XAE and XEA robustness of max–min matrices

Discrete Applied Mathematics 267 (2019) 142–150

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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam

X AE and X EA robustness of max–min matrices Helena Myšková, Ján Plavka

∗

Department of Mathematics and Theoretical Informatics, Technical University, B. Němcovej 32, 04200 Košice, Slovakia

article

info

Article history: Received 17 April 2018 Received in revised form 15 December 2018 Accepted 19 April 2019 Available online 17 May 2019 Keywords: Robustness max–min matrix Interval vector

a b s t r a c t A max–min (fuzzy) matrix A (operations max and min are denoted by ⊕ and ⊗, respectively) is called X − robust if the orbit x, A ⊗ x, A2 ⊗ x, . . . of any initial vector x belonging to the interval X = {x : x ≤ x ≤ x} ends up with a max–min algebraic eigenvector of A. The X -robustness of a fuzzy matrix is extended to interval vectors X using forall–exists quantification of their interval entries (so-called X AE -robustness and X EA -robustness). A complete characterization of X AE -robustness and X EA -robustness of fuzzy matrices is presented. Moreover, O(n4 ) algorithms for checking the both types of robustness are described. © 2019 Elsevier B.V. All rights reserved.

1. Introduction In max–min algebra (fuzzy algebra) the standard pair of operations plus and times is substituted by the pair of operations maximum and minimum, which are involved in many optimization problems. Fuzzy matrix operations 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 [21] or medical diagnosis [18]. The problem studied in [18] and recently adapted in [15,17] has interesting applications in informatics and leads to the problem of finding the invariants and the greatest invariant of the fuzzy system. As a motivation for the research of fuzzy algebra and later the robustness can be considered adapting max–plus interaction systems [1–3,20]. In these systems we have n entities (processors, servers, machines, etc.) which work in stages, and in the algebraic model of their interactive work, entry xi (k) of a vector x(k), represents the state of entity i after some stage k (start-time of the kth stage on entity i (i = 1, . . . , n)), and the entry aij of a matrix A encodes the influence of the work of entity j in the previous stage on the work of entity i in the current stage. For simplicity, the system is assumed to be homogeneous, so that A does not change from ⨁ stage to stage. Summing up all the influence effects multiplied by the results of previous stages, we have xi (k + 1) = j aij ⊗ xj (k). The summation is often interpreted as waiting till all works of the system are finished and all the necessary influence constraints are satisfied. Thus the orbit x, A ⊗ x, . . . , Ak ⊗ x, where Ak = A ⊗ · · · ⊗ A, represents the evolution of such a system. The problem of finding the set of vectors for which multi-entity interaction system stabilizes (an eigenproblem A ⊗ x = x) belongs to the most intensively studied questions. The matrices of the fuzzy system for which an eigenspace is reached with any initial vector x belonging to the interval X = {x : x ≤ x ≤ x} are called X -robust. Such matrices have been studied in [14]. The robustness of matrices in max-plus algebra was studied in [2]. Attraction cones of nonnegative irreducible matrices in max-algebra were studied in [20]. The matrices for which an eigenspace is reached only if a starting vector is an eigenvector of the fuzzy matrix are called weakly robust. Efficient characterizations of such matrices are described in [16]. ∗ Corresponding author. E-mail addresses: [email protected] (H. Myšková), [email protected] (J. Plavka). https://doi.org/10.1016/j.dam.2019.04.021 0166-218X/© 2019 Elsevier B.V. All rights reserved.

