

∗

b

1, 2

0, 1, 2

rep

₃

(N)

N

is

3

-recognizable

27 9 3 1

ε 0

1 1

2 2

1 0 3

1 1 4

1 2 5

2 0 6

2 1 7

2 2 8

1 0 0 9

(5)

Integer base

b

≥ 2

U = (b

i

₎

i≥0

rep

U

, Σ

U

→

rep

b

, Σ

b

Σ

b

= {0, · · · , b − 1}

L

b

= rep

b

(N) = Σ

∗

b

\ 0 Σ

∗

b

1

2 0, 2

0, 2

1

1 rep

U

3 (

2 N

)

2 N

is

3

-recognizable

27 9 3 1

ε

0 1 1

2

2 1 0 3

1 1

4 1 2 5

2 0

6 2 1 7

2 2

8 1 0 0 9

(6)

Let

F = (F

i

)

i≥0

= (1, 2, 3, 5, 8, 13, 21, . . .)

bedened by

F

0 = 1, F

1 = 2

and

∀i ∈ N, F

i+2

= F

i+1

+ F

i

.

Σ

F

= {0, 1}

The factor

11

isforbidden:

1

0

1 rep

_F

(N) = 1{0, 01}

∗

_{∪ {ε}}

N

is

F

-recognizable

13 8 5 3 2 1

ε 0

1 1

1 0 2

1 0 0 3

1 0 1 4

1 0 0 0 5

1 0 0 1 6

1 0 1 0 7

1 0 0 0 0 8

(7)

Let

F = (F

i

)

i≥0

= (1, 2, 3, 5, 8, 13, 21, . . .)

bedened by

F

0 = 1, F

1 = 2

and

∀i ∈ N, F

i+2

= F

i+1

+ F

i

.

0

1

0

1

0

1

0

1

0

0 rep

_F

(

2 N

)

2 N

is

F

-recognizable

13 8 5 3 2 1

ε

0 1 1

1 0

2 1 0 0 3

1 0 1

4 1 0 0 0 5

1 0 0 1

6 1 0 1 0 7

1 0 0 0 0

8

(8)

U

-recognizability of

N

Is the set

N

U

-recognizable? Otherwisestated, isthe numeration language

rep

U

(N)

regular? Not necessarily:

Theorem(Shallit 1994)

Let

U

be aPNS. If

N

is

U

-recognizable,then

U

islinear,i.e.,it satises alinearrecurrence relationover

Z.

Loraud (1995)and Hollander (1998)gave sucientconditions for the numeration language to be regular: The characteristic polynomial of therecurrence relationhas aparticularform.

(9)

U

-recognizability of arithmetic progressions

Proposition

Let

U = (U

i

)

i≥0

be alinearnumerationsystem andlet

p, q ∈ N

. If

N

is

U

-recognizable,then

p + N q

is

U

-recognizableand a DFA accepting

rep

U

(p + N q)

can beobtained eciently.

(10)

An abstractnumeration system(ANS) isatriple

S = (L, Σ, <)

where

L

isan innite regularlanguage over atotallyordered alphabet

(Σ, <)

.

By enumeratingthe words of

L

w.r.t. the radix order

<

rad

induced by

<

,we denea bijection:

rep

_S

: N → L

val

S

= rep

−1

S

: L → N.

A set

X ⊆ N

is

S

-recognizableif

rep

(11)

L = {a, b}

∗

_{Σ = {a, b}}

_{a < b}

n 0 1 2

3

4

5

6

7 · · ·

rep(n) ε a b aa ab ba bb aaa · · ·

L = a

∗

b

∗

Σ = {a, b}

a < b

n 0 1 2

3

4

5

6 · · ·

rep(n) ε a b aa ab bb aaa · · ·

(12)

ANS generalize PNShavinga regularnumeration language: Let

U

be aPNSand let

x, y ∈ N

. We have

x < y ⇔ rep

U

(x) <

rad

rep

U

(y).

