Decidability through first-order logic and regular sequences

sequen es

ÉmilieCharlier

DépartementdeMathématique,UniversitédeLiège

I. b-automati sequen esandb-re ognizablesets

II. Chara terizingb-re ognizablesetsvialogi

III. Appli ationstode idabilityquestionsforautomati sequen es

IV. Enumeration: ountingb-denablepropertiesof b-automati sequen esis b-regular

Let b

≥

2beaninteger(theintegerbase).

Anaturalnumbernis representedbythenite word

rep b

(

n

)

=

ℓ

· · ·

1 0 over thealphabetA

b

= {

0

,

1

, . . . ,

b

−

1

}

obtainedfromthegreedy algorithm: n

=

ℓ

X

i

=

0 i b i

.

Takeb

=

2and onsiderthefollowingDFAO:

0 1

0 0

1

1

Forea hn ,theDFAOreads rep 2

(

n

)

andoutputs0or1a ordingtothe laststate thatis rea hed.

WeobtaintheThue-Morsesequen e

Asequen ex

: N

d

→ N

issaidtobeb-automati ifthereexistsaDFAO withinputalphabetA

b

su hthatforea hn

∈ N

d

,x

(

n

)

isthesymbol outputted bytheDFAOafterreadingrep

b

(

n

)

.

Tworemarks:

◮

Ab-automati sequen e antakeonlynitelymanyvalues.

◮

We anworkin anydimensiond:

rep 2





5 3 10





=





101 11 1010





0

=





0101 0011 1010





=





0 0 1









1 0 0









0 1 1









1 1 0





.

AsetX

⊆ N

d

isb-re ognizableifthelanguage

rep b

(

X

)

= {

rep b

(

n

) :

n

∈

X

}

is regular.

Itisequivalenttosaythatits hara teristi sequen e

χ

X

: N

d

→ {

0

,

1

}

is b-automati : thereexistsa DFAOthatoninputrep

b

(

n

)

ouputs1if n

∈

X,andoutputs0otherwise.

The setofevilnumbers

{

0

,

3

,

5

,

6

,

9

,

10

,

12

,

15

,

17

,

18

,

20

,

23

, . . .}

,i.e.the naturalnumbershavinganevennumberof1inbase 2,is 2-re ognizable. Its hara teristi sequen eistheThue-Morsesequen e.

Semi-linearsetsof

N

d

areniteunionsofsetsoftheform

p 0

+

p 1

N

_{+ · · · +}

p

ℓ

N

where p 0

,

p 1

, . . . ,

p

ℓ

∈ N

d .

Theorem(Cobham 1969, Semenov 1977)

Let bandb

′

bemultipli ativelyindependentintegerbases. Ifasubsetof

N

d

issimultaneouslyb-re ognizableandb

′

-re ognizable, thenitissemi-linear.

There existseveralequivalentdenitionsofb-re ognizablesetsof integers using

◮

logi

◮

b-uniformmorphisms

◮

nitenessoftheb-kernel

◮

algebrai formalseries

◮

re ognizable/rationalformalseries

Theorem(Bü hi 1960, Bruyère 1985)

AsubsetX of

N

d

Let

S

bealogi alstru ture whosedomainisS. AsetX

⊆

S

d

is denablein

S

ifthereexistsarst-order formula

ϕ(

x 1

, . . . ,

x d

)

of

S

su hthat X

= {(

s 1

, . . . ,

s d

) ∈

S d

: S  ϕ(

s 1

, . . . ,

s d

)}.

Arst-orderformulais denedre ursively from

◮

variablesx 1

,

x 2

,

x 3

, . . .

des ribingelementsofthedomainS

◮

theequality

=

◮

therelationsandfun tionsgiven inthestru ture

S

◮

the onne tives

¬, ∨, ∧, =⇒ , ⇐⇒

◮

Presburger arithmeti

hN, +i

x

≤

y isdenable by

(∃

z

) (

x

+

z

=

y

)

. Nottruein

hZ, +i

.

x

=

0isdenable byx

+

x

=

x. OK in

hZ, +i

.

x

=

1isdenablebyx

6=

0

∧ ((∀

y

) (

y

=

0

∨

x

≤

y

))

. Nottruein

hZ, +i

. Indu tively,x

=

is denablefor every

∈ N

.