H. Myšková and J. Plavka / Discrete Applied Mathematics 267 (2019) 142–150

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In practice, the values of the matrix entries are not exact numbers and usually they are contained in some intervals. Interval arithmetic is an efficient way to represent matrices in a guaranteed way on a computer. Max–min algebra is a convenient algebraic setting to deal with some types of optimization problems, see [6]. Hence, considering matrices and vectors with interval coefficients in a max–min algebra has great practical importance, see [14,15]. The aim of this paper is to characterize X -robustness of a fuzzy matrices extended to interval vectors using forall–exists quantification of interval entries (so-called X AE -robustness and X EA -robustness). In this paper we suppose that X can be split into two subsets according to forall–exists quantification of its interval entries, i.e., X = X ∀ ⊕ X ∃ . The properties of vectors with interval coefficients are studied and complete characterization of both types of robustness, i.e., the X EA -robustness (there is at least one vector of X ∃ such that for any vector of X ∀ their vector maximum reaches an invariant of A) and X AE -robustness (for any vector of X ∀ there is at least one vector of X ∃ such that their vector maximum reaches an invariant of A), in fuzzy algebra are presented. The practical meaning of the paper aim in correspondence with the above motivation example is that some start-times xi (k) are considered for each interval value and for some of them it suffices to consider only for at least one value. Moreover polynomial algorithms for verifying the equivalent conditions and the corresponding properties of interval fuzzy vectors are suggested. The paper is organized as follows. In Section 2 we give definitions and basic results used in the paper. The known results describing the equivalent conditions of the X -robustness in fuzzy algebra are given in Section 3. Section 4 presents results on the X AE -robustness and X EA -robustness. 2. Background of the problem The fuzzy algebra B is a triple (B, ⊕, ⊗), where (B, ≤) is a bounded linearly ordered set with binary operations maximum and minimum, denoted by ⊕, ⊗, 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 N0 the set N0 = N ∪{0}. The greatest common divisor of a set S ⊆ N is denoted by gcd S. For a given natural n ∈ N, we use the notation N for the set of all smaller or equal positive natural numbers, i.e., N = {1, 2, . . . , n}. 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 rth power of a matrix A ∈ B(n, n) is denoted by Ar , with elements arij . For A, C ∈ B(n, n) we write A ≤ C (A < C ) if aij ≤ cij (aij < cij ) holds for all i, j ∈ N. A digraph is a pair G = (V , E), where V , the so-called vertex set, is a finite set, and E, the so-called edge set, is a subset of V × V . A digraph G ′ = (V ′ , E ′ ) is a subdigraph of the digraph G (for brevity G ′ ⊆ G ), if V ′ ⊆ V and E ′ ⊆ E. A path in the digraph G = (V , E) is a sequence of vertices p = (i1 , . . . , ik+1 ) such that (ij , ij+1 ) ∈ E for j = 1, . . . , k. If all vertices in a path p are distinct then p is called an elementary path. If a path p is not elementary then p is called a non-elementary path. The number k is the length of the path p and is denoted by ℓ(p). If i1 = ik+1 , then p is called a cycle. A digraph G = (V , E) without cycles is called acyclic. If G = (V , E) contains at least one cycle then G is called cyclic. For a given matrix A ∈ B(n, n) the symbol G (A) = (N , E) stands for the complete, edge-weighted digraph associated with A, i.e., the vertex set of G (A) is N, and the capacity of any edge (i, j) ∈ E is aij . In addition, for given h ∈ B, the threshold digraph G (A(h) ) is the digraph G = (N , E ′ ) with the vertex set N and the edge set E ′ = {(i, j); i, j ∈ N , aij ≥ h}. The following lemma describes the relation between matrices and corresponding threshold digraphs and follows from the transitivity of ordering. Lemma 2.1 ([14]). Let A ∈ B(n, n). Let h1 , h2 ∈ B. If h1 < h2 then G (A(h2 ) ) ⊆ G (A(h1 ) ). By a strongly connected component of a digraph G (A(h) ) = (V , E) we mean a subdigraph K = (VK , EK ) generated by a non-empty subset VK ⊆ V such that any two distinct vertices i, j ∈ VK are contained in a common cycle, EK = E ∩(VK ×VK ) and VK is a maximal subset with this property. A strongly connected component K of a digraph is called non-trivial, if there is a cycle of positive length in K. For any non-trivial strongly connected component K the period of K is defined as per K = gcd { ℓ(c); c is a cycle in K, ℓ(c) > 0 }. If K is trivial, then we define per K = 1. By SCC⋆ G we denote the set of all non-trivial strongly connected components of G. Definition 2.1. 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 ∈ N0 ) whose initial vector is x(0) = x and successive members are defined by the formula x(r + 1) = A ⊗ x(r). The ith coordinate of x(r) is denoted by xi (r). The ith coordinate orbit is the sequence Oi (A, x) = (xi (r); r ∈ N0 ). Definition 2.2. 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 : S(k + p) = S(k) for all k ≥ 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).

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Both operations in fuzzy algebra are idempotent, so no new numbers are created in the process of generating an orbit. Therefore any orbit in fuzzy algebra contains only a finite number of different vectors. Thus an orbit is always ultimately periodic. The same holds true for the power sequence (Ak ; k ∈ N). Hence a power sequence, an orbit O(A, x) and a coordinate orbit Oi (A, x) are always ultimately periodic sequences. Their periods will be called the period of A, the orbit period and coordinate-orbit period of O(A, x), 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. It is known that the defect of any orbit in max–min algebra is at most (n − 1)2 + 1 and that a periodic member of an orbit can be found in O(n3 log n) time (see [19]). For a given matrix A ∈ B(n, n), the number λ ∈ B and the n-tuple x ∈ B(n) are the so-called eigenvalue of A and eigenvector of A, respectively, if A ⊗ x = λ ⊗ x. The eigenspace V (A, λ) is defined as the set of all eigenvectors of A with associated eigenvalue λ, i.e., V (A, λ) = {x ∈ B(n); A ⊗ x = λ ⊗ x}. Define attraction set attr(A, λ) as follows attr(A, λ) = {x ∈ B(n); O(A, x) ∩ V (A, λ) ̸ = ∅}. In case λ = I let us denote attr(A, I) by abbreviation attr(A). Lemma 2.2.

Let A ∈ B(n, n) be a given matrix. Then

attr(A) = {x ∈ B(n): Ak ⊗ x = Ak+1 ⊗ x for some k ≥ def(A)}. Proof. By definition x ∈ attr(A) if and only if As+1 ⊗ x = As ⊗ x for some s, hence Ak ⊗ x = Ak+1 ⊗ x for some k ≥ def(A) is sufficient for x ∈ attr(A). ′ ′ For the converse implication suppose that As+1 ⊗ x = As ⊗ x. Then we have As +1 ⊗ x = As ⊗ x for some s′ ≥ max(s, def(A)) s′ k k+1 k and such that A = A . Hence A ⊗ x = A ⊗ x. □ Lemma 2.3.