Example (Fibonacci)

rep

_F

(N) = 1{0, 01}

∗

∪ {ε}

6 < 7

and

10

0 1 <

rad

10

1

0

(samelength)

6 < 8

and

1001 <

rad

10000

(dierent lengths)

13 8 5 3 2 1

ε 0

1 1

1 0 2

1 0 0 3

1 0 1 4

1 0 0 0 5

1 0 0 1

6 1 0 1 0

7 1 0 0 0 0 8

(13)

U

be aPNSand let

x, y ∈ N

. We have

x < y ⇔ rep

U

(x) <

rad

rep

U

(y).

Example (Fibonacci)

rep

_F

(N) = 1{0, 01}

∗

∪ {ε}

6 < 7

and

10

0 1 <

rad

10

1

0

(samelength)

6 < 8

and

1001 <

rad

10000

(dierent lengths)

13 8 5 3 2 1

ε 0

1 1

1 0 2

1 0 0 3

1 0 1 4

1 0 0 0 5

1 0 0 1

6 1 0 1 0 7

1 0 0 0 0

8

(14)

U

be aPNSand let

x, y ∈ N

. We have

x < y ⇔ rep

U

(x) <

rad

rep

U

(y).

Example (Fibonacci)

rep

_F

(N) = 1{0, 01}

∗

_{∪ {ε}}

rep

_F

(n) n

ε 0

1 ₁

10 ₂

100 ₃

101 ₄

1000

5 1001

₆

1010

₇

10000

₈

(15)

Let

L

be alanguage orderedw.r.t. the radixorder. If

w

0 < w

1 < · · ·

arethe elements of

L

and

X ⊆ N

,then

L[X]

= {w

n

: n ∈ X}.

If

S = (L, Σ, <)

,then

L[X] = rep

S

(X)

.

If

L[X]

isaccepted by aniteautomaton, whatdoesitimplyon

X

? What conditions on

X

insuresthat

L[X]

isregular?

(16)

ANS are ageneralizationof allusualPNSlike integerbase

numeration systems orlinearnumerationsystems,and even rational numeration systems.

Thanks tothis generalpoint ofviewon numeration systems,we try to distinguishresultsthatdeeply dependon the algorithmused to represent the integersfromresultsthat onlydepend on theset of representations.

Due to thegeneral settingofANS, somenew questionsconcerning languages arisenaturally fromthisnumeration pointof view.

(17)

I

Rec. setsin agivenANS?

I

Rec. setsin allANS?

I

Arethere subsetsof

N

that arenever recognizable?

I

Givena subsetof

N

can we buildan ANSforwhichitis rec.?

I

How do rec. dependon the choiceof the numeration?

I

For whichANS do arithmeticoperationspreserve rec.?

I

Operationspreservingrec. in agivenANS?

I

How torepresent real numbers?

I

Can we deneautomaticsequences inthat context?

I

Logical characterization ofrec. sets?

I

Extensions to themultidimensionalsetting?

I

(18)

I

Rec. setsin allANS?

I

Arethere subsetsof

N

I

Givena subsetof

N

I

(19)

I

Rec. setsin allANS?

I

Arethere subsetsof

N

I

Givena subsetof

N

I

(20)

I

Rec. setsin allANS?

I

Arethere subsetsof

N

I

Givena subsetof

N

I

(21)

I

Rec. setsin allANS?

I

Arethere subsetsof

N

I

Givena subsetof

N

I

(22)

I

Rec. setsin allANS?

I

Arethere subsetsof

N

I

Givena subsetof

N

I

(23)

I

Rec. setsin allANS?

I

Arethere subsetsof

N

I

Givena subsetof

N

I

(24)

I

Rec. setsin allANS?

I

Arethere subsetsof

N

I

Givena subsetof

N

I

(25)

I

Rec. setsin allANS?

I

Arethere subsetsof

N

I

Givena subsetof

N

I

(26)

I

Rec. setsin allANS?

I

Arethere subsetsof

N

I

Givena subsetof

N

I

(27)

I

Rec. setsin allANS?

I

Arethere subsetsof

N

I

Givena subsetof

N

I

(28)

I

Rec. setsin allANS?

I

Arethere subsetsof

N

I

Givena subsetof

N

I

(29)

(30)

b

-automatic words

An innite word

x = (x

n

)

n≥0

is

b

-automaticifthere existsaDFAO

A = (Q, q

0 , Σ

b

, δ, Γ, τ )

s.t. forall

n ≥ 0

,

x

n

= τ (δ(q

0 , rep

b

(n))).

