A decision problem for ultimate periodicity in non-standard numeration systems

non-standard numeration systems

ÉmilieCharlier UniversitélibredeBruxelles

Let's startwithclassical

k

-ary numeration system,

k ≥ 2

:

n =

`

X

i=0

d

i

k

i

, d

`

6= 0,

rep

k

(n) = d

`

· · · d

0

∈ {0, . . . , k − 1}

∗

A set

X ⊆ N

is

k

-recognizable,ifthe language

rep

_k

(X) = {rep

_k

(x) : x ∈ X}

Examples of

k

-recognizable sets

I

Inbase

2

,the evenintegers:

rep

2

(2N) = 1{0, 1}

∗

0 ∪ {ε}

I

Inbase

2

,the powers of

2

:

rep

2

({2

i

|i ∈ N}) = 10

∗

I

Inbase

2

,the Thue-Morseset:

{n ∈ N : rep

₂

(n)

contains aneven numbers of

1

s

}

I

Givena

k

-automaticsequence

(x

n

)

n≥0

over an alphabet

Σ

, then, forall

σ ∈ Σ

,the set

{i ∈ N : x

i

= σ}

is

k

-recognizable.

Divisibility criteria

If

X ⊆ N

is ultimatelyperiodic,then

X

is

k

-recognizable

∀k ≥ 2

.

X = (3N + 1) ∪ (2N + 2) ∪ {3},

Preperiod

= 4,

Period

= 6

χ

X

=



  

|





  







  



· · ·

Twointegers

k, ` ≥ 2

aremultiplicativelyindependentif

k

m

= `

n

⇒ m = n = 0.

Theorem(Cobham 1969)

Let

k, ` ≥ 2

be multiplicativelyindependent integers.

Theorem(J. Honkala 1986)

It is decidableif a

k

-recognizableset is ultimatelyperiodic. Sketchof Honkala's decisionprocedure:

I

Theinput isaDFA

A

X

accepting

rep

k

(X)

.

I

Thenumberof statesof

A

X

produces an upperbound on the possible(minimal) preperiodand period for

X

.

I

Consequently, therearenitelymany candidates tocheck.

I

For each pair

(a, p)

ofcandidates, produce aDFAforall possiblecorrespondingultimately periodicsetsand compareit with

A

X

.

I

Anumeration system(NS) isan increasingsequence of integers

U = (U

n

)

n≥0

such that

I

U

0

= 1

and

I

C

_U

:= sup

n

≥0

dU

n+1

/U

n

e < +∞

.

I

_U

islinearifitsatises alinearrecurrence relationover

Z

.

I

Let

n ∈ N

. Aword

w = w

`−1

· · · w

0

over

N

represents

n

if

`−1

X

i=0

w

i

U

i

= n.

I

I

Arepresentation

w = w

`−1

· · · w

0

ofan integeris greedyif

∀j,

j−1

X

i=0

w

i

U

i

< U

j

.

I

Inthat case,

w ∈ {0, 1, . . . , C

U

− 1}

∗

.

I

rep

U

(n)

isthe greedyrepresentationof

n

with

w

`−1

6= 0

.

I

X ⊆ N

is

U

-recognizable

∆

⇔ rep

_U

(X)

is acceptedby anite automaton.

I

_rep

It is basedon the sequence

F = (F

i

)

i≥0

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

dened by

F

0

= 1, F

1

= 2

and

F

i+2

= F

i+1

+ F

i

forall

i ≥ 0.

1 1 8 10000 15 100010 2 10 9 10001 16 100100 3 100 10 10010 17 100101 4 101 11 10100 18 101000 5 1000 12 10101 19 101001 6 1001 13 100000 20 101010 7 1010 14 100001 21 1000000 The pattern

11

is forbidden,

A

F

= {0, 1}

.

The

`

-bonacci numeration system

0

1

1

1

0

0

0

I

_U

n+`

= U

n+`−1

+ U

n+`−2

+ · · · + U

n

I

_U

_i

_{= 2}

i

,

i ∈ {0, . . . , ` − 1}

I

A

_U

accepts allwordsthat do notcontain

1

`

Proposition

Let

U = (U

i

)

i≥0

be aNS s.t.

N

is

U

-recognizable. Any ultimately periodic

X ⊆ N

is

U

-recognizableand aDFAaccepting

rep

U

(X)

can be obtainedeectively.

NB: If

N

is

U

-recognizable,then

U

islinear.

Periodicityproblem: Given

U

s.t.

N

is

U

-recognizable and a

U

-recognizable set

X ⊆ N

. Isitdecidable if

X

is ultimately periodic?