The following equalities hold true:

(i) attr(A) = {x ∈ B(n); (∃k ∈ N) Ak ⊗ x ∈ V (A)}, (ii) attr(A) = {x ∈ B(n); per(A, x) = 1}. According to [7] we define SCC⋆ (A) = ∪{SCC⋆ (G (A(h) )); h ∈ {aij ; i, j ∈ N }} Theorem 2.1 ([7]). Let A ∈ B(n, n). Then per A = lcm{per K; K ∈ SCC⋆ (A)}. We define the period of the threshold digraph G (A(h) ) as follows per G (A(h) ) = lcm { per K; K ∈ SCC⋆ (G (A(h) ))}. Definition 2.3. Let A = (aij ) ∈ B(n, n), λ ∈ B. A is called λ-robust if attr(A, λ) = B(n). A λ-robust matrix with λ = I is called robust matrix. It is easy to see that if A = (aij ) is ultimately periodic with period p and λ ≥ maxi,j∈N aij then λ ≥ maxi,j∈N akij and Ak+p = λ ⊗ Ak = Ak for sufficiently large k. Thus without loss of generality we can consider λ equal to I. The results we shall formulate for λ = I, as well as the methods used to prove them, can be generalized for arbitrary λ ∈ [maxi,j∈N aij , I ]. The robustness of matrices in max-plus algebra was studied in [2,3]. Attraction sets of nonnegative irreducible matrices in max-plus algebra were studied in [20]. Lemma 2.4 ([17]). Let A = (aij ) ∈ B(n, n) and λ ≥ maxi,j∈N aij . Then A is λ-robust if and only if per(A, λ) = 1. It was proved in [5] that a matrix is robust (called stationary in [5] or strongly stable in [3]) if and only if the greatest common divisor of the lengths of all cycles in any non-trivial strongly connected component of the threshold digraph G (A(h) ) is 1 (per G (A(h) ) = 1 for any value of h ∈ {aij ; ∈ j ∈ N }). Note that an O(n3 ) algorithm for finding per A is presented in [7]. 3. X -robustness of matrices The robustness of a matrix A means that the members of O(A, x) achieve the eigenspace of A starting at arbitrary initial vector x ∈ B(n). In the following we shall deal with the so-called X -robustness of matrix which requires achieving the eigenspace starting at each vector from a given interval vector X . In this section we shall deal with vectors with interval elements. Similarly to [6,8,9,11–15] we define interval vector X .

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Definition 3.1. Let x, x ∈ B(n) be given with x ≤ x. An interval vector X = (x1 , . . . , xn )T is defined as follows X = [x, x] =

{

x ∈ B(n); x ≤ x ≤ x

}

and xi = [xi , xi ] for i ∈ N. Definition 3.2. Let A ∈ B(n, n) and let X be a given interval vector. A is called X -robust if X ⊆ attr(A). Corollary 3.1. Let A ∈ B(n, n) and an interval vector X be given. The following assertions are equivalent (i) A is X -robust, (ii) for each x ∈ X there exists k ∈ N such that Ak ⊗ x ∈ V (A), (iii) per(A, x) = 1 for each x ∈ X . Corollary 3.2. If per A = 1 then A is X -robust. For given index i ∈ N we define vector x(i) by putting for every k ∈ N (i) xk

{ =

xi , xk ,

for k = i otherwise

(1)

The vector x(i) will be called a generator of X . Lemma 3.1. xi ≤ β i ≤ xi .

Let x ∈ B(n) and A ∈ B(n, n). Then x ∈ X if and only if x =

⨁

i∈N

βi ⊗ x(i) for some values βi ∈ B with

Proof. For the proof of statement, let us suppose that x ∈ X , i.e., the inequalities xi ≤ xi ≤ xi hold for every i ∈ N. Denoting βi = xi we get βi ⊗ xi = xi ⊗ xi = xi and βi ⊗ xi = xi ⊗ xi = xi ≤ xi for every i ∈ N. It can be easily verified that ⨁ β ⊗ x(i) = x. The converse implication is trivial. □ i i∈N Theorem 3.1 ([14]). Let A ∈ B(n, n), X be given. Then A is X -robust if and only if per(A, x(j) ) = 1 for each j ∈ N. 4. X AE -robustness and X EA -robustness If each element of X is associated either with the universal, or with the existential quantifier, then we can split the interval vector as X = X ∀ ⊕ X ∃ , where X ∀ is the interval vector comprising universally quantified coefficients and X ∃ concerns existentially quantified coefficients. Denote the set of indices of X ∀ corresponding with universal quantifier by N ∀ and the set of indices of X ∃ corresponding ∃ ∀ with existential quantifier by N ∃ . In the other words, x∃i = xi = O for each i ∈ N ∀ and x∀i = xi = O for each i ∈ N ∃ . Definition 4.1. A matrix A is called

• X EA -robust if (∃x∃ ∈ X ∃ )(∀x∀ ∈ X ∀ ) per(A, x∃ ⊕ x∀ ) = 1, • X AE -robust if (∀x∀ ∈ X ∀ )(∃x∃ ∈ X ∃ ) per(A, x∀ ⊕ x∃ ) = 1. Let A be a matrix and let X be an interval. Then there is x ∈ X such that per(A, x) = 1 if and only if system ⨁ 2 2 (i) An ⊗ β = An +1 ⊗ β is solvable with β ∈ X whereby x = (see Lemmas 2.2 and 3.1). i∈N βi ⊗ x Notice that for C , D ∈ B(m, n) a polynomial method for solving a general two-sided system C ⊗ y = D ⊗ y of max–min linear equations is presented in [10]. The method finds the maximum solution of the system. Computational complexity of the proposed method is O(mn · min(m, n)). 4.1. X EA -robustness Put x(n+1) := x∀ and N˜ ∀ := N ∀ ∪ {n + 1}. Theorem 4.1.