Theorem(Cobham 1972)

Let

b ≥ 2

. Aninnite wordis

b

-automatic iitis the imageunder a codingofan innitexed point ofa

b

-uniformmorphism.

(31)

Let

S = (L, Σ, <)

bean ANS.

An inniteword

x = (x

n

)

n≥0

is

S

-automatic ifthere existsaDFAO

A = (Q, q

0 , Σ, δ, Γ, τ )

s.t. forall

n ≥ 0

,

x

n

= τ (δ(q

0 , rep

S

(n))).

Theorem(Rigo-Maes 2002)

An innite wordis

S

-automaticforsomeANS

S

iitisthe image under acodingofan innitexed pointof amorphism, i.e. a

(32)

The setof primesisnever

S

-recognizable.

Itscharacteristicsequence isnot morphic (Mauduit 1988).

Corollary

The factorcomplexityof an

S

-automatic word is

O(n

2 ₎

(33)

Example (Morphic

→ S

-Automatic)

Consider the morphism

µ

dened by

a 7→ abc

;

b 7→ bc

;

c 7→ aac.

We have

µ

ω

_{(a) = ab}

_cb

_c

_aacbc

_{aacabcabcaacbcaacabcabc · · ·}

. One canonicallyassociatesthe DFA

A

µ,a

a

b

c

0

1

2

0

1

2 0, 1

L

µ,a

= {ε, 1,

2 ,

10 , 11,

20, 21, 22

,

100, 101

, 110, 111, 112, 200, . . .}

If

S = (L

µ,a

, {0, 1, 2}, 0 < 1 < 2)

,then

(µ

ω

_(a))

(34)

Example (

S

-Automatic

→

Morphic)

S = (L, {0, 1, 2}, 0 < 1 < 2)

where

L = {w ∈ Σ

∗

_{: |w|}

1

is odd

}

minimalautomatonof

L

DFAO generating

x

I

F

0, 2

1 0, 2

1 a

_b

2 _{0, 1}

2 0, 1

n

0 1 2 3 4 5 6 7 8

· · ·

rep

_S

(n)

1 01 10 12 21 001 010 012 021

· · ·

x

b a a b b b b a a

· · ·

(35)

Ia

Ib

Fb

Fa

2

0

1

2

0

1

2

0

1

2

0

1 f : α 7→ αI

b

F

b

I

a

F

a

7→ F

b

I

b

F

a

g : α, I

a

, I

b

7→ ε

I

a

7→ I

b

F

b

I

a

F

b

7→ F

a

I

a

F

b

F

a

7→ a

I

b

7→ I

a

F

a

I

b

F

b

7→ b

L

⊆ Σ

∗

_ε

0 1 2 00 01 02 10 11 12 20

f

ω

_(α)

_α

_I

b

F

b

I

a

I

a

F

a

I

b

F

a

I

a

F

b

I

b

x

b a a b

g(f

ω

(α)) = x

(36)

A

d

-dimensional inniteword over an alphabet

Σ

is amap

x : N

d

_{→ Σ}

. We usenotation like

x

n

1 ,...,n

d

or

x(n

1 , . . . , n

d

)

to denote the valueof

x

at

(n

1 , . . . , n

d

)

.

If

w

1 , . . . , w

d

are nitewords over thealphabet

Σ

,

(w

1 , . . . , w

d

)

#

:= (#

m−|w

1 |

w

1 , . . . , #

m−|w

d

|

w

d

)

where

m = max{|w

1 |, . . . , |w

d

|}

.

Example

(37)

A

d

-dimensional inniteword over an alphabet

Γ

is

b

-automatic if there existsa DFAO

A = (Q, q

0 , (Σ

b

)

d

, δ, Γ, τ )

s.t. forall

n

1 , . . . , n

d

≥ 0

,

τ

δ

q

0 , (rep

b

(n

1 ), . . . , rep

b

(n

d

))

0 = x

n

1 ,...,n

d

.

