WEAK STABILITY OF INTERVAL ORBITS OF CIRCULANT MATRICES IN FUZZY ALGEBRA

Fuzzy algebra is an algebraic structure in which classical addition and multiplication are replaced by ⊕ and ⊗, where a⊕ b = max{a,b}, a⊗ b = min{a,b}. An orbit of A generated by x is called stable if per(A,x) = 1. An interval orbit of an interval matrix A and an interval vector X and the weak stability of an interval orbit are defined. A necessary and sufficient condition for the weak stability of interval orbits of circulant matrices is introduced and justified.


INTRODUCTION
Matrices in fuzzy algebra are useful for expressing applications of fuzzy discrete dynamic systems, graph theory, scheduling, knowledge engineering, cluster analysis, fuzzy systems and for describing diagnosis of technical devices [18], [19], medical diagnosis [15], [16] or fuzzy logic programs [10].The problem studied in [15] leads to the problem of finding the greatest invariants of the fuzzy system (the greatest eigenvector of the fuzzy matrix corresponding to the greatest eigenvalue).
In practice, matrix and vector inputs are rather contained in some intervals than exact values.Considering matrices and vectors with interval coefficients is therefore of great practical importance, see [2], [8], [14].The aim of this paper is to describe matrices and vectors with inexact data (interval matrices and vectors) for which there exists a stable orbit, i.e., an orbit with period equal to one, for some matrix and some vector from the given interval vector and interval matrix.The main result is concentrated in Theorem 5.1 which gives a necessary and sufficient condition for the weak stability of an interval orbit of circulant matrix which can be checked in O(n 2 log n) arithmetic operations.

PRELIMINARIES
The fuzzy algebra B is the triple (B, ⊕, ⊗), where (B, ≤) is a bounded linearly ordered set with binary operations maximum and minimum, denoted by ⊕ and ⊗, respectively.The least element in B will be denoted by O, the greatest one by I.
By N we denote the set of all natural numbers and by N 0 the set N 0 = N ∪ {0}.The greatest common divisor of a set S ⊆ N is denoted by gcd S and the least common multiple by lcm S. For a given natural number n ∈ N, we use the notations N = {1, 2, . . ., n} and N 0 = {0, 1, . . ., n − 1}.
For any n ∈ N, B(n, n) denotes the set of all square matrices of order n and B(n) the set of all n-dimensional column vectors over B. The matrix operations over B are defined formally in the same manner (with respect to ⊕, ⊗) as matrix operations over any field.The r-th power of a matrix A is denoted by A r , with elements (A r ) i j . For By digraph we understand a pair G = (V G , E G ), where V G is a non-empty finite set, called the node set, and A cycle is elementary if all nodes except the terminal node are distinct.A digraph is called strongly connected if any two distinct nodes of G are contained in a common cycle.By a strongly connected component of G we mean a maximal strongly connected subdigraph of G .A strongly connected component K = (V K , E K ) is called non-trivial if there is a cycle of positive length in K .The strongly connected component of G containing node i will be denoted by G [i].By SCC G we denote the set of all non-trivial components of G .