Let A ∈ B(n, n) and an interval vector X = [x, x] be given. Then A is X EA -robust if and only if

(∃x∃ ∈ X ∃ )(∀i ∈ N˜ ∀ ) per(A, x∃ ⊕ x(i) ) = 1. 2

Proof. Suppose that there is x∃ ∈ X ∃ such that An ⊗ (x∃ ⊕ x(i) ) = An auxiliary interval vector Xˆ = (xˆ 1 , . . . , xˆ n )T as follows:

{ xˆ i =

∃ ∃ [[xi , xi]], xi , xi ,

for i ∈ N ∃ . for i ∈ N ∀

2 +1

⊗ (x∃ ⊕ x(i) ) holds for all i ∈ N˜ ∀ . Define the

(2)

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H. Myšková and J. Plavka / Discrete Applied Mathematics 267 (2019) 142–150

Notice that vectors xˆ (i) of Xˆ have the form

⎧ ⎨ xi , (i) x, xˆ k = ⎩ xi∃ , k

for k = i ∧ i ∈ N ∀ for k ̸ = i ∧ k ∈ N ∀ for k ∈ N ∃ ,

(3)

or equivalently, xˆ (i) = x∃ ⊕ x(i) for each i ∈ N ∀ and xˆ (i) = xˆ = x∃ ⊕ x(n+1) for each i ∈ N ∃ . It is easy to see that A is X EA -robust if and only if A is Xˆ -robust. By Lemma 3.1 an arbitrary vector xˆ ∈ Xˆ is defined as the max–min linear ⨁ 2 n2 ˆ (i) combination xˆ = ⊗ xˆ = An +1 ⊗ xˆ holds for each xˆ ∈ Xˆ . Thus, i∈N ∀ βi ⊗ x . Then we shall prove that the equality A we get n2

A

n2

⊗ xˆ = A ⊗

⨁

⎞ ⎛ ⨁ ⨁ βi ⊗ xˆ (i) ⊕ βi ⊗ xˆ (i) ⎠ = βi ⊗ xˆ (i) = A ⊗ ⎝ n2

i∈N ∃

i∈N

i∈N ∀

⎛ ⎞ ⨁ ⨁ n2 (i) A ⊗⎝ βi ⊗ xˆ ⊕ βi ⊗ xˆ ⎠ = i∈N ∃

i∈N ∀

⎛ ⎞ ⎛ ⎞ ⨁ ⨁ 2 2 ⎝ βi ⊗ An ⊗ xˆ ⎠ ⊕ ⎝ βi ⊗ An ⊗ xˆ (i) ⎠ = i∈N ∃

i∈N ∀

⎞ ⎛ ⎞ ⎛ ⨁ ⨁ 2 2 ⎝ βi ⊗ An ⊗ (x∃ ⊕ x(i) )⎠ = βi ⊗ An ⊗ (x∃ ⊕ x(n+1) )⎠ ⊕ ⎝ i∈N ∀

i∈N ∃

⎞ ⎛ ⎞ ⎛ ⨁ ⨁ 2 +1 2 +1 n ∃ (n + 1) n ∃ (i) ⎝ βi ⊗ A ⊗ (x ⊕ x )⎠ ⊕ ⎝ βi ⊗ A ⊗ (x ⊕ x )⎠ = i∈N ∃

i∈N ∀

⎞

⎛ An

2 +1

⊗⎝

⨁

βi ⊗ xˆ (i) ⊕

i∈N ∃

⨁

2 +1

βi ⊗ xˆ (i) ⎠ = An

⊗ xˆ .

i∈N ∀

The reverse implication is trivial.

□

W.l.o.g. suppose that N ∃ = {1, 2, . . . , k} and N ∀ = {k + 1, k + 2, . . . , n}, i.e., X ∃ = ([x1 , x1 ], . . . , [xk , xk ], [O, O] . . . , [O, O])T and X ∀ = ([O, O] . . . , [O, O], [xk+1 , xk+1 ], . . . , [xn , xn ])T .

ˆ , k + j) for k + j ∈ N˜ ∀ and D(r) ˆ Define the matrix D(r as follows ˆ , k + j) = (Ar ⊗ x(1) D(r

...

Ar ⊗ x(k)

O

...

Ar ⊗ x(k+j)

O

O

...

O),

ˆ ˆ , k + 1), . . . , D(r ˆ , n + 1))T = D(r) = (D(r

⎛

Ar ⊗ x(1) ⎜ Ar ⊗ x(1) ⎜

⎜ ⎝

Theorem 4.2.

.. . Ar ⊗ x(1)

... ...

Ar ⊗ x(k) Ar ⊗ x(k)

Ar ⊗ x(k+1) O

O Ar ⊗ x(k+2)

O O

... ...

O O

...

Ar ⊗ x(k)

O

O

O

...