The setsa

N

+

baredenable: a

N

+

b

= {

x

: (∃

y

) (

x

=

a

·

y

+

b

)}

where a

·

y standsfory

+

y

+ · · ·

y (atimes).

Infa t,asubsetX

⊆ N

isdenable in

hN, +i

i itisaniteunionof arithmeti progressions,orequivalently,ultimatelyperiodi .

AsubsetX

⊆ N

d

AsetX

⊆ N

d

isb-denableifitis denableinthestru ture

hN, +,

V b

i

, where

◮

+(

x

,

y

,

z

)

is theternaryrelationdenedbyx

+

y

=

z,

◮

V

b

(

x

)

istheunaryfun tiondened asthelargestpowerofb dividing x ifx

≥

1andV

b

(

0

) =

1.

Forexample,thesetX

= {

x

∈ N :

x isapowerofb

}

is denableby V

b

(

x

) =

x.

It anbeshownthatthestru tures

hN, +,

V b

i

and

hN, +,

P b

i

arenot equivalent, whereP b

(

Theorem(Bü hi 1960, Bruyère 1985)

AsubsetX of

N

d

isb-re ognizablei itisb-denable. Moreover,both dire tionsareee tive.

Sket hoftheproof.

◮

From aDFAa eptingrep b

(

X

)

, onstru tarst-order formula

ϕ

of thestru ture

hN, +,

V

b

i

deningX, i.e.su hthat

X

= {(

n 1

, . . . ,

n d

) ∈ N

d

: ϕ(

n 1

, . . . ,

n d

)

is true

}.

◮

Conversely,given arst-order formula

ϕ

ofthestru ture

hN, +,

V b

i

deningX,builda DFAa eptingrep

b

(

X

)

.

The rstorder theoryof

hN, +,

V b

i

isde idable

Proof.

◮

Wehavetoshow that,given any losedrst-order formulaof

hN, +,

V b

i

,we ande idewhetheritistrueorfalsein

N

.

◮

Sin ethere isno onstantinthestru ture,a losedformula of

hN, +,

V b

i

isne essarilyoftheform

∃

x

ϕ(

x

)

or

∀

x

ϕ(

x

)

.

◮

The set

X

ϕ

= {

n

∈ N : hN, +,

V b

i  ϕ(

n

)}

isb-denable,soitisb-re ognizablebytheBü hi-Bruyèretheorem. Thismeans thatwe anee tively onstru ta DFAa epting rep

b

(

X

ϕ

)

.

◮

The losedformula

∃

x

ϕ(

x

)

istrueifrep b

(

X

ϕ

)

isnonempty,and falseotherwise.

◮

Astheemptinessofthelanguagea eptedbyaDFAisde idable, we ande ideif

∃

x

ϕ(

x

)

is true.

◮

The ase

∀

x

ϕ(

x

)

redu estothepreviousonesin e

∀

x

ϕ(

x

)

is logi allyequivalentto

¬∃

x

¬ϕ(

x

)

. We an onstru taDFA a eptingthebase-brepresentationsof

X

¬ϕ

= N \

X

ϕ

.

Ifwe anexpressapropertyP

(

n

)

usingquantiers,logi aloperations, addition,subtra tion, omparison,andelements ofsomeb-automati sequen es,then

∃

nP

(

n

), ∃

∞

In parti ular, what about the property x

(

i

) =

x

(

j

)

?

Ifx

: N

d

→ N

is ab-automati sequen ethen,foralllettersao urring in x,thesubsetsx

−

1

(

a

)

of

N

d

areb-re ognizable.

Hen e theyaredenablebysomerst-order formulae

ψ

a

of

hN, +,

V b

i

(byBü hi-Bruyère theorem):

ψ

a

(

n

)

istrueix

(

n

) =

a .

