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

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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. Overview.

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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

Overview

• 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

### Two-dimensional (2dim) Languages

Problem: generalizing formal language theory

from 1dim to 2dim

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 22) such that, for any p in L, any sub-picture 22of 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

0

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

REC family II

• 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

b

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

b

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

0

1

0

1

0

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

0

1

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.

Scanning strategies (I)

By column

Diagonal (“OTA”)

1

2

4

1

4

5

3

13

2

5

3

6

6

14

15

Scanning strategies (II)

Snake-like

Free

1

2

3

4

5

4

6

3

5

1

2

Scanning strategy

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)

By column scanning strategy

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

1

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)

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)

Tiling Automata (3)

• 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(TA) = REC = L(OTA)

• 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)

Sketch proof item 3)

• 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)

Conclusions

• 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.

b

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)

b

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???)

w with border

#

0

1

0

0

1

#

0

0

1

0

0

1

Θ =

0

1

#

#

### Local (string) languages…(TOGLIERE

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

b0

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