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Tiling Automata: a computational model for recognizable two-dimensional languages

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### Two-dimensional (2dim) Languages

### Local (string) languages…(TOGLIERE

Tiling Automata: a computational model for recognizable two-dimensional languages

Marcella Anselmo Dora Giammarresi Maria Madonia

Univ. of Salerno Univ. Roma Tor Vergata Univ. of Catania

ITALY

- Topic:recognizability of 2dim languages (picture languages) by finite devices
- Motivation:define a computational device for 2dim languages based on tiling systems
- Results:definition of tiling automata, comparison of deterministic and non-deterministic models, with OTA, 4NFA

Two-dimensional string (or picture)over a finite alphabet:

a

b

b

c

a

c

b

a

c

b

b

a

a

b

a

- finite alphabet
- ** all 2dim strings (pictures) over
- L **2dim language

Problem: generalizing formal language theory

from 1dim to 2dim

A unifying point of view (G, Restivo, 1992):

Recognizability by tiling system (= local language + projection) REC family

Several attempts since 60s:

- Automata (4NFA and OTA, AFA, … )
- Logics (monadic second-order, first-order, existential monadic second-order)
- Grammars (matrix, image, array, TRG,… grammars)
- Operations (column-, row- concatenation, stars, …)

Definition of different classes of picture languages

First model by Blum & Hewitt (1967)

- Generalization of classical 2-way automata:

They can move: Left, Right,

Up, Down

4-way automata (4NFA)

Transition function (p,a)= (q,d )d{,,,}

- The deterministic model is denoted by 4DFA

- L(4DFA)L(4NFA)

- L(4DFA), L(4NFA) not closed under concatenations and *

OTA: a restricted type of 2dim cellular automata

….computing by diagonal waves

q i-1,j-1

q i-1,j

q i,j-1

- L(DOTA)L(OTA)

/

On-line tesselation automata (OTA)

: Q Q Q 2Q

- DOTA if : Q Q Q Q

p =

p =

- L islocalif there exists a set of tiles (i. e. square pictures of size 22) such that, for any p in L, any sub-picture 22of is in

p

REC family I

- REC family is defined in terms of local languages

- It is necessary to identify the boundary of a picture p
- using a boundary symbol

0

0

0

1

0

0

0

1

0

1

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1

0

0

1

0

0

0

0

0

=

0

0

0

0

1

0

0

0

0

1

1

0

1

0

1

0

p =

#

#

#

#

#

1

0

0

#

1

0

0

#

0

1

0

#

0

1

0

#

p =

0

0

1

#

0

0

1

#

#

#

#

#

#

(Usual) Example of local language

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

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

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

Example: LSq = all squares over = {a}

is recognizable by tiling system.

Set L’=Ld and (1)= (0)= a

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

b0

a0

b1

b0

b0

a0

b2

a1

b1

b2

a1

b1

a1

b1

a0

a1

b2

b2

a1

a0

b1

#

b

b1

b2

a

b

a1

b0

#

Θ=

1

2

0

#

#

#

#

a0

b1

b0

a0

a

(a0)=(a1) =a;

(b0)= (b1) = (b2)=b;

w = b a b a a b a

w’= # b0 a1 b2 a0 a1 b2 a0#

1dim case: from an automaton to a tiling system

L= strings over ={a,b}starting with b and with evenoccurrences of a

b0

b0

b1

b0

b2

a0

a1

a0

b2

b1

a1

a1

a1

b2

b1

b1

b2

a1

a0

b1

a0

#

b

b1

b2

a

b

a1

#

b0

Θ=

1

2

0

#

#

#

#

a0

a0

b0

b1

a

#

#

1dim case: “Computing” by a tiling system

(from a tiling systemto an automaton)

b

w =

w’= # b0 a1 b2 a0 a1

b a b a a

a

a0

b2

Θ corresponds to undirected edges!

To use Θ for a computation we need to decide a scanning strategy + variable to keep current local symbol (state)

0

0

0

1

0

0

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0

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=

0

0

0

0

1

0

0

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0

1

1

0

1

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1

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p =

0

1

0

2dim case: “Computing” by a tiling system

(from a tiling systemto an automaton)

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

#

#

#

#

#

First, decide a scanning strategy!

Recall last computed local symbols!

1

a

a

0

a

#

#

0

a

a

a

#

#

0

a

a

a

#

#

#

#

#

#

#

“Computing” by a tiling system

(from a tiling system to an automaton)

Remark : Tiling system = “undirectional” transitions

For a 2dim finite tiling automaton we need

Tiling system+scanning strategy+data structure

Local picture is the run of the automaton.

We need a “good” one!

- Start in a corner
- Filling-all-the-picture property
- Computable next-position function
- Contiguity property

- Mono-directional (tl2br or tr2bl or br2tl or bl2tr)

Start in a corner

#

#

#

#

#

#

#

#

#

Filling-all-the-picture

1

4

#

#

2

5

Comp. next-position function:

(i, j) (i+1, j)

#

#

3

6

#

#

#

#

#

#

#

Contiguity property

Mono-directional (tl2br)

An example

2

3

- Supports updating operations
- extract 3 local symbols 1,2,3needed to compute
- the next local symbol (ex. )
- insert new local symbol

Data Structure

In 1dim automata the data structure saves the current state (local symbol) and updates it.