Pseudo-result

Let

X

be ultimatelyperiodic withperiod

pX

. Any DFAaccepting

rep

U

(X)

has atleast

f

(

p

X

)

states, where

f

isincreasing.

Pseudo-corollary

Let

X ⊆ N

be a

U

-recognizable set ofintegerss.t.

rep

U

(X)

is accepted by a

d

-stateDFA.

If

X

isultimatelyperiodicwith period

pX

,then

f

(

p

X

) ≤

d

with

(

d

xed

f

increasing.

lim

i→+∞

U

i+1

− U

i

= +∞.

(1) Mostsystems arebuilt on anexponential sequence

(U

i

)

i≥0

.

Lemma

Let

U = (U

i

)

i≥0

be aNS satisfying (1 ). If

w

isagreedy

U

-representation, thenso is

10

r

_w

Let

N

U

(m)

∈ {1, . . . , m}

denotes the number ofvaluesthat are taken innitelyoftenby the sequence

(U

i

mod m)

i≥0

.

Example (Zeckendorf system)

(F

i

mod 4) = (1, 2, 3, 1, 0, 1, 1, 2, 3, . . .)

,so

N

F

(4) = 4

.

(F

i

mod 11) = (1, 2, 3, 5, 8, 2, 10, 1, 0, 1, 1, 2, 3, . . .)

,so

N

F

(11) = 7

.

If

U = (U

i

)

i≥0

isalinearsystemoforder

k

,then,forall

m ≥ 2

,we have

k

pπ

U

(m) ≤ N

U

(m) ≤ π

U

(m),

Let

U

be aNS satisfying (1). If

X ⊆ N

isan ultimatelyperiodic

U

-recognizable set ofperiod

p

X

,then any DFAaccepting

rep

U

(X)

has atleast

N

U

(

pX

)

states.

Corollary

Let

U

be aNS satisfying (1). Assumethat

lim

m→+∞

N

U

(m) = +∞.

Then the period ofan ultimatelyperiodic set

X ⊆ N

s.t.

rep

U

(X)

is accepted by a

d

-stateDFA isbounded by the smallestinteger

M

s.t.

N

U

(m) >

d

forall

m ≥

M

,whichiseectively computable.

If

U = (U

i

)

i≥0

satisesa recurrencerelationof the kind

U

i+k

= a

1

U

i+k−1

+ · · · + a

k

U

i

,

(2)

with

a

k

= ±1

,then

lim

m→+∞

N

U

(m) = +∞

.

Proposition

Let

U = (U

i

)

i≥0

be anincreasingsequence satisfying (2 ). The following assertionsareequivalent:

I

lim

m→+∞

N

U

(m) = +∞

;

I

forallprime divisors

p

of

a

k

,

lim

v→+∞

N

U

(p

v

_{) = +∞}

Let

Q

U

(x)

denote thecharacteristicpolynomialof the shortest recurrence relationsatised by

U

;and let

P

U

(x)

= x

k

_Q

U

(

1

_x

)

, where

k = deg(Q

U

(x))

. Theorem(Bell-C-Fraenkel-Rigo 2009) We have

N

U

(p

v

_{) → +∞}

as

v → +∞

ifand only if

P

U

(x) = A(x)B(x)

with

A(x), B(x) ∈ Z[x]

suchthat:

I

_{B(x) ≡ 1 (mod p Z[x])}

;

I

_A(x)

Theorem(Muchnik 1991)

The ultimateperiodicityproblem isdecidable forallNS with a regularnumerationlanguage,provided thatadditionis recognizable.

Example (

U

i+4

= 3U

i+3

+ 2U

i+2

+ 3U

i

for all

i

≥ 0

,

(U

0

, U

1

, U

2

, U

3

) = (1, 2, 3, 4)

)

Additionisnotcomputable by aniteautomaton (duetoFrougny). Nevertheless,

N

U

(3

v

_{) → +∞}

as

v → +∞

because

P

U

= 1 − 3x − 2x

2

− 3x

4

Theorem(C-Rigo 2008)

Let

U

be aNS satisfying (1)and

X ⊆ N

bean ultimatelyperiodic

U

-recognizable set ofperiod

pX

. If

1

occursinnitelymany times in

(U

i

mod

p

X

)

i≥0

thenany DFA accepting

rep

U

(X)

hasatleast

pX

states.

Theorem(Zeckendorf system)

Let

X ⊆ N

be ultimately periodicwith period

p

X

(and preperiod

a

X

). Any DFAaccepting

rep

F

(X)

has atleast

pX

states.

