Applied Mathematical Sciences, Vol. 8, 2014, no. 106, 5269 - 5275 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.46500

Infinite Triangular Arrays and Recognizability V. Devi Rajaselvi Researh Scholar, Sathyabama University, India T. Kalyani Department of Mathematics St. Joseph’s Institute of Technology, Chennai - 600 119, India D.G. Thomas Department of Mathematics Madras Christian College, Chennai - 600 059, India c 2014 V. Devi Rajaselvi, T. Kalyani and D. G. Thomas. This is an open Copyright access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract In this paper the concept of local languages is extended to infinite triangular arrays and ωω local languages are defined. ωω-recognizable languages of infinite triangular arrays accepted by two direction online tesselation automata are considered. Properties of these languages are studied.

Mathematics Subject Classification: 68Q Keywords: Local languages, recognizable languages, infinite triangular arrays, two direction online tessellation automata

5270

1

V. Devi Rajaselvi, T. Kalyani and D. G. Thomas

Introduction

Infinite triangular arrays or images are digitized images of symbols occupying a triangular grid of a plane. They can be thought of as extensions of infinite triangular tiles to three dimensions. The motivation for considering infinite triangular arrays lies in the fact that pictures of functions can be considered as infinite digitized images. An early work of infinite arrays is by Nakamura and Ono. Also infinite arrays have been the object of study in [5]. Recently Giammarresi and Restivo have generalized the concept of local and recognizable string languages to two-dimensional picture languages of finite arrays. In this paper we extend the concept of local languages to infinite triangular arrays. We define ωω-local languages by requiring a set of windows of size 2 to occur in the indefinite triangular arrays. This family coincides with the family of adherences of the local picture languages of infinite triangular arrays We then consider an extended notion of ωω-local languages by requiring certain windows of size 2 occur infinite by often in the infinite triangular arrays. The resulting class strictly contains the ωω-local language family. We define the notion of a ωω-recognizable language of infinite triangular arrays as a coding of an extended ωω-local language. This language is accepted by the two-direction online tessellation automaton reading infinite triangular arrays. We study the concept of Muller recognizability in the context of infinite images. The analysis of infinite images considered here helps us to understand better their structures.

2

Basic Definitions

Let Σ be a finite alphabet. A triangular array p over Σ of size m (also called a finite triangular array or finite triangular image) is a triangular arrangement of symbols over Σ. The set of all triangular pictures over the alphabet Σ ∗∗ is denoted by Σ∗∗ T . A triangular picture language over Σ is a subset of ΣT . Given a triangular picture p the number of rows (counting from the bottom to top) denoted by r(p) is the size of the triangular picture The empty picture is denoted by Λ. We adopt the convention that the bottom most row is the first row and the right slanding line is the first column. The set of all triangular arrays is denoted by Σ∗∗ T . The set of all triangular subpictures p of size k for k ≤ i where i is the size of the triangular picture p is denoted by ΣkT . An infinite triangular array (also called infinite image) has an infinite number of rows and an infinite number of columns (right slanding lines). The collection of all infinite triangular arrays over Σ is denoted by Σωω T . th th If a triangular array p has entry aij in the i row and j column, aij ∈ Σ then we write p = (aij ), 1 ≤ i ≤ m, 1 ≤ j ≤ m if p ∈ Σ++ and has size m and T p = (aij ), i = 1, 2, . . . , j = 1, 2, . . . , if p ∈ Σωω . T

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ωω For p ∈ Σ∗∗ T ∪ ΣT , p = (aij ), a prefix of p is an triangular array q = (aij ), i = 1, 2, . . . , `, j = 1, 2, . . . , r; 1 ≤ `, r < ∞, and ` ≤ m, r ≤ n, if p is of size ωω (m, n) We then write q ≤ p and if q 6= p, we write q < p. For p ∈ Σ∗∗ T ∪ ΣT , the set of all prefixes of p is denoted by F G(p). For L ⊆ Σ∗∗ T , we define