Ar ⊗ x(n+1)

⎞ ⎟ ⎟ ⎟. ⎠

Let A and X = [x, x] be given. Then

ˆ 2 ) ⊗ β = D(n ˆ 2 + 1) ⊗ β (∃x∃ ∈ X ∃ )(∀j ∈ N˜ ∀ ) per(A, x∃ ⊕ x(j) ) = 1 ⇔ D(n 2

2 +1

is solvable with xi ≤ βi ≤ xi for i ∈ N ∃ and maxs∈N ((An ⊕ An Proof. By Lemma 3.1, if xi ≤ βi ≤ xi for i ∈ N ∃ , then x∃ = ⨁k (i) xi ≤ βi ≤ xi for i ∈ N ∃ such that x∃ = i=1 βi ⊗ x .

⨁k

) ⊗ x(j) )s ≤ βj ≤ I for j ∈ N˜ ∀ .

i=1

βi ⊗ x(i) belongs to X ∃ , and if x∃ ∈ X ∃ , then we can find

H. Myšková and J. Plavka / Discrete Applied Mathematics 267 (2019) 142–150

147

ˆ 2 ) ⊗ β = D(n ˆ 2 + 1) ⊗ β is solvable with xi ≤ βi ≤ xi for i ∈ N ∃ and We also have that the system D(n n2 n2 +1 (j) ∀ maxs∈N ((A ⊕ A ) ⊗ x )s ≤ βj ≤ I for j ∈ N˜ if and only if the following equivalences hold true: ˆ 2 ) ⊗ β = D(n ˆ 2 + 1) ⊗ β ⇔ D(n

⎡ ⨁ 2 2 (∀j ∈ N˜ ∀ ) ⎣ (An ⊗ x(i) ⊗ βi ) ⊕ (An ⊗ x(j) ⊗ βj ) = i∈N ∃

⎤ ⨁

n2 + 1

(A

⊗ x ⊗ βi ) ⊕ (A (i)

n2 +1

⊗ x ⊗ βj ) ⎦ ⇔ (j)

i∈N ∃

(∀j ∈ N˜ ∀ ) ⎣A

⎞

⎛

⎡ n2

⊗⎝

⨁

x(i) ⊗ βi ⊕ x(j) ⎠ =

i∈N ∃

⎛ 2 +1

An

⊗⎝

⎞⎤ ⨁

x(i) ⊗ βi ⊕ x(j) ⎠⎦ ⇔

i∈N ∃ 2

(∃x ∈ X )(∀j ∈ N˜ ∀ )[An ⊗ (x∃ ⊕ x(j) ) = An ∃

∃

2 +1

⊗ (x∃ ⊕ x(j) )]. □

Theorem 4.3. Suppose a matrix A and an interval vector X = [x, x] are given. The recognition problem of whether A is X EA -robust is solvable in O(n4 ) time. Proof. According to Theorem 4.2, the recognition problem of X EA -robustness of A is equivalent to recognizing whether ˆ 2 ) ⊗ β = D(n ˆ 2 + 1) ⊗ β is solvable with xi ≤ βi ≤ xi for i ∈ N ∃ and maxs∈N ((An2 ⊕ An2 +1 ) ⊗ x(j) )s ≤ βj ≤ I system D(n for j ∈ N˜ ∀ . The computation of a system C ⊗ y = D ⊗ y needs O(mn · min(m, n)) time (see [10]), where C , D ∈ B(m, n). ˆ 2 ) ⊗ β = D(n ˆ 2 + 1) ⊗ β is done in O(n2 · n · n) = O(n4 ) time. □ Therefore, the computation of D(n 4.2. X AE -robustness In this section we shall deal with X AE -robustness of fuzzy matrices for which we suggest efficient equivalent conditions. Definition 4.2. 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 (VG (A) , EG (A) ) with VG (A) = N, EG (A) = {(i, j); aij = I} and by G (A, x) we understand the corresponding node-weighted digraph obtained from G (A) by appending weight xi to each node i. A path in G (A, x) is called an orbit path if the weight of its terminal node is I. Definition 4.3. 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:

{ (A(h) )ij =

I O

if aij ≥ h, otherwise.

(4)

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 is equal to the threshold vector x(r)(h) . Similarly, the threshold coordinate-orbit Oi (A(h) , x(h) ) is scalar sequence (xi (r)(h) ; r ∈ N0 ). For given A ∈ B(n, n), x ∈ B(n), denote H(A) = {aij ; i, j ∈ N } and H(A, x) = H(A) ∪ {xi ; xi ≤ max H(A)}. Theorem 4.4 ([4]). Let A ∈ B(n, n), x ∈ B(n), O < h ∈ B, r ∈ N and i, j ∈ N be given. Then (i) (Ar )ij ≥ h if and only if there is a path in G (A(h) ) from i to j of length r, (ii) Oi (A, x)(r) ≥ h if and only if there is an orbit path in G (A(h) , x(h) ) starting at i of length r. per

Denote by Oper (A, x) and Oi (A, x) the periodic part of O(A, x) and Oi (A, x), respectively, i.e., per

Oper (A, x) = (x(r); r > def(A, x)) ∧ Oi

(A, x) = (xi (r); r > def(A, x, i)).

By (O) and (I) we understand the infinite sequences of the same elements O and I, respectively.

