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Dal non-determinismo al determinismo

( nei linguaggi 2dim ): alcune riflessioni

Marcella Anselmo, Dora Giammarresi, Maria Madonia, Antonio Restivo

Riunione Prin. Varese, luglio 2006

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finite alphabet ** all 2dim rectangular words (pictures) over • L ** 2dim language• p L has size (m,n)

• Column concatenation

• Row concatenation

• Column/Row star

2dim Languages

p qp q =

p q =p

q

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Local 2dim Languages

L is local if there exists a finite set of tiles that contains all allowed subpictures of size (2,2,), i.e.

p L if and only if any 22 sub-picture of is in p

• tile: a square picture of size (2,2)

• bordered picture p:

p =

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Ld = the set of square pictures with symbol “1” in all main diagonal positions and symbol “0” in the other positions

(Usual) Example of local language

1001

10

0010

0000

01

1

10

00

00

01

0

00

00

0

1

=0100

100

010

001p =

#####

#100#

#010#

#001#

#####

p =

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L is recognizable by tiling system if L= (L’) where L’ is a local language and is a mapping from the alphabet of L’ to the alphabet of L

Recognizable 2dim Languages

REC is the family of two-dimensional languages recognizable by tiling system

REC is closed almost under all operations but it is not closed under complement

(, , , ) , where L’=L(), is called tiling system

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(Usual) Example

LSq = all squares over = {a}.

LSq is recognizable by tiling system.

Set L’=Ld and (1)= (0)= a

100

010

001 Ld

aaa

aaa

aaa

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# ###

(Usual) Example

LSq = squares over {a}. Use L’=Ld (1)= (0)= a

aaa

aaa

aaa

1001

10

0010

0000

01

1

10

00

00

01

0

00

00

0

1

=0100

#####

##

##

##

#

p =

1 0 0

0 0

1

100

“Computing” by a tiling system(from a tiling system to an automaton)

First, decide a scanning strategy!

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“Computing” by a tiling system(from a tiling system to an automaton)

Remark :Tiling system = “undirectional” transitions

Definition: A 2dim finite automaton is

Tiling system + scanning procedure

Local picture is the run of the automaton.

Remark :

All 2dim finite automata “correspond” to family REC (i.e. scanning procedure does not matter!)

Ex: 2OTA (2dim on-line tesselation automata)

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Scanning strategies (I)

# # # # # # # #

# #

# #

# #

# #

# #

# #

# # # # # # # #

1 2

3

4

5

6

7

8

9

10

34

35 36

Diagonal (“2OTA”)

# # # # # # # #

# #

# #

# #

# #

# #

# #

# # # # # # # #

By column

1

2

3

4

5

6

7

8

9

10

11

12

31

32

33

34

35

36

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Scanning strategies (II)

# # # # # # # #

# #

# #

# #

# #

# #

# #

# # # # # # # #

Snake-like

1 2 3 4 5 6

789101112

1314

343536

Free

# # # # # # # #

# #

# #

# #

# #

# #

# #

# # # # # # # #

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3

4

5

6

7

8

9

36

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A remark about REC

• A tiling system (= local language + projection) “generalizes” to 2 dim a non-deterministic finite automaton.• Family REC is not closed under complement

Definition of REC is intrinsically non-deterministic and it is not possible to eliminate non-determinism without getting a smaller class!

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From non-determinism to determinism.. ..

• Non-determinism • Possible accepting computations: “several” • Possible backtracking steps at each step of

computation: linear in the size of input

[if pictures: =O(mn)]

• Determinism • Possible accepting computations: 1 • Possible backtracking steps at each step of

computation: 0

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Remark on 1DIM case :

In string languages 2 definitions possible:

- Determinism from left to right

- Co-determinism from right to left

Correspond to same class!....choose one definition…

??

??

Remark: Languages recognized by automata that are both deterministic and co-deterministic are smaller class!

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A tiling system is Top-Left-deterministic if a,b,c and s unique tile such that (s)=d.a b

c d

(Analogously TR-,BL-,BR-deterministic tiling system)

??

There is an unique way to fill this position with a symbol of

L is deterministic if it has a TL- or TR- or BL- or BR-

deterministic tiling system

Deterministic Recognizable Languages (DREC)

Classical definition (only a bit extended)

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• Lcol-1n REC

aababaaa

bbbbabbb

aaaaaaaa

bbabbbbb

p’=

Lcol-1n = {p | first col = last col }{a,b}**

New Example

Local alphabet: = {xy}

Projection “erase” subscripts: (xy) = x

• Lcol-1n DREC

Lcol-1i = {p | 1<in , first col = col i}DREC

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From non-determinism to determinism:what can we define in beetween?

• Non-determinism

• Possible accepting computations: “several”

• Possible backtracking steps at each step of computation: linear in the size of p (mn)

• Determinism

• Possible accepting computations: 1

• Possible backtracking steps at each step of computation: 0

• Unambiguity

one

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Unambiguous Recognizable Languages (UREC)

Def [GR92] A tiling system (, , , ) is unambiguous for L ** if the projection π is injective on L() (i.e. for any pL there is a unique p’ L’ such that (p’)=p).

UREC: all unambiguous recognizable 2dim languages.

L ** is unambiguous if it admits an unambiguous tiling system.