Theorem(Salon 1987)

Let

b ≥ 2

and

d ≥ 1

. A

d

-dimensional inniteword is

b

-automatic

iitis the imageunderacodingof axedpoint ofa

b

-uniform

(38)

Let

d ≥ 1

. The

d

-dimensional innite wordis

S

-automatic forsome ANS

S = (L, Σ, <)

where

ε ∈ L

iitis the imageunderacoding of ashape-symmetric innite

d

-dimensionalword.

(39)

µ(a) = µ(f ) =

a

b

c

d

; µ(

b

) =

e

c

; µ(

c

) =

e

b ; µ(d) = f

µ(e) =

e

b

g

d

; µ(

g

) =

h

b ; µ(

h

) =

h

b

c

d

.

µ

ω

(a) =

a

b

e

e b

e

b

e

· · ·

c

d c g d g d c

e

b f

e b h b f

e

b

e a b

e

b

e

g

d c

c d g d c

e

b

e

e b a b

e

g

d c g d c d c

h

b f

e b

e

b f

. . . . . .

(40)

Consider the morphism

µ

1

dened by

a 7→ ab ; b 7→ e ; e 7→ eb.

We have

µ

ω

1 (a) = abeebebeebeebebeebebeebeeb · · ·

. One canonicallyassociatesthe DFA

A

µ

1 ,a

a

_b

e

0

1

0

1

0 L

µ

1 ,a

= {ε, 1, 10, 100, 101, 1000, 1001, 1010, 10000, . . .}

(41)

I

If

S

and

T

aretwo ANS,

(S, T )

-automaticwords are

bidimensional innitewords

(x

m,n

)

m,n≥0

forwhichthere exists aDFAO

A = (Q, (Σ ∪ {#})

d

_{, δ, q}

0 , Γ, τ )

s.t.

∀m, n ∈ N

,

x

m,n

= τ (δ(q

0 , (rep

S

(m), rep

T

(n))

#

)).

Can these

(S, T )

-automatic words becharacterizedby iteratingmorphisms?

(42)

b

-kernel

An innite word

(x

n

)

n≥0

is

b

-automatic iits

b

-kernel

{(x

b

e

_n+r

)

_n≥0

: e, r ∈ N, r < b

e

}

is nite. The

b

-kernelcan be rewritten

{(x

_b

|w|

_n+val

_b

_(w)

)

n≥0

: w ∈ Σ

∗

b

}.

8 4 2 1

ε 0

1 1 0

6 1 1 0 0 12

1 1

1 1 1 7

1 1 0 1 13

1 0

2 1 0 0 0 8

1 1

1 0

14 1 1 3

1 0 0 1 9

1 1 1 1 15

1 0 0 4

1 0

10 1 0 0 0 0 16

1 0 1 5

1 0 1 1 11

1 0 0 0 1 17

NB:

b

|w|

_{n + val}

b

(w)

isthe base-

b

valueofthe

(n + 1)

-thwordin

L

b

having

w

as asux.

(43)

The

S

-kernel of

(x

n

)

n≥0

is

{(x

f

w

(n)

)

n≥0

: w ∈ Σ

∗

_}

where

f

w

(n)

isthe

S

-valueofthe

(n + 1)

-thword in

L

having

w

as a sux.

Theorem(Rigo-Maes 2002)

An innite wordis

S

-automaticiits

S

-kernel isnite.

I

Doesa similarcharacterization holdin the multidimensional setting?

(44)

(45)

It is anexercise toshow thatallultimately periodicset are

b

-recognizableforall

b ≥ 2

.

Theorem(Cobham 1969)

Let

k, ` ≥ 2

be twomultiplicativelyindependent integers. A subsetof

N

is both

k

-recognizable and

`

-recognizableiitis ultimately periodic.

Two numbers

k

and

`

aremultiplicativelyindependent if

k

m

_{= `}

n

and

m, n ∈ N

implies

m = n = 0

.

Corollary

A subsetof

N

is

b

-recognizableforall

b ≥ 2

iitis ultimately periodic.

(46)

Theorem(Lecomte-Rigo2001, Krieger et al. 2009)

Ultimately periodicsetsare

S

-recognizable forallANS

S

.

Corollary

A subsetof

N

is

S

-recognizableforallANS

S

iitisultimately periodic.