ORBIT PERIODICITY
The notions of an orbit of A generated by x and known properties of the orbit periodicity are introduced in this section.A necessary and sufficient condition for the orbit period to be equal to one is proved.Definition 3.1.For any A ∈ B(n, n) and x ∈ B(n) the orbit of A generated by x is the vector sequence O(A, x) = (x(r); r ∈ N 0 ) whose initial vector is x(0) = x and successive members are defined by the formula x(r + 1) = A ⊗ x(r).The i-th coordinate of x(r) is denoted by x i (r).The i-th coordinate orbit is the sequence O i (A, x) = (x i (r); r ∈ N 0 ).Definition 3.2.The sequence S = (S(r); r ∈ N) is ultimately periodic if there is a natural number p such that the following holds for some natural number R : The smallest natural number p with the above property is called the period of S, denoted by per(S).The smallest R with the above property is called the defect of S, denoted by def(S).
Both operations in fuzzy algebra are idempotent, so no new elements are created in the process of exponentiation of a Unauthenticated | 194.138.39.60 Download Date | 1/15/14 3:09 AM matrix.Therefore any power sequence (A k ; k ∈ N) contains only finite number of different matrices.The same holds true for any orbit.So a power sequence, an orbit O(A, x) and a coordinate orbit O i (A, x) are always ultimately periodic sequences.Their periods will be called the period of A, the orbit period and the coordinate-orbit period, in notation per(A), per(A, x) and per(A, x, i).Analogous notations def(A), def(A, x) and def(A, x, i) will be used for the defects.
A matrix (vector) is called binary if a i j ∈ {O, I} (x j ∈ {O, I}) for each i, j ∈ N.
Definition 3.3.Let A ∈ B(n, n) be a binary matrix and x ∈ B(n) be a binary vector.Then by G (A) we understand the digraph {(i, j); a i j = I} and by G (A, x) we understand the corresponding node-weighted digraph obtained from G (A) by appending weight x i to each node i.A path in G (A, x) is called an orbit path if the weight of its terminal node is I.
Remark 3.1.We shall say that two strongly connected components K 1 ∈ SCC G (A, x) and K 2 ∈ SCC G (B, y) are identical, in notation , the evaluation of vertices is not relevant.
Definition 3.4.For A ∈ B(m, n) and h ∈ B, the threshold matrix A (h) corresponding to the threshold h is a binary matrix of the same type as A, defined as follows: The associated digraphs G (A (h) ) and G (A (h) , x (h) ) will be called the threshold digraphs corresponding to the threshold h.
Since any vector is viewed as an (n × 1) matrix, the above definition concerns also vectors.
For A ∈ B(n, n) and x ∈ B(n), the threshold orbit O(A (h) , x (h) ) corresponding to a threshold h ∈ B is a vector sequence whose rth member equals to the threshold vector x(r) (h) .Similarly, the threshold coordinate-orbit from i to j of length r, ii) O i (A, x)(r) ≥ h if and only if there is a an orbit path in G (A (h) , x (h) ) starting at i of lenght r.
Lemma 3.1.[17] The decomposition of a matrix over B to its threshold matrices has the following properties:

Orbit stability
In this part we shall deal with the so-called stable orbits.We give a necessary and sufficient condition for the stability of an orbit.
. By (O) and (I) we understand the infinite sequences of the same elements O and I, respectively.
For a given Remark 3.2.Gavalec in [3] has shown that the computation of the coordinate-orbit period is NP-hard, which is because the period of a matrix may be exponential large.In the following we shall deal with the class of circulant matrices, which have the period bounded by n.Thus the computation of the coordinate-orbit period becomes polynomial.
In [17], we can find the O(n 3 log n) algorithm for computing the orbit period in general case.
x.An interval matrix A and interval vector X with bounds A, A and x, x, respectively, are defined as follows In like manner we can define the notion of an interval orbit.

ORBITS OF CIRCULANT MATRICES
In this section we shall deal with the special class of matrices, the circulant matrices.We prove the assertions needed in the next section. .
For a given circulant matrix A = A(a 0 , a 1 , . . ., a n ) let us denote by h 1 , h 2 , . . ., h r the elements of H(A) in the following way: It is well known that the threshold digraphs G (A, h) for each h ∈ H(A) consist from isomorphic strongly connected components.Moreover we can compute the vertex sets of these components in the following way (for details see [9]).
For each j ∈ {1, 2, . . ., r} denote Then the number of strongly connected components of and we denote them by K j 1 , K j 2 , . . ., K j m j .For each j ∈ {1, 2, . . ., r}, i ∈ {1, 2, . . ., m j } the vertex set of K j i is given by the formula The cardinality of the vertex set V j i is v j = n m j for each i ∈ {1, 2, . . ., m j }.Remark 4.1.Let us note, that if k, l are not lying in the same strongly connected component of G (A (h) ), then there is not edge from k to l in G (A (h) ).
According to [5] the computational complexity of ( 4) is O(n).
Corollary 4.1.Let A be a circulant matrix.Then per(A) ≤ n.
The structure of the eigenspace of circulant matrices is described in [7].
[17] Let A ∈ B(n, n) and x ∈ B(n) be binary.
Then the periods of coordinate-orbits to the nodes of the same strongly connected component are equal.