• UREC REC • Generalization in 2dims of unambiguous automata

for strings

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UREC and REC

Lcol-ij REC Lcol-ij UREC

Lcol-ij = ** Lcol-1n ** and

REC is closed with respect to

• UREC REC

Lcol-ij =

i j

i,j:

col i = col j

Necess. Cond. for UREC

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Properties of UREC

Proposition UREC is not closed under row/column concatenation/closure.

Proposition UREC is closed under intersection and rotation operations.

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From non-determinism to determinism:what can we define in beetween? (2)

• Non-determinism

• Possible accepting computations: “several”

• Possible backtracking steps at each step of computation: linear in the size of p (mn)

• Determinism

• Possible accepting computations: 1

• Possible backtracking steps at each step of computation: 0

one dimension of p (m or n)

• “line”-unambiguity

one

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A tiling system is Left-Right Column-Unambiguous if, after having computed the local symbols in an entire column, the local symbols on the next column are univocally determined.

??

??

??

??

L is Col-UREC if L has a tiling system that is LR- or RL- column unambiguous.

Column-Unambiguos Languages (Col-UREC)

Remark: Backtracking at each step of possibly O(m) steps.

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A tiling system is Top-Down Row-Unambiguous if, after having computed the local symbols in an entire row of a picture, the local symbols on the next row are univocally determined.

?? ???? ??

L is Row-UREC if L has a tiling system that is TD- or DT- column unambiguous.

Row-Unambiguos Languages (Row-UREC)

Remark: Backtracking at each step of possibly O(n) steps.

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(A new) Example

= {a, b}

LSq-cent-a = odd-side squares with a in

center positionabbabbabaaaababb

b b

b

ba

aa aa

a0b0b0a2

b1b0a2b0

a0a1a0a0

b2a0b1b0

b1 b0

b0

b1

a0

a0

a0 a2a0 LSq-cent-a Col-UREC, Row-UREC

LSq-cent-a DRECBy an “old” proof

by Inoue et al.

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Col-UREC and UREC

Lcol-1ij n UREC Lcol-1ijn Col-UREC

Lcol-1ijn = Lcol-1j Lcol-in and

UREC is closed with respect to

• Col-UREC UREC

i,j:

col 1 = col j

col i = col nLcol-1ij n=

1 i j n

Necess. Cond. for Col-UREC

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A necessary condition for unambiguity

Theorem Let L **. There is a k such that, for all m,

1. If L Col-UREC then Row(ML(m)) km

2. If L UREC then RankQ(ML(m)) km

L(m) L is the subset of all pictures with m rows. It can be viewed as a string language over the columns alphabet.

S*, regular string language. MS is the boolean matrix MS=|a| *, * where a= 1 iff L. The number of different rows , Row(MS ), is finite.

Idea of ProofUse Matz’s Theorem and Hromkovic et al. Theorem

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From 2dim to 1dim

Theorem [Matz 97] Let L **. If L REC, then there is a k such that, for all m, there is a finite string automaton Am with km states for L(m).

Fact

If L UREC, then Am is an unambiguous automaton with km states for L(m).

If L Col-UREC, then Am is a deterministic automaton with km states for L(m).

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Theorem of Hromkovic et al.

Theorem (Hromkovic et al.) For every regular string language S*,

d(S) = Row(MS)

uns(S) RankQ(MS).

d(S) the size of the minimal deterministic automaton accepting S

uns(S) the size of a minimal unambiguous non-deterministic automaton accepting S.

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The following inclusions are all strict:

DREC Col-UREC UREC REC

Collecting all classes…

a

1 i j n

i j

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A separation result

Theorem

Whatever we choose a definition of deterministic 2dim finite automaton, the family of corresponding languages is strictly included in UREC.

Proof: By previous strict inclusions results (Col-UREC UREC )

Det-REC is strictly included in UREC , for any definition of Det-REC we choose.

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An undecidability result for UREC

Theorem

Given a tiling system (, , , ) for L **, it is undecidable whether it is unambiguous.

Proof: By reduction from the undecidable 2dimensional Unique Decipherability Problem.

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A decidability result for Col-UREC

(Row-UREC)Theorem Given a tiling system T = ( , , , ) for

L **, it is decidable whether it is col-unambiguous.

Proof: Let M=Card {(,) : , }.

T col-unambiguous

No pair of pictures

sp with • p, s, t n,1

• s t (s) = (t)• Any 22 sub-picture of p s,

p t in • n M

tp

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A tiling system is Top-Left Diagonal-Unambiguous if, after having computed the local symbols in an entire diagonal of a picture, the local symbols on the next diagonal are univocally determined.

??

??

??

??

L is Diag-UREC if L has a tiling system that is TL-, TD-, BL- or BR- diagonal unambiguous.

REMARK: Diag-Unambiguos Languages

Remark: NO backtracking at each step

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Conjecture: If L REC\UREC then L REC

• Is UREC largest subset in REC closed under complement?

• Is UREC (Col-UREC) closed under complement?

Open Problems

• Is L(4NFA) UREC?

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Conclusioni alle riflessioni…

Tiling systems are a “compact” way to represent classes of finite state automata on 2 dims.

Unambiguos languages are a strict intermediate class between non-deterministic and deterministic families.

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riflettere a mente fresca…