Therefore,we anexpressx

(

i

) =

x

(

j

)

bytherst-order formula

ϕ(

x 1

, . . . ,

x d

,

y 1

, . . . ,

y d

)

of

hN, +,

V b

i

:

ϕ(

i

,

j

) ≡

_

a

(ψ

a

(

i

) ∧ ψ

a

(

j

)).

Considerthepropertyofhavinganoverlap.

A(unidimensional)sequen ex hasanoverlapbeginningatpositioni i

(∃ℓ ≥

1

) (∀

j

≤ ℓ)

x

(

i

+

j

) =

x

(

i

+ ℓ +

j

)

.

Nowsupposethatx is b-automati .

GivenaDFAOM 1

generatingx,werst reateanNFAM 2

thatoninput

(

i

, ℓ)

a eptsif

(∃

j

≤ ℓ)

x

(

i

+

j

) 6=

x

(

i

+

j

+ ℓ)

.

Todothis,M 2

guessesthebase-brepresentationofj digit-by-digit, veries thatj

≤ ℓ

, omputes i

+

j andi

+

j

+ ℓ

onthey, anda eptsif x

(

i

+

j

) 6=

x

(

i

+

j

+ ℓ)

.

2 3

inverse thenal statusofea hstate. Thus, M 3

a eptsthosepairs

(

i

, ℓ)

su hthat

(∀

j

≤ ℓ)

x

(

i

+

j

) =

x

(

i

+

j

+ ℓ)

.

Nowwe reateanNFAM 4

thatoninput iguesses

ℓ

≥

1anda eptsi M

3

a epts

(

i

, ℓ)

.

Aswe ande ideifM 4

a eptsanything,wehaveobtained:

Proposition

◮

Itisde idablewhetherab-automati sequen ehask-powers (fora xedk).

◮

Itisde idablewhetherab-automati sequen eisultimately periodi .

◮

Giventwob-automati sequen esx andy,itis de idablewhether Fa

(

x

) ⊆

Fa

(

y

)

.

◮

The predi ate

∀

n

∃

p

≥

1

∀ℓ

x

(

n

) =

x

(

n

+ ℓ

p

)

is notarst orderformulain

hN, +,

V b

i

. Why? Isthisproperty b-denable? Whataboutthe ase wheretheperiods parerestri tedto powersofthebaseb?

Ifx is anarbitraryb-automati sequen e,thenthepredi ate

“

x

[

i

,

i

+

2n

−

1

]

is anabeliansquare

is notexpressibleinthelogi altheory

hN, +,

V b

i

Intheworst ase,wehaveatowerofexponentials: 2 2

·

·

·

2 n

where nisthenumberofstatesofthegivenDFAOandtheheightof the toweristhenumberofalternatingquantiersiftherst-order predi ate.

This pro edurewasimplementedbyMousavi,givingbirththeWalnut software.

Inpra ti e,Go ,Henshall,Mousavi,Shallit andotherswereabletorun their programsinordertoprove(and/orreprove)manyresultsabout b-automati sequen es.

Proposition

Let x

: N → N

beab-automati sequen eandlety

: N → N

bedened asy

(

i

) =

1ifx hasanoverlapatpositioni,andy

(

i

) =

0otherwise. Then y isb-automati .

Inthesamevein,we anprovethat ountingb-denablepropertiesof a b-automati sequen egiverisetoab-regularsequen e.

Let K bea ommutativesemiring. Asequen ex

: N

d

→

K is

(

K

,

b

)

-regularif thereexist

◮

anintegerm

≥

1

◮

ve tors

λ

∈

K 1

×

m and

γ

∈

K m

×

1

◮

amorphismofmonoids

µ: ((

A b

)

d

)

∗

_→

K m

×

m su hthat

∀

n

∈ N

d

,

x

(

n

) = λµ

rep b

(

n

)
γ.

The triple

(λ, µ, γ)

is alled alinearrepresentationofx andmisits

Theorem

Foranyb-denablesubsetX of

N

d

+

1 ,thesequen ea

: N

d

→ N ∪{∞}

dened by a

(

n 1

, . . . ,

n d

) =

Card

{

m

∈ N : (

n 1

, . . . ,

n d

,

m

) ∈

X

}

is

(N ∪{∞},

b

)

-regular. Ifmoreovera

(N

d

) ⊆ N

,thenais

(N,

b

)

-regular.