Lemma 4.2. [17]
Let A ∈ B(n, n) and x ∈ B(n) be binary.Let (v 0 , v 1 , . . ., v l ) be path in a non-trivial strongly con- Definition 4.2.Let A be a circulant matrix and h ∈ B. We say that a strongly connected component K ∈ SCC G (A (h) , x (h) ) has the property (P), if for each i ∈ V K there exists r ∈ N 0 such that for each k > r there exists an orbit path from i in K of length k .Theorem 4.2.Let A(a 0 , a 1 , . . ., a n−1 ) be a circulant matrix and x ∈ B(n).An orbit O(A, x) is stable if and only if for each i ∈ N the property (P) holds for the strongly connected component If for each i ∈ N the strongly connected component K i has the property (P) then per(A (h(A,x,i)) , x (h(A,x,i)) , i) = 1 for each i ∈ N by Theorem 3.2ii).In view of Lemma 3.2 we get per(A, x) = 1.
For the converse implication suppose that there exists i ∈ N such that K i does not have the property (P).Then there exists j ∈ V K i such that for each r ∈ N there exists k > r such that there is no orbit path of length k from j in K i .In view of Remark 4.1 there is no orbit path from j to node which does not lie in K i .Thus O per j (A (h(A,x,i)) , x (h(A,x,i)) ) = (I) and, according to Corollary 4.2, O per j (A (h(A,x,i)) , x (h(A,x,i)) ) = (O) which implies per(A (h(A,x,i)) , x (h(A,x,i)) , j) = 1.Then, by Lemma 4.1, per(A (h(A,x,i)) , x (h(A,x,i)) , i) = 1.In view of Lemma 3.2 we get per(A, x) = 1.Lemma 4.3.Let A, C ∈ B(n, n) and x, y ∈ B(n) be binary and K 1 ∈ SCC G (A, x), K 2 ∈ SCC G (C, y) be such that K 1 ⊆ K 2 , x i ≤ y i for each i ∈ V K 1 , and K 1 have the property (P).Then K 2 has the property (P).
Proof.Let i ∈ V K 2 .If i ∈ V K 1 then the assertion trivially holds.If i ∈ V K 2 \V K 1 then there exists a path P from i to j, where j ∈ V K 1 .The existence of an orbit path from j of length k in K 2 implies the existence of an orbit path from i of length k + |P|.If r ∈ N 0 is such that for each k > r there exists an orbit path from j in K 1 of length k, then there exists s = r + |P| such that for each l > s there exists an orbit path from i in K 2 of length l.Thus K 2 has the property (P).

INTERVAL CIRCULANT MATRICES
The notions of an interval circulant matrix and a weak solvability of interval orbit of interval circulant matrix are defined in this section.A necessary and sufficient condition for the weak solvability of interval orbit of interval circulant matrix is proved.The notions of the possible and universal X-robustness of an interval circulant matrix are defined and the polynomial algorithms for checking of them are introduced in this section.Definition 5.1.An interval circulant matrix A C is the set of all circulant matrices A ∈ A where , and a i = [a i , a i ] for each i ∈ N. We denote an interval circulant matrix by A C (a 0 , . . ., a n−1 ).
There are matrices, that are not circulant, in A, so A = A C .On the other hand A, A ∈ A C , therefore the set A C is always non-empty.Definition 5.2.An interval orbit O(A C , X) is weakly stable if there exist A ∈ A C and x ∈ X such that O(A, x) is stable.