Lim(L) = {p ∈ Σωω T /F G(p) ∩ L is infinite} and adh(L) = {p ∈ Σωω T /F G(p) ⊆ F G(L)} where F G(L) =

[

F G(p).

p∈L

3

Two-direction Online Tesselation Automata

In this section, the notion of acceptance of languages of infinite arrays [1] by online tessellation automata is extended to the acceptance of infinite triangular arrays by the two direction online tessellation automata. A nondeterministic two direction online tesselatin automaton (2DOTA) is A = (Σ, Q, Q0 , F, δ) where Σ is an input alphabet, Q is a finite set of states, Q0 ⊆ Q is a set of initial states, F ⊆ Q is a set of final states, and δ : Q × Q × Σ → 2Q is a transition function. A computation by a two-direction OTA on an infinite triangular array p where · · · #

b(p) = # # #

· a31

a21 a11

#

·

a22 a12

#

· a32

a33 a23

a13 #

a24 a14

#

with aij ∈ Σ and # is a special symbol not in Σ is done as follows. At time t = 0, an initial state q0 ∈ Q0 is associated with all the positions of b(p) holding #. At time t = 1, a state from δ(q0 , q0 , a11 ) is associated with the position (1, 1) holding a11 . At time t = 2, states are associated simultaneously with positions (1, 2) and (2, 1) respectively holding a12 and a21 . If s11 is the state associated with the position (1, 1) then the state associated with the position (2, 1) is an element of δ(q0 , s11 , a21 ) and to the positions (1, 2) is an element of δ(s11 , q0 , a12 ). We then proceed to the next diagonal. The states associated with each position (i, j) by the transition function δ, depends on the states already associated with the position (i, j − 1), (i − 1, j) where the entry is aij . Let sij be the state associated with the position (i, j) where the

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V. Devi Rajaselvi, T. Kalyani and D. G. Thomas

entry is aij . A run (or a computation) of an infinite triangular array is an element of QωT . A run for an infinite triangular array is a sequence of states s11 s12 s21 s13 s22 s31 . . . and it is denoted by r(p). If A is non deterministic, a run for an infinie triangular array is a set, but in the deterministic case it is a singleton set. If p ∈ Σωω T , the set of runs of p is denoted by R(p). For s ∈ QωT , we define inf(s) as the set of all states which repeat infinitely many times in s. The language of infinite triangular arrays recognized by the non determinωω istic two direction online tessellation automaton A is Lωω T (A) = {p ∈ ΣT : inf(r(p)) ∩ F 6= φ, for some r(p) ∈ R(p)}. Here we note that if L(A) is the set of finite triangular arrays generated by A, then L(A) ⊆ F G(Lωω T (A)).

4

ωω-local Triangular Array Languages

We say that a set of triangular arrays L ⊆ Σ∗∗ T is local if there exists a set ∗∗ Q ⊆ Σ2×2 such that L = {p ∈ Σ : B (p) ⊂ Q}. Let L be the family of all 2,2 T T local languages. We say that a set of infinite triangular arrays L ⊆ Σωω T is ωω-local if there 2×2 ωω exists a set I ⊆ ΣT such that L = {p ∈ ΣT : B2,2 (p) ⊆ I}. Let Lωω T be the family of all ωω-local languages.

Proposition 4.1. If LT is the family of all local languages of finite triangular arrays, then Lωω T = adh(LT ). 0 ωω Proof. Let L ∈ Lωω T and L ∈ L be such that L = {p ∈ ΣT : B2,2 (p) ⊆ I} and 2×2 0 L0 = {x ∈ Σ∗∗ T , B2,2 (x) ⊆ I} for some I ⊆ ΣT . We prove that L = adh(L ), 0 0 0 p ∈ adh(L ) iff F G(p) ⊆ F G(L ) iff B2,2 (F G(p)) ⊆ B2,2 (F G(L )) iff B2,2 (p) ⊆ I iff p ∈ L. Hence L = adh(L0 ). This implies Lωω T ⊆ adh(LT ). Similarly we can ωω prove that adh(LT ) ⊆ LT .