Theorem(Krieger et al. 2009,Angrand-Sakarovitch 2010)

Let

m, r ∈ N

with

m ≥ 2

and

0 ≤ r ≤ m − 1

and let

S = (L, Σ, <)

be an ANS.If

L

is acceptedby a

n

-stateDFA,then the minimalDFAof

rep

S

(mN + r)

hasat most

nm

n

(47)

A subset

X

of

N

d

is

b

-recognizable ifthe language

(rep

b

(X))

#

over

({0, 1, . . . , b − 1} ∪ {#})

d

isregular, where

rep

_b

(X) = {(rep

_b

(n

1 ), . . . , rep

b

(n

d

)) : (n

1 , . . . , n

d

) ∈ X}.

Theorem(CobhamSemenov, Semenov 1977)

Let

k, ` ≥ 2

be twomultiplicativelyindependent integers. Asubset of

N

d

isboth

k

-recognizableand

`

-recognizableiitis semi-linear. A set

X ⊆ N

d

islinearifthere exist

v

0 , v

1 , · · · , v

t

∈ N

d

suchthat

X = v

0 + N v

1 + N v

2 + · · · + N v

t

. Aset

X ⊆ N

d

issemi-linearif it isanite unionof linearsets.

(48)

9

b

8

b

7

b

6

b

5

b

4

b

3

b

2

b

1

b

0

b

0 1 2 3 4 5 6 7 8 9

{(n, m) : n, m ∈ N

and

n ≥ m} = N(1, 0) + N(1, 1)

(49)

Corollary

A subsetof

N

d

is

b

-recognizable forall

b ≥ 2

iitis semi-linear.

In the one-dimensionalcase,we havethe following equivalences: semi-linear

⇔

ultimatelyperiodic

⇔ 1

-recognizable.

(50)

One mightthereforeexpect that thesemi-linearsetsare

recognizable inallANS.However,this failsto be the case,as the following exampleshows.

Example

The semi-linearset

X = {n(1, 2) : n ∈ N} = {(n, 2n) | n ∈ N}

is

not

1

-recognizable. Consider the language

{(a

n

_#

n

_{, a}

2n

_{) | n ∈ N}}

, consisting of theunary representations ofthe elementsof

X

. Use thepumping lemmato showthat thisis notaccepted by a nite automaton.

(51)

Let

S = (L, Σ, <)

bean ANS. A subset

X

of

N

d

is

S

-recognizable ifthe language

(rep

S

(X))

#

over

(Σ ∪ {#}))

d

is regular,where

rep

_S

(X) = {(rep

_S

(n

1 ), . . . , rep

S

(n

d

)) : (n

1 , . . . , n

d

) ∈ X}.

It is

1

-recognizable ifitis

S

-automatic fortheANS

S

built on

a

∗

(52)

Multidimensional

1

-recognizable sets

Theorem(C-Lacroix-Rampersad2012)

A subsetof

N

d

is

S

-recognizableforallANS

S

iitis

1

-recognizable.

Theorem(C-Lacroix-Rampersad2012)

The multidimensional

1

-recognizablesetsarethe niteunions of setsof the form

(a

1 + b

1 N

)v

1 + · · · + (a

t

+ b

t

N

)v

t

,

where

I

Supp(v

1 ) ⊇ Supp(v

2 ) ⊇ · · · ⊇ Supp(v

t

)

I

(53)

Another well-studiedsubclass ofthe classof semi-linearsetsisthe classof recognizablesets.

A subset

X

of

N

d

isrecognizableif theright congruence

∼

X

has nite index (

x ∼

X

y

if

∀z ∈ N

d

_{(x + z ∈ X ⇔ y + z ∈ X))}

. When

d = 1

,we haveagain thefollowing equivalences:

recognizable

⇔

ultimately periodic

⇔ 1

-recognizable.

(54)

Theorem(Mezei)

The recognizablesubsetsof

N

2

arepreciselyniteunions of setsof the form

Y × Z

,where

Y

and

Z

areultimatelyperiodic subsets of

N.

In particular, the diagonalset

D = {(n, n) | n ∈ N}

isnot recognizable.

However, theset

D

isclearlya

1

-recognizable subset of

N

2

. So weseethat for

d > 1

,the class of

1

-recognizable sets

correspondsneither to theclass of semi-linearsets,norto the class of recognizablesets.