Syntactical complexity of periodic sets

ÉmilieCharlier UniversitélibredeBruxelles

1stJointConferenceoftheBelgian,RoyalSpanishandLuxembourg MathematicalSocieties,June2012,Liège

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

. . .

The set

2N

ofeven integersis

F

-recognizable or

F

-automatic,i.e., the language

rep

F

(2N) = {ε, 10, 101, 1001, 10000, . . .}

isaccepted by somenite automaton.

Remark (in terms of the Chomsky hierarchy)

With respect tothe Zeckendorfsystem,any

F

-recognizable setcan be considered asaparticularlysimple set ofintegers.

0

1

0

I

F

_n+2

= F

_n+1

+ F

_n

I

F

₀

= 1

,

F

1

= 2

I

A

_F

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

`

U

-recognizability of arithmetic progressions

Proposition

Let

U = (U

i

)

i≥0

be anumeration systemand let

m, r ∈ N

.

If

N

is

U

-recognizable,then

m N +r

is

U

-recognizable and,given a DFAaccepting

rep

U

(N)

,aDFA accepting

rep

U

(m N +r)

can be obtained eectively.

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

.

The conditionis notsucient:

Example (

U

i

= (i + 1)

2

for all

i

∈ N

)

It is linear:

U

i+3

= 3U

i+2

− 3U

i+1

+ U

i

forall

i ∈ N

,but:

rep

_U

(N) ∩ 10

∗

10

∗

= {10

a

₁₀

b

_{: U}

a+b+1

+ U

b

< U

a+b+2

}

= {10

a

10

b

: b

2

< 2a + 4}

Thus,

rep

What is the best automaton we can get?

0

1

2

3

0

1

0

1

0

1

0

1

0

1

3

0

1

₀

1

0

1

DFAsacceptingthebinaryrepresentationsof

4N + 3

.

Question

The generalalgorithmdoesn't provide aminimalautomaton. What is the statecomplexity of

rep

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

Let

m, r ∈ N

with

m ≥ 2

and

r < m

. If

rep

U

(N)

isaccepted by a

n

-stateDFA, thenthe minimal automaton of

rep

U

(

m

N

+ r)

has atmost

n

m

n

states.

NB: Thisresultremainstrue forthe larger classof abstract numeration systems.

Theorem(Alexeev 2004)

Let

b, m ≥ 2

. Let

N, M

besuch that

b

N

_{< m ≤ b}

N +1

and

(m, 1) < (m, b) < · · · < (m, b

M

_{) = (m, b}

M +1

₎

.

The minimalautomaton recognizing

m

N

inbase

b

hasexactly

m

(

m

, b

N +1

₎

+

inf

{N,M −1}

X

t=0

b

t

(

m

, b

t

₎

states.

In particular, if

m

and

b

are coprime,then thisnumberisjust

m

. Further, if

m = b

n

Given any niteautomaton recognizingaset

X

of integerswritten in base

b

,itis decidablewhether

X

isultimatelyperiodic.

I

J.Honkala,Adecisionmethodfortherecognizabilityofsetsdenedbynumber systems,Theor. Inform. Appl. 20(1986).

I

A.Muchnick,ThedenablecriterionfordenabilityinPresburgerarithmetic anditsapplications,TCS290(2003).

I

J.Leroux,APolynomialTimePresburgerCriterionandSynthesisforNumber DecisionDiagrams,LICS2005(2005).

I

J.-P.Allouche,N.Rampersad,J.Shallit,Periodicity,repetitions,andorbitsof anautomaticsequence,TCS410(2009).

I

J.Bell,ÉC,A.Fraenkel,M.Rigo,Adecisionproblemforultimatelyperiodic setsinnon-standardnumerationsystems,IJAC19(2009).

I

F.Durand,DecidabilityoftheHD0Lultimateperiodicityproblem,arXiv(2011).

I

I.Mitrofanov,AproofforthedecidabilityofHD0Lultimateperiodicity,arXiv (2011).

Consider alinearnumerationsystem

U

such that

N

is

U

-recognizable.

How many states doesthetrim minimalautomaton

A

U,m

recognizing

m N

contain?

1. Give upper/lower bounds?

2. Study specialcases, e.g.,Zeckendorfnumeration system.

3. Get informationon the trim minimalautomaton

AU

recognizing

N

.

Theorem(C-Rampersad-Rigo-Waxweiler2011)

Let

U

be any numerationsystem (notnecessarily linear). The numberof statesof

A

U,

m

is atleast

| rep

U

(

m

)|

.

I

Let

U = (U

n

)

n≥0

bea linearnumeration system.

I

Let

k = kU,m

be the lengthofthe shortest linearrecurrence relationsatised by

(U

i

mod m)

i≥0

.

I

For

t ≥ 1

dene

H

t

:=













U

0

U

1

· · ·

U

t−1

U

1

U

2

· · ·

U

t

. . . . . . . . . . . .

U

t−1

U

t

· · ·

U

2t−2













.

I

For

m ≥ 2

,

kU,m

isalso the largest

t

suchthat

det H

t

6≡ 0

I

Let

SU,m

denote thenumberof

k

-tuples

b

in

{0, . . . , m − 1}

k

such thatthe system

H

k

x

≡ b

(mod m)

has atleast onesolution

x

= (x

1

, . . . , x

k

)

.

I

S

_U,m

≤ m

k

Calculating

S

U,m

I

U

_n+2

= 2U

_n+1

+ U

_n

,

(U

0

, U

1

) = (1, 3)

I

(U

_n

)

n≥0

= 1, 3, 7, 17, 41, 99, 239, . . .