Hence the family of ωω-local language is the family of adherences of local languages. Let p ∈ Σωω T and inf(B2,2 (p)) be the collection of all triangular subarrays of size (2, 2) in B2,2 (p) which occur as triangular sub arrays infinitely many times in p.

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Example 4.2. Let L = {Pn,m /n, m ≥ 1} where · · · · · · ·

#

· b

b ·

#

Pn,m =

· · · b

b ·

·

· ·

· ·

·

b b

·

b ·

·

· ·

· ·

·

b · # a a a a a a b b · # a a a a a a b b · · # a a a a a a b b · · · # # # # # # # # # · · · · #

a

a

a

a

a

a

In Pn,m the entries in the (i, j)th positions 1 ≤ i ≤ n, 1 ≤ j ≤ m are a. b If p ∈ L then ∈ inf(B2,2 (p)). b b a a But , 6∈ inf(B2,2 (p)). a a a a A set of infinite triangular arrays L ⊆ Σωω T is called an ωω-local language if there exists a set I ⊆ Σ2×2 and I ⊆ I such that 1 T L = {p ∈ Σωω T : B2,2 (p) ⊆ I and inf(B2,2 (p)) ∩ I 6= φ}. Let Lωω ET be the family of all extended triangular ωω-local languages. Let L = {Pn,m |n, m ≥ 1} ∪ {Qm |m ≥ 1} ∪ {Rn |n ≥ 1}. Let  I=

# # 

and I1 =

a

b

, , , # # # # # # b

b b

#

a

a

b

b

, , , , , a a a a b b b a a #

#

b



, b a b

 .

In the above example we note that L is an extended triangular ωω-local language, but is not a ωω-local language because if it were ωω-local then an infinite triangular array in which all the entries are ‘a’ will be in L. Let π be a strict alphabetic morphism from Σ to Γ where Σ and Γ are finite alphabets. We call π a projection. If p ∈ Σωω T , then π(p) = (π(aij )), i = 1, 2, . . . , j = 1, 2, . . . . We say that L is ωω-recognizable if and only if there exists a projection π and an extended triangular ωω-local language L0 such that L = π(L0 ).

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V. Devi Rajaselvi, T. Kalyani and D. G. Thomas

Proposition 4.3. If a language L ⊆ Σωω T is recognizable by a two-direction online tessellation automaton, then it is a ωω-recognizable language. Proof. Let L ⊆ Σωω T be recognized by a two-direction online tessellation automaton A = (Σ, Q, Q0 , F, δ), i.e., L = Lωω T (A). We define an extended triangular ωω-local language corresponding to A. Let Γ = (Σ ∪ {#}) × Q. I = I(1) ∪ I(2) ∪ I(3) ∪ I(4) ∪ I(5) where I(1) =

(#,q )

n

0

(#,q ) 0

I(2) =

0

I(3) =

(#,q ) 0

I(4) =

n

I(5) =

n

q0 ∈ δ(q, q0 , #), a 6= #

.

t ∈ δ(r, s, c), a, b, c 6= #

.

p3 ∈ δ(p1 , p2 , c), a, b, c 6= #

0

(c,t)

(a,p1) (b,p2)

.

(#,q )

(a,r) (b,s)

q ∈ δ(q0 , q0 , a), a 6= #, q0 ∈ Q0

(a,q)

(a,q)

n

.

0

0

(#,q )

q0 ∈ δ(q0 , q0 , #)

(#,q )

(#,q )

n

.

