HRL-Local Infinite Triangular Array Languages

ABSTRACT

A new subclass of infinite triangular arrays called hrl-local infinite triangular arrays is introduced. We introduce infinite triangular domino systems to recognize the infinite triangular picture language. Also we introduce strictly domino testable ωω-triangular array languages.

Key words:
hrl-local infinite triangular arrays; infinite triangular domino systems; strictly dommo testable ωω-triangular array languages

INTRODUCTION

Infinite triangular pictures are the digitized images which occur in the triangular grid of two dimensional plane. Infinite triangular picture p is a triangular array of elements of the terminal alphabet. It is useful to introduce the notation for the set of all infinite triangular pictures over the same alphabet (. Infinite triangular picture has infinite number of rows and infinite number of right slanting lines and infinite number of left slanting lines. The size |p| of a picture p is specified by the pair (|p|_row, |p|_rsline, |p|_lsline) of its number of rows and right slanting lines and left slanting lines. A pixel p(i, j, k), 1 ≤ i ≤ |p|_row, 1 ≤ j ≤ |p|_rsline, 1 ≤ j ≤ |p|_lsline is the element at position (i, j, k) in the triangular array P. Conventionally the indices grow from bottom to top for the rows and from right to left for left slanting lines and from left to right for right slanting lines.

For convenience we usually consider the bordered version of picture p obtained by surrounding the picture with the special boundary symbol # which is assumed not to be in the alphabet. In ²domino recognizability of triangular picture languages and hrl-domino systems are defined. Also in ²we define the overlapping of iso-triangular pictures.

In ¹we learn about recognizable infinite array languages. In ⁶we study the recognizability of infinite arrays and in ³we define the recognizability of infinite triangular pictures.

In this paper we consider another formalism hrl-domino system to recognize infinite triangular picture languages.

INFINITE TRIANGULAR PICTURE AND DOMINO SYSTEMS

Infinite triangular picture has infinite number of rows and infinite number of right slanting and left slanting lines. The set of all infinite triangular arrays over ( is denoted by .

In ⁴,⁵ hv-local picture languages are defined, where the square tiles of side 2 are replaced by dominoes that correspond to two kinds of tiles (i) horizontal dominoes of size (1,2) and (ii) vertical dominoes of size (2, 1). As in ² here we consider dominoes of the following types

Definition 1. is hrl-local if there exists a finite set ( of dominoes over such that L = {p ( / B_2,1() ( (}. In this case we write L = L(().

Definition 2. An infinite triangular domino system is a 4-tuple ITD = ((, (, (, () where ( and ( are two finite alphabets, ( is a finite set of dominoes over the alphabet ( ( and ( : ( ( ( is a projection.

An infinite triangular domino system recognizes an infinite triangular picture language L over the alphabet ( and is defined as L = ((L() where L( = L(() is the hrl-local infinite triangular picture language over (. The family of infinite triangular picture languages recognizable by infinite triangular domino systems is denoted by L(ITDS).

Proposition 1. If is a hrl-local infinite triangular picture language then L is a local infinite triangular picture language. That is L(ITDS) ( L(ITTS).

Proof. Let is a hrl-local infinite triangular picture language. Then L = L(Δ) where ∆ is a finite set of dominoes. We will construct a finite set θ of ωω-triangular arrays of size 2 and show that L = L(θ).

We now show that L( = L. Let p ( L(. Then by definition B1,2() ( (. This implies that B2,1() ( B2,1(B2,1()) ( B2,1(θ) ( (. Hence p ( L.

Conversely let p ( L and q ( B1,2(). Then B2,1(q) ( B2,1() ( (. Therefore q ( θ and p ( L(. Hence L = L(. □

Lemma 1. Let L be a local ωω-triangular picture language over an alphabet Σ. Then there exists an hrl local language L( over an alphabet and a mapping ( : ( ( ( such that L = ((L().

Proof. Let L = L(θ) where θ is a finite set of ωω-triangular arrays over ( (

Let S1, S2, S3 ( be three strings languages over (. The hrl-combination of S₁ , S₂ , S₃ is an triangular array language L_T = S₁ ( S₂ ( S₃ ( such that p ( L if and only if the strings corresponding to the hrl overlapping of p belong to S₁ , S₂ and to S₃ respectively. Also we have to prove that the class of all hrl-local triangular array languages is the hrl combination of the class of all local string languages. □

Theorem 1. hrl-TLOC = TLOC ( TLOC ( TLOC.