Weak stability of interval orbits of interval circulant matrices
Denote a = max k∈N 0 a k .Let us define the circulant matrix ) as follows: for each i ∈ M. For a given vector y ∈ Y let us denote J(y) = {i ∈ M; y i = max j∈M y j }.
Proof.If J(y * ) = M, then J(y) ⊆ J(y * ) trivially holds.If J(y * ) = {i ∈ M; y i ≥ y} = M then for each i / ∈ J(y * ) the inequality y i < y holds true.Consequently max i / ∈J(y * ) y i < y.Let y ∈ Y and r ∈ J(y) be arbitrary.We get For a given matrix A(a 0 , a 1 , . . ., a n−1 ) and an interval vector X we define the vector x(A) by the following algorithm.
In Algorithm A , an auxiliary vector p ∈ {0, 1} n will be used.The vector p will be used to register the entries of x(A) which are assigned a final value, i.e., p k = 0 till x k (A) is not assigned the final value.
x k = 6 < h 1 , the entries p k and x k (A) for k ∈ V 1 2 will be not changed in this step.
Proof.Suppose that per(A, x(A)) = 1.We will prove that per(A, x) = 1 for each x ∈ X.
Let l ∈ N be such that G (A (h(A,x(A),l)) , x(A) (h(A,x(A),l)) )[l] does not have the property (P).Let j ∈ {1, 2, . . ., r} be the least number such that max k∈V j i x(A) k ≥ h j where i ∈ {1, 2, . . ., m j } is such that l ∈ V j i .We shall distinguish two cases.
Case 1.If j = 1 then for each x ∈ X there exists k ∈ V j i such that and G (A (h(A,x(A),l)) , x(A) (h(A,x(A),l)) )[l] does not have property (P), then for each x ∈ X the strongly connected component G (A h(A,x,l) , x h(A,x,l) )[l] does not have property (P) by Lemma 4.3.Consequently per(A, x) = 1.
Case 2. If j > 1 then h(A, x(A), l) = h(A, x(A), i) = max k∈V j i x(A) k .Let x ∈ X be arbitrary but fixed.Let us define the vectors y = (x i 1 , x i 2 , . . ., x i v j ) and y = (x i 1 , x i 2 , . . ., x i v j ), where According to Algorithm A and (6) the equality y * = (x(A) i 1 , x(A) i 2 , . . ., x(A) i v j ) holds true.
We have h(y * ) = h(A, x(A), i) and h(y) ≥ h(y * ) for each y ∈ [y, y].Then there exists u ∈ V j i such that y u = h(y) and h Moreover, by Lemma 5.3, the inequality y (h(y)) ≤ y * (h(y * )) holds true which implies x k (h(A,x,iu)) ≤ x k (h(A,x(A),l)) for each k ∈ V G (A h(A,x,iu) ,x h(A,x,iu) )[i u ] .Since G (A (h(A,x(A),l)) , x (h(A,x(A),l)) )[l] does not have the property (P) then G (A h(A,x,i u ) , x h(A,x,i u ) )[i u ] does not have the property (P) for each x ∈ X by Lemma 4.3.Consequently per(A, x) = 1.
Further, we need to compute the vector x(A * ).Whereas the matrix A * is equal to the matrix A from Example 5.1

Corollary 4 . 2 .
Let A ∈ B(n, n) and x ∈ B(n) be binary and K ∈ SCC (A, x).If i, j ∈ V K and O per i (A, x) = (O) then O per j (A, x) = (O) .

For
each y ∈ B(m) denote h(y) = max i∈M y i .From the previous the next lemma follows.Lemma 5.3.For each y ∈ Y the inequality y (h(y)) ≤ y * (h(y * )) is satisfied.Proof.It follows directly from Lemma 5.2.

Theorem 5 . 1 .
An interval orbit O(A C , X) is weakly stable if and only if per(A * , x(A * )) = 1, where A * is the matrix defined by(5).Proof.If there exists x ∈ X and A ∈ A C such that per(A, x) = 1 then, by Lemma 5.1, there exists x ∈ X such that per(A * , x) = 1.In view of Lemma 5.4 we get per(A * , x(A * )) = 1, where the vector x(A * ) is given by Algorithm A with the input matrix A * .The converse implication is trivial.Theorem 5.2.Let A C and X be given.There is an algorithm which decides whether the interval orbit O(A C , X) is weakly stable in O(n 2 log n) arithmetic operations.Proof.The complexity of checking the weak stability of O(A C , X) using Theorem 5.1 consists of O(n) operations needed for computing the matrix A * by formula (6), O(n 2 ) operations needed for computing of the vector x(A * ) according to Algorithm A and O(n 2 log n) operations, which are necessary for determination the orbit period of O(A * , x(A * )).Thus the complexity of the complete algorithm is O(n) + O(n 2 ) + O(n 2 log n) = O(n 2 log n).