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H. Myšková and J. Plavka / Discrete Applied Mathematics 267 (2019) 142–150 (r)

For any r ∈ N and i, j ∈ N by Pij (A) denote the set of all paths of length r in G (A) with initial node i and terminal node j. The set of all paths in G (A) with initial node i and terminal node j will be denoted by Pij (A). Similar notation is used for the set of orbit paths in G (A, x): (r)

Pi (A, x) = {P ; P is orbit path of length r beginning at i}. (r)

Theorem 4.5 ([19]). Let A be a binary matrix, x be a binary vector. Then per(A, x) = 1 ⇔ (∀i ∈ N)(∀r ≥ def(A, x))[Pi (A, x) ̸ = ∅]. per

Lemma 4.1. Let A ∈ B(n, n), A ̸ = O be a binary matrix, x ∈ B(n) be a binary vector and i ∈ N. Then Oi (A, x) ̸ = (O) if and only if there exists a non-elementary orbit path p in G (A, x) beginning at i. Proof. Suppose that p is a non-elementary orbit path p in G (A, x) beginning at i and ending at j, i.e., p = (i, v1 , . . . , vt , vt +1 . . . , vt +s , vt , vt +s+1 , . . . , vt +s+r , j) where c = (vt , vt +1 . . . , vt +s ) is a cycle with ℓ(c) = s ≥ 1. Then there exist orbit paths with length |P | + ks for each k ∈ N. According to Theorem 4.4(ii) xi (r) = I for each per r ∈ {|P | + ks}, so Oi (A, x) ̸ = (O). For the converse implication suppose that there is no non-elementary orbit path from i. Then there is no orbit path per from i of length k ≥ n, so Oi (A, x) = (O). □ per

Lemma 4.2. Let A ∈ B(n, n), A ̸ = O be a binary matrix, x ∈ B(n) be a binary vector and i ∈ N. Then Oi (A, x) ̸ = (I) if and (r) only if for any k ∈ N, k ≥ def(A, x, i) there exists r ≥ k such that Pi (A, x) = ∅. Proof. The equivalence follows from the definition of the orbit, Theorems 4.4 and 4.5. □ Theorem 4.6 ([14]). Let A ∈ B(n, n), x ∈ B(n) be given and i ∈ N. Then per(A, x) = 1 if and only if per(A(h) , x(h) , i) = 1 for each i ∈ N and for each h ∈ H(A, x). Theorem 4.7.

Suppose a matrix A and an interval vector X = [x, x] are given. Then A is X AE -robust if and only if

(∀i ∈ N˜ ∀ )(∃z i ∈ X ∃ ) per(A, z i ⊕ x(i) ) = 1. Proof. Suppose that A is not X AE -robust, i.e., (∃x∀ ∈ X ∀ )(∀x∃ ∈ X ∃ ) per(A, x∀ ⊕ x∃ ) ̸ = 1. By Theorem 4.6 there exist i ∈ N and the value h ∈ H(A, x∀ ⊕ x∃ ) such that per(A(h) , (x∀ ⊕ x∃ )(h) , i) ̸ = 1. We shall prove that there exists j ∈ N˜ ∀ such that per(A(h) , (x(j) ⊕ x∃ )(h) , i) ̸ = 1. Observe that if per(A(h) , (x∀ ⊕ x∃ )(h) , i) ̸ = 1, then per

Oi

per

(A(h) , (x∀ ⊕ x∃ )(h) ) ̸ = (O) ∧ Oi (A(h) , (x∀ ⊕ x∃ )(h) ) ̸ = (I). per

By Lemma 4.1 if Oi (A(h) , (x∀ ⊕ x∃ )(h) ) ̸ = (O), then there exists a non-elementary orbit path p = (i, v1 , . . . , vt , vt +1 . . . , vt +s , vt , vt +s+1 , . . . , vt +s+r , j) in G (A(h) , (x∀ ⊕ x∃ )(h) ) beginning at i and ending at j. (j) If j ∈ N ∀ consider the vector x(j) . It is clear that x(h) = I and p is again non-elementary orbit path in G (A(h) , (x(j) ⊕ x∃ )(h) ) per beginning at i and ending at j. Hence, by Lemma 4.1 the inequality Oi (A(h) , (x(j) ⊕ x∃ )(h) ) ̸ = (O) holds true. (j) If j ∈ N ∃ , x(h) = I then p is non-elementary orbit path in G (A(h) , (x(n+1) ⊕ x∃ )(h) ) beginning at i and ending at j. Hence, per the inequality Oi (A(h) , (x(n+1) ⊕ x∃ )(h) ) ̸ = (O) is fulfilled. per If Oi (A(h) , (x∀ ⊕ x∃ )(h) ) ̸ = (I), then by Lemma 4.2 for any k ∈ N, k ≥ def(A, x, i) there exists r ≥ k such that (r) Pi (A(h) , (x∀ ⊕ x∃ )(h) ) = ∅. Moreover, the inclusion (r)

(r)

Pi (A(h) , (x(j) ⊕ x∃ )(h) ) ⊆ Pi (A(h) , (x∀ ⊕ x∃ )(h) ) (r)

per

which holds true for each j ∈ N˜ ∀ implies Pi (A(h) , (x(j) ⊕ x∃ )(h) ) = ∅. Hence Oi (A(h) , (x(j) ⊕ x∃ )(h) ) ̸ = (I). per per We have shown that Oi (A(h) , (x(j) ⊕ x∃ )(h) ) ̸ = (O) and Oi (A(h) , (x(j) ⊕ x∃ )(h) ) ̸ = (I), which implies per(A(h) , (x(j) ⊕ x∃ )(h) , i) ̸ = 1. The reverse implication is trivial. □