Corollary

Foranyb-automati sequen ex

: N → N

,thefa tor omplexityofx is

(N,

b

)

-regular.

◮

Letx

: N → N

beab-automati sequen e.

◮

Forall n

∈ N

,letp x

(

n

)

denotethenumberoflength-n fa torsof x.

◮

Then p x

(

n

) = #{

i

∈ N : ∀

j

<

i

,

x

[

j

,

j

+

n

−

1

] 6=

x

[

i

,

i

+

n

−

1

]}

.

◮

ConsiderX

= {(

i

,

n

) ∈ N

2

: ∀

j

<

i

,

x

[

j

,

j

+

n

−

1

] 6=

x

[

i

,

i

+

n

−

1

]}

.

◮

Sin ex is b-automati ,thesetX isb-denable.

◮

By hoi eofX,wehave p x

(

n

) = #{

i

∈ N : (

i

,

n

) ∈

X

}

.

◮

What aboutthe ountingthenumberofre tangularfa torsofsize

(

m

,

n

)

in abidimensionalb-automati sequen e? Isthe orresponding

Let F

= (

F

i

)

i

≥

0

= (

1

,

2

,

3

,

5

,

8

, . . .)

bethesequen eobtainedfromthe rules: F 0

=

1

,

F 1

=

2andF i

+

2

=

F i

+

1

+

F i fori

≥

0

.

Anaturalnumbernis representedbythenite word

rep F

(

n

)

=

ℓ

· · ·

1 0 over thealphabetA

F

= {

0

,

1

}

obtainedfromthegreedyalgorithm:

n

=

ℓ

X

i

=

0 i F i

.

The greedyalgorithmimposes,inadditiontohavinganonzeroleading digit

ℓ

,thatthevalidrepresentationsdo not ontaintwo onse utive digits1. The setofall possiblerepresentationsis

L

F

=

1

{

0

,

01

}

Let U

= (

U

i

)

i

≥

0

= (

1

,

2

,

3

,

5

,

8

, . . .)

bea basesequen e,thatis,an in reasingsequen eofpositive integerssatisfying:

U 0

=

1 and C U

=

sup i

≥

0 U i

+

1 U i

<

+∞.

Anaturalnumbernis representedbythenite word

rep U

(

n

)

=

ℓ

· · ·

1 0 over thealphabetA

U

= {

0

,

1

, . . . ,

⌈

C U

⌉ −

1

}

obtainedfromthegreedy algorithm: n

=

ℓ

X

i

=

0 i U i

.

Twoproblems:

◮

Ingeneral,

N

is notU-re ognizable.

◮

APisotnumberis analgebrai integer

>

1su hthatall ofitsGalois onjugateshave absolutevalue

<

1.

WorkingHypothesis(WH): U satiesalinearre urren ewhose hara teristi polynomialistheminimalpolynomialofaPisotnumber.

Forsu hsystems,Frougnyshowedthat

N

andtheadditionare re ognizablebyniteautomata.

U-denablesetsaresubsetsof

N

d

thataredenable inthelogi al stru ture

hN, +,

V

U

i

,where

◮

+(

x

,

y

,

z

)

is theternaryrelationdenedbyx

+

y

=

z,

◮

V

U

(

x

)

istheunaryfun tiondenedasthesmallestU i

orresponding

toanonzerodigitinrep

U

(

x

)

ifx

≥

1,andV U

(

0

) =

1.

Theorem(Bruyère-Hansel 1997)

Under WH,theU-re ognizablesetsofintegers oin idewiththe U-denablesetsofintegers.

The rstorder theoryof

hN, +,

V U

i

is de idable

This resultimpliesthatthereexistalgorithmstode ideU-denable propertiesforU-automati sequen es.

Asanappli ation,one anprove(andreprove,orverify) manyresults abouttheFibona i inniteword

f

=

01001010010010100101001001010010

· · ·

(whi his thexedpoint of0

7→

01

,

1

7→

0).

0 1

0 1

Manon Stipulanti)

What isaU-regularsequen e? Several hoi esofdenitionsarepossible.