I

Consider the system

(

1 x

1

+ 3 x

2

≡ b

1

(mod 4)

3 x

1

+ 7 x

2

≡ b

2

(mod 4)

I

2x

₁

≡ b

₂

− b

₁

(mod 4)

I

For each valueof

b

1

there areatmost

2

values for

b

2

.

I

Theorem

Let

m ≥ 2

bean integer. Let

U = (U

n

)

n≥0

be alinearnumeration system suchthat

(a)

N

is

U

-recognizable and

A

U

satises (H.1)and (H.2),

(b)

(U

n

mod m)

n≥0

ispurelyperiodic.

The numberof statesof

A

U,

m

fromwhichinnitelymany words areaccepted is

|C

U

| S

U,

m

.

(H.1)

A

U

has asinglestronglyconnected component

CU

.

(H.2) For allstates

p, q

in

C

U

with

p 6= q

,thereexists aword

x

pq

such that

δ

U

(p, x

pq

) ∈ C

U

and

δ

U

(q, x

pq

) 6∈ C

U

,orvice-versa.

Corollary

If

U

satisesthe conditions ofthe previoustheorem and

A

U

is strongly connected,then the numberof statesof

A

U,

m

is

Result for the

`

-bonacci system

0

1

1

1

0

0

0

Corollary

For

U

the

`

-bonacci system,the numberof states of

A

U,

m

is

`

m

`

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

. . .

I

Analyze the structure of

A

U

forsystemswith no dominant root.

I

Remove the assumptionthat

(U

n

mod m)

n≥0

ispurely periodic in the statecomplexity result.

I

Let

N

U

(m) ∈ {1, . . . , m}

denote 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, . . .)

and

N

F

(4) = 4

.

(F

i

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

and

N

F

(11) = 7

.

Theorem(C-Rigo 2008)

Let

U = (U

i

)

i≥0

be anumeration systemsatisfying

lim

i→+∞

U

i+1

− U

i

= +∞

.

If

X ⊆ N

is anultimately periodic

U

-recognizable set ofperiod

p

, then any DFA accepting

rep

U

(X)

has atleast

N

U

(

p

)

states.

I

If

N

U

(m) → +∞

as

m → +∞

,thenwe obtain adecision procedure tothe periodicityproblem.

I

If

U

isaLNS satisfying

U

i+k

= a

1

U

i+k−1

+ · · · + a

k

U

i

, i ≥ 0,

with

a

k

= ±1,

then

lim

m→+∞

N

U

(m) = +∞

.

I

WorksfortheZeckendorf system.

I

Not trueforinteger baseb:

N (b

n

_{) = 1}

I

Theformulaforthe state complexity of

m N

forthe Zeckendorf systemismuch simplerthan the formulafor integer base

b

systems.

I

Inthis point ofview, statecomplexityis not completely satisfying.

I

Hope: Findacomplexity that wouldhandle allthesesystems in akind ofuniformway.

I

Let

L

be alanguage over the nitealphabet

Σ

.

I

Myhill-Nerode equivalencerelationfor

L

:

u ∼

L

v

means that forall

y ∈ Σ

∗

,

uy ∈ L ⇔ vy ∈ L

.

I

Leads tothe minimalautomaton of

L

:

|A

L

| = |Σ

∗

_/∼

L

|

is the state complexity of

L

.

I

Syntacticcongruence for

L

:

u ≡

L

v

means that forall

x, y ∈ Σ

∗

,

xuy ∈ L ⇔ xvy ∈ L

.

I

Leads tothe syntactic monoidof

L

:

|H

L

| = |Σ

∗

_/≡

L

|

isthe

syntactic complexity of

L

.

Theorem

A language

L

is regularifand only if

Σ

∗

_/≡

The syntacticcomplexity of

X ⊆ N

isthe syntacticcomplexityof the language

0

∗

_rep

U

(X)

.

Let

ord

m

(b) = min{j ∈ N

0

: b

j

_{≡ 1 (mod m)}}

.

Theorem(Rigo-Vandomme2011)

I

Let

m

, b ≥ 2

becoprimeintegers.

If

X ⊆ N

isperiodicof minimalperiod

m

,thenthe syntactic complexityof

X

is equal to

m

ord

m

(b)

.

I

Let

b ≥ 2

and

m

= b

n

with

n ≥ 1

.

(a) The syntacticcomplexityof

m

N

is equalto

2

n

+ 1

.

(b) If

X ⊆ N

isperiodicofminimalperiod

m

,thenthesyntactic complexityof

X

is

≥

n

+ 1

.

I

Let

b ≥ 2

and

m

= b

n

_q

with

n ≥ 1

and

(b, q) = 1

. Then the syntacticcomplexity of

m

N

is equalto

Theorem(Lacroix-Rampersad-Rigo-Vandomme,to appear)

Let

b ≥ 2

and

m

=

d

b

n

_q

with

n ≥ 1

and

(b, q) = 1

and where

n

and

q

arechosen to be maximal.

If

X ⊆ N

is periodic ofminimalperiod

m

,thenthe syntactic complexity of

X

is

≥ max



q

ord

q

(b),

γ + 1

q

ord

q

(b)



,

where

γ → +∞

as

n

or

d

→ +∞

.

Theorem

The syntacticcomplexity of

m N

is

4m

2

p

F

(m) + 2

where

p

F

(m)

isthe minimalperiod of

(F

i

mod m)

i≥0

.

Further work forsyntactic complexity:

I

Tryto estimatethe syntacticcomplexity ofperiodic setsfora larger classof numerationsystems.

Syntactic complexity seemsto allowusto handleinteger basesand the Zeckendorf systematonce.