ˆ 2 , i) ⊗β = D(n ˆ 2 + 1, i) ⊗β Theorem 4.8. Let A, X be given. Then A is X AE -robust if and only if for each i ∈ N˜ ∀ the equation D(n ∃ n2 n2 +1 (i) ) ⊗ x )s ≤ βi ≤ I. is solvable with xj ≤ βj ≤ xj for j ∈ N and maxs∈N ((A ⊕ A

H. Myšková and J. Plavka / Discrete Applied Mathematics 267 (2019) 142–150

Proof. By Lemma 3.1, if xj ≤ βj ≤ xj for j ∈ N ∃ then x∃ =

⨁k

⨁k

j=1

149

βj ⊗ x(j) belongs to X ∃ , and if x∃ ∈ X ∃ then we can find

j=1 βj ⊗ x . ∀ ˜ ˆ 2 , i) ⊗ β = D(n ˆ 2 + 1, i) ⊗ β is solvable with xj ≤ βj ≤ xj for j ∈ N ∃ We also have that for each i ∈ N the equation D(n n2 n2 +1 (i) and maxs∈N ((A ⊕ A ) ⊗ x )s ≤ βi ≤ I if and only if the following equivalences for an arbitrary i ∈ N˜ ∀ hold true:

xj ≤ βj ≤ xj for j ∈ N such that x = ∃

∃

(j)

ˆ 2 , i) ⊗ β = D(n ˆ 2 + 1, i) ⊗ β ⇔ D(n 2

⨁

2

(An ⊗ x(j) ⊗ βj ) ⊕ (An ⊗ x(i) ⊗ βi ) =

j∈N ∃

⨁

(An

2 +1

2 +1

⊗ x(j) ⊗ βj ) ⊕ (An

⊗ x(i) ⊗ βi ) ⇔

j∈N ∃

⎞ ⎛ ⨁ 2 (x(j) ⊗ βj ) ⊕ x(i) ⎠ ⇔ (x(j) ⊗ βj ) ⊕ x(i) ⎠ = An +1 ⊗ ⎝ ⎞

⎛ 2

An ⊗ ⎝

⨁

j∈ N ∃

j∈N ∃

∃

∃

(∃x ∈ X )[A

n2

(i)

∃

⊗ (x ⊕ x ) = A

n2 +1

⊗ (x ⊕ x(i) )]. □ ∃

Theorem 4.9. Suppose a matrix A and an interval vector X = [x, x] are given. The recognition problem of whether A is X AE -robust is solvable in O(n4 ) time. Proof. According to Theorem 4.8, the recognition problem of X AE -robustness of A is equivalent to recognizing whether ˆ 2 , i) ⊗ β = D(n ˆ 2 + 1, i) ⊗ β is solvable with xj ≤ βj ≤ xj for j ∈ N ∃ and for each i ∈ N˜ ∀ the system D(n 2

2 +1

maxs∈N ((An ⊕ An

) ⊗ x(i) )s ≤ βi ≤ I. The computation of a system

ˆ 2 , i) ⊗ β = D(n ˆ 2 + 1, i) ⊗ β D(n needs O(n3 ) time (see [10]). Therefore, the computation of at most n such systems is done in n · O(n3 ) = O(n4 ) time.

□

Example 4.1. Suppose that B = [0, 10], consider matrix A and interval vector X which have the forms

⎛

0 ⎜ 1 A=⎝ 2 1

1 0 1 2

⎞

2 1 0 1

1 2 ⎟ , 1 ⎠ 0

⎛

⎞

⎞

⎛

1 ⎜ 1 ⎟ x=⎝ , 1 ⎠ 0

2 ⎜ 3 ⎟ x=⎝ 2 ⎠ 1

with N ∃ = {1, 2}, N ∀ = {3, 4} and N˜ ∀ = {3, 4, 5}. Then

⎛

x(3)

⎞

0 ⎜ 0 ⎟ =⎝ , 2 ⎠ 0

⎛

x(4)

⎞

0 ⎜ 0 ⎟ =⎝ , 1 ⎠ 1

⎛

x(5)

⎞

0 ⎜ 0 ⎟ =⎝ . 1 ⎠ 0

and attr(A) = {x ∈ X ; A16 ⊗ y = A17 ⊗ y}, where

⎛

A16

2 ⎜ 1 =⎝ 1 1

1 2 1 1

1 1 2 1

⎞

1 1 ⎟ , 1 ⎠ 2

⎛

A17

1 ⎜ 1 =⎝ 2 1

1 1 1 2

2 1 1 1

⎞

1 2 ⎟ . 1 ⎠ 1

Then for x(3) there is z 3 ∈ X ∃ , z 3 = (2, 1, 0, 0)T , for x(4) there is z 4 ∈ X ∃ , z 4 = (1, 1, 0, 0)T and for x(5) there is also z 5 ∈ X ∃ , z 5 = (1, 1, 0, 0)T such that A16 ⊗ (x(i) ⊕ z i ) = A17 ⊗ (x(i) ⊕ z i ) holds for i = 3, 4, 5. Hence by Theorem 4.8 A is X AE -robust. On the other hand by Theorem 4.2 we will show that A is not X EA -robust. The equality