I

_w

−1

_{L = {u : wu ∈ L} ↔}

statesof minimalautomatonof

L

I

(F

_i

mod p

_X

)

_i≥0

is purelyperiodic.

I

If

i, j ≥ a

X

and

i 6≡ j (mod p

X

)

then thereexists

t < p

X

s.t. either

i + t ∈ X

and

j + t 6∈ X

,or

i + t 6∈ X

and

j + t ∈ X

.

I

_∃n

₁

_{, . . . , n}

_pX

_{, ∀t}

,

0 ≤ t < p

X

,the words

10

n

pX

_{· · · 10}

n

2

₁₀

n

1

₀

| rep

F

(p

X

−1)|−| rep

F

(t)|

rep

F

(t)

I

Moreover

n

1

, . . . , n

p

X

canbe chosen s.t.

∀j

,

1 ≤ j ≤ p

X

,

val

F

(10

n

j

· · · 10

n

1

+| rep

F

(pX−1)|

) ≡

j

(mod p

X

)

and

val

F

(10

n

1

+| rep

F

(pX

−1)|

) ≥ a

_X

.

I

For

i, j ∈ {1, . . . , p

X

}

,

i 6= j

, thewords

10

ni

_{· · · 10}

n

1

and

10

nj

_{· · · 10}

n

1

willgenerate dierentstates in theminimalautomaton of

rep

F

(X)

. Thiscan beshown by concatenating someword of length

| rep

w

−1

_{L = {u : wu ∈ L} ↔}

states ofminimalautomaton of

L

X

= (11N + 3) ∪ {2}, a

X

= 3, p

X

= 11, | rep

F

(10)| = 5

Workingin

(F

i

mod 11)

i≥0

:

· · ·

21 1 01 10 28 53 21 10 110 28 53 21 1 0 00 0 00 00 00 1 1 0 00 0 00 00 01 0 00 0 00 00 00 2 1 0 00 0 00 00 10 1+2

∈ X

1 0 00 0 00 00 01 0 00 0 00 00 10 2+2

∈ X

/

⇒ (

10

5

)

−1

rep

_F

(X) 6= (10

9

10

5

)

−1

rep

_F

(X)

Forasequence

U = (U

i

)

i≥0

of integers,if

(U

i

mod m)

i≥0

is ultimately periodic,wedenote its(minimal) preperiodby

ι

U

(m)

.

Theorem(C-Rigo 2008)

Let

U = (U

i

)

i≥0

be alinearnumerationsystem. Let

X ⊆ N

be ultimately periodic withperiod

p

X

and preperiod

a

X

. Thenany DFAaccepting

rep

U

(X)

has atleast

| rep

U

(aX

− 1)|

−

ι

U

(p

X

)

states.

Theorem(C-Rigo 2008)

It is decidableif a

U

-recognizableset is ultimatelyperiodic for numeration systems

U = (U

i

)

i≥0

s.t.

I

N

is

U

-recognizable;

I

lim

i→+∞

U

i+1

− U

i

= +∞

;

I

lim

m→+∞

N

U

(m) = +∞

.

Remark

Whenever

gcd(a

1

, . . . , a

k

) = g ≥ 2

, we have

U

i

≡ 0 (mod g

n

₎

for all

n ≥ 1

and forall

i

large enough;hence

N

U

(m) 6→ +∞

.

Examples

I

Integerbases:

U

n+1

= k U

n

I

U

_n+2

= 2U

_n+1

+ 2U

_n

Learnmoreaboutlinearrecurrentsequences mod

m

...

I

H.T.Engstrom, On sequences denedby linearrecurrence relations,Trans. Amer. Math. Soc. 33(1931).

I

M.Ward, Thecharacteristicnumber ofasequence ofintegers satisfyinga linearrecursion relation, Trans. Amer. Math. Soc. 35 (1933).

I

M.Hall,An isomorphismbetweenlinearrecurringsequences and algebraic rings,Trans. Amer. Math. Soc. 44(1938).

I

Main ideas foranautomata-resolutionofthe periodicity problem:

I

If

X ⊆ N

isultimately periodic,thenthe state complexity of the associated minimalDFAshould grow withthe period and preperiod of

X

.

I

Analysethe inner structure ofDFAsaccepting the

F

-representations of even numbers

0

1

0

0

1

0

1

0

1

0

0

0

13 8 5 3 2 1

1 0

2

1 0 1

4

1 0 0 1

6

1 0 0 0 0

8

1 0 0 1 0 10

1 0 1 0 1 12

. .