(c,p ) 3

I(4) ∪ I(5) . i.e., set of infinitely often repeating triangular subarray of size (2 × 2). Let L0 be the extended triangular ωω-local language corresponding to the sets I1 and I defined earlier. Let π : Γ → Σ such that π(a, r) = a for a ∈ Σ ∪ {#} and r ∈ Q. Then π(L0 ) = L. Hence L is ωω-recognizable. Proposition 4.4. If a language L is ωω-recognizable, then it is recognized by a two-direction online tessellation automaton. Proof. Let L be an extended triangular ωω-local language. We now define a two-direction online tessellation automaton A = (Σ, Q, Q0 , F, δ) such that Lωω T (A) = L. Let L be an extended triangular ωω-local language such that 2×2 L = {x ∈ Σωω T : B2×2 (x) ⊆ I and inf(B2×2 (x)) ∩ I1 6= φ}, for some I ⊆ Σ and I1 ⊆ I. We define A corresponding to it. Let Q = I. o n # # : # 6∈ Σ and ∈I Q0 = #

F =

#

#

a

n b

a

:

c

b

c

∈ I1

#

o

Q

δ : Q × Q × Σ → 2 such that for a b

d c

,

e

a

∈ Q, x ∈ Σ

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Infinite triangular arrays and recognizability

δ

a

 b

d c

,

e

a

 n ,x =

x a

x f

:

a

f

∈Q

o

Then L = Lωω T (A). Since language accepted by two direction OTA is closed under alphabetic morphism, we have the result. Definition 4.5. We define that a sequence of triangular arrays {xi } is a prefix preserving sequence of xi is a prefix of xi+1 for i = 1, 2, . . . . Let L ⊆ Σωω T . We say that L is row bounded if there exists a prefix preserving increasing sequence of triangular arrays {xi } in L such that the row size of each xi is less than some given positive number N . Similarly column bounded triangular array language can be defined A language L is bounded if it is either row bounded or column bounded or both. Proposition 4.6. Let L0 = L(A) where A is a two-direction OTA and let 0 L be an unbounded triangular array language. Then Lωω T (A) = Lim(L ). 0

Proof. Let x ∈ Lωω T (A). Then there exists a run s = s1 s2 . . . for x such that inf(s) ∩ F 6= φ where A = (Σ, Q, Q0 , F, δ). Let us assume that in s, elements of F occur in (mi , ni ) positions of the triangular arrays. Then obviously x[mi , ni ] ∈ L0 for i = 1, 2, . . . . Since L0 is an unbounded language, {x[mi , ni ]} is an unbounded increasing sequence of triangular arrays. Let x = lim(x[mi , ni ]). 0 0 ωω This implies x ∈ lim(L0 ). Hence Lωω T (A) ⊆ lim(L ). Hence LT (A) ⊆ lim(L ). Let x ∈ lim(L0 ). This implies x = lim(xi ) where x1 < x2 < . . . . Hence we can get a run for x which contains infinitely many states of F so that x ∈ Lωω T (A). 0 ωω 0 ωω Hence lim(L ) ⊆ LT (A). Thus lim(L ) = LT (A).

References [1] V.R. Dare, K.G. Subramanian, D.G. Thomas and R. Siromoney, Infinite arrays and recognizability, Int. J. Pattern Recognition and Artificial Intelligence, 14(4) (2000), 525–536. [2] D. Giammarresi and A. Restivo, Recognizable picture languages, Int. J. Pattern Recognition and Artificial Intelligence, 6 (1992), 241–256. [3] K. Inoue and A. Nakamura, Some properties of two dimensional online tessellation acceptor, Inf. Sci., 13 (1977), 95–121. [4] A. Nakamura and H. Ono, Pictures of functions and their acceptability by automata, Theoret. Comput. Sci., 23 (1983), 37–48. [5] R. Siromoney, V.R. Dare and K.G. Subramanian, Infinite arrays and infinite computation, Theoret. Comput. Sci., 24 (1983), 195–205. Received: June 7, 2014