Proof. Let L ( hrl-TLOC. Then , for a finite set of dominoes over ( ( {#}. Let

Let L1, L2 and L3 be the hrl-local string languages generated by the local sys-tems (I1, C1, J1), (I2, C2, J2) and (I3, C3, J3) respectively. Then L = L1 ( L2 ( L3 ( TLOC ( TLOC ( TLOC.

Conversely let L ( TLOC ( TLOC ( TLOC. Let L = L1 ( L2 ( L3 where L1, L2 and L3 are the local string languages generated by the local systems (I1, C1, J1), (I2, C2, J2) and (I3, C3, J3) respectively. Let

Let ∆ = ∆1 ( ∆2 ( ∆3 ( ∆4 ( ∆5 ( ∆5 ( ∆6 ( ∆7 ( ∆8 ( ∆9. Then and therefore L ( hrl-TLOC.

Example 1.

is the hrl-local language which is represented by the following set of dominoes.

If L₁, L₂ and L₃ are the local languages generated by the local systems S₁ = {I₁, C₁, J₁} and S₂ = {I₂, C₂, J₂} respectively where

I₁ = {a}, C₁ = {ab}, J₁ = {a},

I₂ = φ, C₂ = {ab}, J₂ = {a},

I₃ = {a}, C₃ = {ab}, J₃ = φ,

L₁ = {(ab)^*}, L₂ = {(ab)^*a}. Then L = L₁ ( L₂ ( L₃ .

Definition 3. Let , we denote by Bh,k,l(p), domino testable, if there exists a finite set of dominoes of size (1, 1, l + 1), (1, k + 1, 1) and (h + 1, 1, 1) such that

L = {p ( : B_1,1,l+1 () ( B_1,k+1,1 () ( B_h+1,1,1 () ( ∆}

and we write . The family of all strictly (h, k, l)-domino testable ωω-triangular array languages is denoted by . Let is strictly domino testable, if , for some h, k > 0. The family of all strictly domino testable ωω-triangular languages is denoted by SDTTωω. Here we notice that

Example 2. Let ( = {a, b} and

In Lh,k,l the entries in the (i, j, k)th position for i > h and j > k and k > l are a and b and other entries are a1 and b1 then

Thus we have

Theorem 2.

Proof. Let

Let L1, L2 and L3 be the members of , and generated by local systems (I₁, C₁), (I₂, C₂) and (I₃, C₃) respectively. Let

Let ∆ = ∆₁ ( ∆₂ ( ∆₃ ( ∆₄ ( ∆₅ ( ∆₆. Then L₁ ( L₂ ( L₃ = where L₁ ( L₂ ( L₃ ( . Thus we proved. □

CONCLUSION

In this paper the notion of recognizability of infinite triangular pictures by a new formalism called hrl-domino systems have been investigated. Learning algorithm and automata characterization of hrl-local ωω-triangular array languages will be studied. The learning of infinite triangular pictures and unary infinite triangular picture languages and their complexity deserve to be studied further.

REFERENCES

• ¹
  Dare VR, Subramanian KG, Thomas DG, Siromoney R. Infinite arrays and recognizabiltiy. Int. J. Pattern Recognition and Artificial Intelligence. 2000; 14: 525-536.
• ²
  Devi Rajaselvi V, Kalyani T, Thomas DG. Domino recognizability of triangular picture languages. International Journal of Computer Applications. 2012; 57(15).
• ³
  Devi Rajaselvi V, Kalyani T, Thomas DG. Infinite triangular arrays and recognisability. Applied Mathematical Sciences. 2014; 8(106): 5269-5275.
• ⁴
  Gnanasekaran S, Dare VR. Infinite arrays and domino systems. Electronic Notes in Discrete Mathematics. 2003; 12.
• ⁵
  Gnanasekaran S, Dare VR. hv-local and prefix picture languages. Proc. National Conference on Recent Trends in Computational Mathematics. 2004.
• ⁶
  Gnanasekaran S, Dare VR. On recognizable infinite array languages. Lecture Notes in Computer Science. 2004; 3322; 204-213.

Publication Dates

• Publication in this collection
  2016

History

• Received
  03 Feb 2016
• Accepted
  14 July 2016