INTRODUCTION
We shall give some definitions that are natural extensions to the twodimensional
case of similar notions that are wellknown in the onedimensional case (Lind
and Marcus, 1995; Kitchens, 1998). We begin with the
dynamical system and its subsystems which are of interest to us.
Modeling and simulation of thermal dynamic system has been studied by Bhatti
et al. (2006). Whereas Shabbak et al.
(2011) proposed multivariate control chart which is less sensitive to the
sustained shift in mean process. Another related work has been proposed by AbuShawiesh
et al. (2009) and Goegebeur et al. (2005).
Huffman code has been widely used in text, image and video compression. For
example, it is used to compress the result of quantization stage in JPEG (Hashemain,
1995). The simplest data structure used in the Huffman decoding is the Huffman
tree. Array data structure (Chen et al., 1999)
has been used to implement the corresponding complete ternary tree for the Huffman
tree.
Let A be a finite set and the set be
the set of all maps x: Z^{2}→A. For every we denoted by the
projection ,
,
where, x_{E} is the restriction of x: Z^{2}→A to
E if and
nZ^{2}
then x_{n} is the nth coordinate of x, i.e., the value of x at n. We
define, for every mZ^{2},
a homomorphism σ_{m} of by
setting (σ_{m}x)_{n} = x_{m+n}.
A closed subset is
shift invariant (or a subshift of )
if σ_{m} (X) = X,mZ^{2}.
If there exist a finite set F⊂Z^{2} and a set P⊂A^{F}
such that .
The set P⊂A^{F} is the collection of permissible (or allowed) words
of X.
Example 1: Let, F = {(0,0), (1,0), (0,1)}⊂Z^{2} and P =
{(t_{(0,0)}), (t_{(1,0)}), (t_{(0,1)})TF:
t_{(0,0)}+t_{(1,0)}+t_{((0,1))} = 0}. Here the alphabet
set A = {0, 1} and addition is W.r.t modulo 2 arithmetic. Then:
Given 2 directed graph G_{H} and G_{V} with the same set of edges ε we now define the edge shift:
Definition 2: Let X = X_{(GH, GV)} be an edge shift. Also let λ:ε→A be a map where A is some finite set, a sofic shift S with presentation (G_{H}, G_{V}, λ) is defined by S = {λ^{∞} (ξ): ξ∈X (G_{H}, G_{V})} and λ^{∞} (ξ)_{(i, j)} = λ(ξ_{(i, j)}).
Definition 3: Let X_{1}, X_{2} is topological space
and Z^{2} acts on X_{1}, X_{2} by a commuting homomorphism.
For each n∈Z^{2}, the action is written as n.x where x∈X_{1}
to represent .
Whenever n = (n_{1}, n_{2}). Similarly for xX_{2},
,
we say X_{2} is a factor of X_{1} if there exist a continuous
onto map φ: X_{1}→X_{2} such that φ (n.x) = n.φ
(x).
When dealing with shift spaces, Curtis HedlundLyudon,s theorem says that the factor map is a sliding block code, i.e., there exist positive integers m, n, r. S and Φ: B_{(n+m+1)x(s+r+1)} (X_{1})→ A (X_{2}):
Here B_{(n+m+1)x(S+r+1) }(X1) denotes the rectangular allowable blocks
of size (n+m+1)x(s+r+1) and A (X_{2}) is the alphabet set of X_{2}.
From the above work and by AlRefaei et al. (2007)
we get this result.
Proposition 4: Every edge shift is a shift of finite type.
Proof: Given X = X(G_{H}, G_{V}), fined F and P such that X = X_{(F, P) }X has alphabet ε = ε (G_{H}) = ε (G_{V}).
Take
Take:
Then P = P_{H}∪P_{V}. In fact, it is a matrix subshift with transition matrix H and V defined as follows:
H and V is of size εH*ε record ε so that each are {e_{1}, e_{2}, e_{3},ÿ, w_{ε}}.
The define:
Similarly for V.
Definition 5: Let X = X (G_{H}, G_{V}) be an edge shift. Also let λ:ε→A be a map where A is some finite set. A sofic shift S with presentation (G_{H}, G_{V}, λ) is defined by S = {λ∞ (ξ): ξ∈X (G_{H}, G_{V})} and:
Proposition 6: If S a sofic system then S is a factor of shift of finite type.
Proof: Let (G_{H}, G_{GV}λ) be presentation of
S .By definition there exist edge shift X = X(G_{H}, G_{V})
and a map λ^{∞}: X→S which is onto. Also λ^{∞}
commutes with the respective shift maps on X and S. Thus S is a factor of X
via λ.
The result follows since X is a shift of finite type.
Proposition 7: Let X be shift of finite type and Y is shift space φ: X→Y be factor map. Then Y is a sofic system.
Proof: First by going to higher block system one can find an edge shift X^{'} which is isomorphic to X. Let Φ be the associated map that induced Φ: X→Y. If we denote by ε the set of edges arising from the graphs of the edge shift X^{'}, then one can define a map λ: ε→A (Y) by λ^{'} (e) = Φ (e), ∀e∈ε. It is then an easy matter to check for commutatively so that λ and the respective graphs is the presentation of the sofic system.
Theorem 8: A twodimensional shift space is
twodimensional sofic system if and only if a factor of twodimensional shift
of finite type.
Proof: Let is sofic shift and be
presentation of X The 1x1block map λ induces a sliding block code λ:χ_{(GH,
GV)}→χ_{S} that is onto by definition of.
Hence X = χ_{S} is factor of edge shift χ_{GH} and
χ_{GV} which has finite type. Conversely, suppose that a
shift space for which there is twodimensional shift of finite type Y and a
factor map φ: Y→X let φ have memory m(r) and anticipation m (s)
then φ is induced by Φ on B_{m(r)+n(s)+1}(Y) by increasing
m(r) if necessary .We can assume that Y is [m(r)+n(s)]step. Define Ψ:
Y→Y^{[m(r)+n(s)+1]} by Ψ(y)_{[i, j]} = y [Im, j+s
and I+n, j+s].
Ψ Almost the same the higher block map B_{m(r)+n(s)+1} except that. We use coordinates from y starting at index Im, j+s rather than i, j. Since y is (m(r)+n(s))step so, there is graphs G_{H}, G_{V} with edges named by block B_{m(r)+n(s)+1 }(Y) So That Y_{[m(r)+n(s)+1] }= X_{(GHGV)}. Define a labeling λ on G_{H}, G_{V} by λ (e) = Φ (e). We will show that S = (χ_{(GH, GV)}, λ) is presentation of X proving that X is Sofic observe that Fig. 1 is commutes for if yεY, then φ (y)[i, j] = Φ (y_{[im, j+s and I+n, j+s]}).

Fig. 1: 
Two dimensional shift 
While λ (Ψ (y)_{[i, j]}) = λ (Ψ (y)_{[i, j]}) since Ψ is conjugacy, the image of φ and of, λ are equal hence X = φ (y) = = λ(χ_{GHV,GV}) so that S is presentation of X.