In 2dim

- Depends on the chosen scanning procedure

Definition: A tiling automaton (TA) of type tl2bris A=(T, S, D0, ) where:

- T = (, , , ) is a tiling system
- S is a tl2br-directed scanning strategy
- D0 initial content of data structure
- :

(1, 2, 3, a) = 4if Θ and (4)=a

1

2

3

4

Tiling Automata (1)

0

0

#

#

0

0

Example

Consider a tiling automatonA=(TSq, Sr, D0, ) where Tsq tiling system for LSq and Sr a scanning strategy that goes row by row (from the left to the right)

0

1

0

(*) = a

a

0

*

Similarly define tiling automata of types tr2bl, br2tl,bl2tr

L(TA-tl2br)= L(TA-tr2bl)= L(TA-br2tl)= L(TA-bl2tr)=

= L(TA) = REC

Acceptance defined as usual

L(TA- tl2br)

Tiling Automata (2)

- Use standard definitions from string case and define:
- Unambiguos Tiling Automata (UTA)
- Deterministic Tiling Automata (DTA)

L(DTA) =L(DTA-tl2br) L(DTA-tr2bl)

L(DTA-br2tl) L(DTA-bl2tr)

- L(DTA) L(DOTA)

Languages of Tiling Automata

Proposition: The following properties hold

- L(DTA) is incomparable with L(4DFA)

- L(UTA) L(DTA)L(4DFA)

- L(DTA) is incomparable with L(4DFA)

Remark : TA are conceptually different from 4NFA

in L(4DFA) but not in L(DTA)

in L(DTA) but not in L(4DFA)

(K. Inoue, A. Nakamura 77)

- Tiling Automata necessary:
- - to use tiling system as computational devices
- - to introduce a “more computational” notion of determinism

- Tiling Automata reduce to classical string automata in the
- case of one-row pictures

Proposition:Deterministic TA with

- next-position function in O(1) time,

- data structure occupies space O(m+n) and

supports the operations in time O(1)

parsing in time O(mn) and O(m+n) extra-space.

Complexity issue

Why Tiling Automata?

A Tiling System does not correspond to an effective procedure of recognition.

b0

b2

a1

b2

a1

b1

b0

a0

b0

a0

b1

b1

a0

b1

a1

b2

a1

a0

a1

b1

b2

#

b

b1

b2

a

b

a1

#

b0

Θ=

1

2

0

#

#

#

#

a0

b1

a0

b0

a

w = b a b a a b a

#

#

1dim case: “Computing” by a tiling system

(from a tiling systemto an automaton)

a0

Θ corresponds to undirected edges!

w’= # b0 a1 b2 a0 a1 b2 a0 #

To use Θ for a computation we need to decide a scanning strategy + variable to keep current local symbol (state)

b0

b0

a1

b2

a1

b0

b1

a0

a0

b1

b2

a1

b1

a1

b1

a1

b2

a0

a0

b1

b2

#

b

b1

b2

a

b

a1

#

b0

Θ=

1

2

0

#

#

#

#

a0

b1

a0

b0

a

w = b a b a a b a

w’= # b0 a1 b2 a0 a1 b2 a0#

1dim case: “Computing” by a tiling system

(from a tiling systemto an automaton)

Θ corresponds to undirected edges!

To use Θ for a computation we need to decide a scanning strategy + variable to keep current local symbol (state)

“Computing” by a tiling system

(from a tiling system to an automaton)

Remark : Tiling system = “undirectional” transitions

For a 2dim finite tiling automaton we need

Tiling system+scanning strategy+data structure

Local picture is the run of the automaton.

Remark :

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

BUT it is necessary to define determinism

(= backtracking 0, where???)

#

0

1

0

0

1

#

0

0

1

0

0

1

Θ =

…

0

1

#

#

string w over Γ= {0, 1}

w=

0

1

0

0

1

finite set of strings of length 2

over Γ #

allowed substrings

Definition: String language L is local if all substrings of length 2 are in a finite set Θ. (L=L(Θ) )

Sono equivalenti ai regular

b1

b0

a1

#

#

a1

a1

a1

b0

b0

b0

b1

Θ =

#

b1

b1

a1

b1

a0

a0

a0

a0

b0

a1

#

a0

b0

a0

#

#

1DIM case:

“Computing” by a tiling system

(from a tiling system to an automaton)

(Usual) Example

a

b

b

#

1

0

a

w= a b b a a b a

Θ corresponds to undirected edges!

w’= # a1 b1 b1 a0 a1 b1 a0#

To use Θ for a computation we need to decide a scanning procedure +variable to keep current local symbol (state)

Diagonal scanning strategy (“2OTA”)

Start in a corner

#

#

#

#

#

#

#

#

#

1

2

4

Filling-all-the-picture

#

#

5

3

13

Comp. next-position function:

(i, j) (i+1, j-1)

#

#

6

14

15

#

#

#

#

#

#

#

Contiguity property

Mono-directional (tl2br)

A second example

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