ˆ ˆ D(16) ⊗ β = D(17) ⊗β

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with xj ≤ βj ≤ xj for j ∈ N ∃ and maxs∈N ((A16 ⊕ A17 ) ⊗ x(i) )s ≤ βi ≤ I for i ∈ N˜ ∀ has the form

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

2 1 1 1 2 1 1 1 2 1 1 1

1 2 1 1 1 2 1 1 1 2 1 1

1 1 2 1 0 0 0 0 0 0 0 0

0 0 0 0 1 1 1 1 0 0 0 0

0 0 0 0 0 0 0 0 1 1 1 1

⎞ ⎟ ⎟ ⎟ ⎟ ⎛ ⎟ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟⊗⎜ ⎟ ⎝ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

⎛

β1 β2 β3 β4 β5

⎜ ⎜ ⎜ ⎞ ⎜ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟=⎜ ⎠ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

1 1 2 1 1 1 2 1 1 1 2 1

1 1 1 2 1 1 1 2 1 1 1 2

2 1 1 1 0 0 0 0 0 0 0 0

0 0 0 0 1 1 1 1 0 0 0 0

0 0 0 0 0 0 0 0 1 1 1 1

⎞ ⎟ ⎟ ⎟ ⎟ ⎛ ⎟ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟⊗⎜ ⎟ ⎝ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

β1 β2 β3 β4 β5

⎞ ⎟ ⎟ ⎟ ⎠

(5)

with 1 ≤ β1 ≤ 2; 1 ≤ β2 ≤ 2; 2 ≤ β3 ≤ 10; 1 ≤ β4 ≤ 10; 1 ≤ β5 ≤ 10. Using the algorithm presented in [10] it is easy to see that the greatest solution of the system (5) is the vector β = (1, 1, 1, 10, 10)T . Since β3 = 1 ∈ / [2, 10] we can conclude that the system is not solvable for the given constraints. Thus, A is not X EA -robust. Acknowledgment The support of the APVV grant 18-0373 is gratefully acknowledged. References [1] P. Butkovič, Max-Linear Systems: Theory and Algorithms, Springer Monographs in Mathematics, Springer-Verlag, 2010. [2] P. Butkovič, R.A. Cuninghame-Green, S. Gaubert, Reducible spectral theory with applications to the robustness of matrices in max-algebra, SIAM J. Matrix Anal. Appl. 21 (2009) 1412–1431. [3] P. Butkovič, H. Schneider, S. Sergeev, Recognizing weakly stable matrices, SIAM J. Control Optim. 50 (5) (2012) 3029–3051. [4] K. Cechlárová, Eigenvectors in bottleneck algebra, Linear Algebra Appl. 175 (1992) 63–73. [5] K. Cechlárová, On the powers of matrices in bottleneck/fuzzy algebra, Linear Algebra Appl. 246 (1996) 97–111. [6] M. Fiedler, J. Nedoma, J. Ramík, J. Rohn, K. Zimmermann, Linear Optimization Problems with Inexact Data, Springer–Verlag, Berlin, 2006. [7] M. Gavalec, Computing matrix period in max–min algebra, Discrete Appl. Math. 75 (1997) 63–70. [8] M. Gavalec, J. Plavka, Monotone interval eigenproblem in max–min algebra, Kybernetika 46 (3) (2010) 387–396. [9] M. Gavalec, K. Zimmermann, Classification of solutions to systems of two-sided equations with interval coefficients, Int. J. Pure Appl. Math. 45 (2008) 533–542. [10] M. Gavalec, K. Zimmermann, Solving systems of two–sided (max, min)–linear equations, Kybernetika 46 (2010) 405–414. [11] H. Myšková, Interval eigenvectors of circulant matrices in fuzzy algebra, Acta Electrotech. Inform. 12 (3) (2012) 57–61. [12] H. Myšková, Robustness of interval toeplitz matrices in fuzzy algebra, Acta Electrotech. Inform. 12 (4) (2012) 56–60. [13] H. Myšková, Weak stability of interval orbits of circulant matrices in fuzzy algebra, Acta Electrotech. Inform. 12 (3) (2012) 51–56. [14] H. Myšková, J. Plavka, X -Robustness of interval circulant matrices in fuzzy algebra, Linear Algebra Appl. 438 (2013) 2757–2769. [15] J. Plavka, On the O(n3 ) algorithm for checking the strong robustness of interval fuzzy matrices, Discrete Appl. Math. 160 (2012) 640–647. [16] J. Plavka, On the weak robustness of fuzzy matrices, Kybernetika 49 (1) (2013) 128–140. [17] J. Plavka, P. Szabó, On the λ-robustness of matrices over fuzzy algebra, Discrete Appl. Math. 159 (5) (2011) 381–388. [18] E. Sanchez, Resolution of eigen fuzzy sets equations, Fuzzy Sets and Systems 1 (1978) 69–74. [19] B. Semančíková, Orbits in max–min algebra, Linear Algebra Appl. 414 (2006) 38–63. [20] S. Sergeev, Max-algebraic cones of nonnegative irreducible matrices, Linear Algebra Appl. 435 (2011) 1736–1757. [21] L.A. Zadeh, Toward a theory of fuzzy systems, in: R.E. Kalman, N. De Claris (Eds.), Aspects of Network and Systems Theory, Hold, Rinehart and Winston, New York, 1971, pp. 209–245.