InManonStipulanti's PhDthesis,itis provedthatsomesequen e S

ϕ

: N → N

isF-regularbyprovingthatthereexist

◮

anintegerm

≥

1

◮

ve tors

λ

∈

K 1

×

m and

γ

∈

K m

×

1

◮

amorphismofmonoids

µ: {

0

,

01

}

∗

_→

K m

×

m su hthat

∀

n

∈ N,

S

ϕ

(

n

) = λµ

0rep U

(

n

)
γ

where,inorder to ompute

µ

0rep

U

(

n

)

,itis understoodthat0rep U

(

Asequen ex

: N

d

→

K is

(

K

,

U

)

-regularifthereexist

◮

anintegerm

≥

1

◮

ve tors

λ

∈

K 1

×

m and

γ

∈

K m

×

1

◮

amorphismofmonoids

µ: ((

A b

)

d

)

∗

_→

K m

×

m su hthat C1

∀

n

∈ N

d

,

x

(

n

) = λµ

rep U

(

n

)
γ

C2

∀

w

∈ ((

A U

)

d

)

∗

_,

x

(

val U

(

w

)) = λµ(

w

)γ

.

Under WH,C1

⇐⇒

C2.

Conje ture (analogue of the useful result)

Under WH,foranyU-denablesubsetX of

N

d

+

1 ,thesequen e a

: N

d

→ N ∪{∞}

denedby a

(

n 1

, . . . ,

n d

) =

Card

{

m

∈ N : (

n 1

, . . . ,

n d

,

m

) ∈

X

}

is

(N ∪{∞},

U

)

-regular. Ifmoreovera

(N

d

) ⊆ N

, thenais

(N,

U

)

-regular.

Ingeneral realnumbersarerepresentedbyinnitewords.

Inthis ontext,we onsiderBü hiautomata. Aninnitewordisa epted when the orrespondingpathgoesinnitelymanytimesthroughan a epting state.

β

-re ognizable and

β

-denable subsets of

R

d

◮

Notionof

β

-re ognizabilityofsubsetsof

R

d , where

β >

1is areal base.

◮

For

β

=

1

+

√

5 2

,the

ω

-languageofthe(quasi-greedy)

β

-representationsof

[

0

,

1

]

isa eptedby

1

0 0

◮

First ordertheory

hR, +, ≤, Z

β

,

X

β

i

leadingtoanotionof

β

-denability.

◮

For

β

Pisot,

β

-re ognizability oin idewith

β

-denability.

◮

For

β

Pisot,thefollowingpropertiesof

β

-re ognizablesubsetsX of

R

d arede idable:

◮

X hasanonemptyinterior:

(∃

x

∈

X

) (∃ε >

0

) (∀

y

) (|

x

−

y

| < ε =⇒

y

∈

X

).

◮

X isopen:

(∀

x

∈

X

) (∃ε >

0

) (∀

y

) (|

x

−

y

| < ε =⇒

y

∈

X

).

◮

X is losed: OKas

R

d

\

X isb-re ognizable.

◮

...

◮

Bruyère-Hansel-Mi haux-Villemaire,Logi andp-re ognizablesetsof integers,Bull. Belg. Math. So . 1(1994)191238.

◮

Bruyère-Hansel,Bertrandnumerationsystemsandre ognizability,TCS 181(1997)1743.

◮

Charlier,First-OrderLogi andNumerationSystems. Sequen es,Groups, andNumberTheory,Chapter3,89141. TrendsinMathemati s, Birkhaüser/Springer,Cham,2018.

◮

Charlier-Leroy-Rigo,AnanalogueofCobham'stheoremforgraphdire ted iteratedfun tionsystems. Adv. inMath. 280(2015)8612.

◮

Charlier-Rampersad-Shallit,Enumerationandde idablepropertiesof automati sequen es. IJFCS23(5)(2012)10351066.

◮

Du-Mousavi-Rowland-S haeer-Shallit,De isionalgorithmsfor Fibona i-automati words,II:Relatedsequen esandavoidability,TCS 657(2017)146162.

◮