Tsallis Entropy of Product MV-Algebra Dynamical Systems.

This paper is concerned with the mathematical modelling of Tsallis entropy in product MV-algebra dynamical systems. We define the Tsallis entropy of order α , where α ∈ ( 0 , 1 ) ∪ ( 1 , ∞ ) , of a partition in a product MV-algebra and its conditional version and we examine their properties. Among other, it is shown that the Tsallis entropy of order α , where α > 1 , has the property of sub-additivity. This property allows us to define, for α > 1 , the Tsallis entropy of a product MV-algebra dynamical system. It is proven that the proposed entropy measure is invariant under isomorphism of product MV-algebra dynamical systems.

1. Introduction

The Shannon entropy [1] is the foundational concept of information theory (cf. [2]). We remind readers that if an experiment has k outcomes with probabilities then its Shannon entropy is defined as the sum where is Shannon’s entropy function defined by: for every ( is defined to be 0). Many years later, the Shannon entropy was exploited surprisingly in a completely different field, namely, in dynamical systems. Recall that by a dynamical system in the sense of classical probability theory, we understand a system where is a probability space, and is a measure preserving transformation. The entropy of dynamical systems was introduced by Kolmogorov and Sinai [3,4] as an invariant for distinguishing them. Namely, if two dynamical systems are isomorphic, then they have the same entropy. In this way Kolmogorov and Sinai showed the existence of non-isomorphic Bernoulli shifts.

The successful using the Kolmogorov-Sinai entropy of dynamical systems has led to an intensive study of alternative entropy measures of dynamical systems. We note that in Reference [5], the concept of logical entropy of a dynamical system was introduced and studied. It has been shown that by replacing Shannon’s entropy function by the function defined by: for every we obtain the results analogous to the case of Kolmogorov-Sinai entropy theory. The logical entropy is invariant under isomorphism of dynamical systems; so it can be used as an alternative instrument for distinguishing them. For some other recently published results regarding the logical entropy measure, we refer, for example, to References [6,7,8,9,10,11,12,13,14,15].

In fact, all of the above mentioned studies are possible in the Kolmogorov probability theory based on the modern integration theory. This allows us to describe and study some of the problems associated with uncertainty. In Reference [16], Zadeh presented another approach to uncertainty when he introduced the concept of a fuzzy set. Whereas the Kolmogorov probability applications are based on objective measurements, the Zadeh fuzzy set theory is based on subjective improvements. One of the first Zadeh papers on the fuzzy set theory was devoted to probability of fuzzy sets (cf. [17]), and therefore, the entropy of fuzzy dynamical systems has also been studied (cf. [18,19,20,21]). We recall that the fuzzy set is a mapping hence, the fuzzy partition of is a family of fuzzy sets such that Again we can meet the Shannon formula: where (cf. [21]).

Anyway the most useful tool for describing multivalued processes is an MV-algebra [22], especially after its Mundici’s characterization as an interval in a lattice ordered group (cf. [23,24]). At present, this structure is being investigated by many researchers, and it is natural that there are results also regarding the entropy in this structure; we refer, for instance, to References [25,26]. A probability theory was also investigated on MV-algebras; for a review see Reference [27]. Of course, in some probability problems it is needed to introduce a product on an MV-algebra, an operation outside the corresponding group addition. The operation of a product on an MV-algebra was introduced independently by Riečan [28] from the point of view of probability and Montagna [29] from the point of view of mathematical logic. We note that the notion of product MV-algebra generalizes some families of fuzzy sets; an example of product MV-algebra is a full tribe of fuzzy sets (see e.g., [30]).

The appropriate entropy theory of Shannon and Kolmogorov-Sinai type for the product MV-algebras was created in References [31,32]. The logical entropy, the logical divergence and the logical mutual information of partitions in a product MV-algebra were studied in Reference [8]. In the present paper, we extend the study of entropy in product MV-algebras to the case of Tsallis entropy.

The concept of Tsallis entropy was introduced in 1988 by Constantino Tsallis [33] as a base for generalizing the usual statistical mechanics. In its form it is identical with the Havrda-Charvát structural —entropy [34], introduced in 1967 in the framework of information theory. If is a probability distribution, then its Tsallis entropy of order where is defined as the number:

The entropic index describes the deviation of Tsallis entropy from the standard Shannon one. Evidently, if we define, for the function by:for every then the Formula (3) can be written in the following form:

Putting in Equation (5), we obtain: which is the logical entropy of a probability distribution defined and studied in Reference [6].

The Tsallis entropy is the most important quantity among Tsallis’ statistics, which form the foundation of nonextensive statistical mechanics of complex systems; for more details, see Reference [35]. The Tsallis statistics are used to describe systems exhibiting long-range correlations, memory, or fractal properties; their applications have been found for a wide range of phenomena in diverse disciplines such as physics, geophysics, chemistry, biology, economics, medicine, etc. [36,37,38,39,40,41,42,43,44,45,46,47,48]. They are also applicable to large domains in communication systems (cf. [49]).

In this article we continue studying entropy in a product MV-algebra, by defining and studying the Tsallis entropy of a partition in a product MV-algebra and the Tsallis entropy of product MV-algebra dynamical systems. The rest of the paper is structured as follows. In the following section, preliminaries and related works are given. Our main results are discussed in Section 3, Section 4 and Section 5. In Section 3, we define and study the Tsallis entropy of a partition in a product MV-algebra. In Section 4, we introduce the concept of conditional Tsallis entropy of partitions in a product MV-algebra and examine its properties. It is shown that the proposed definitions of Tsallis entropy are consistent, in the case of the limit of going to 1, with the Shannon entropy of partitions studied in Reference [31]. Section 5 is devoted to the study of Tsallis entropy of product MV-algebra dynamical systems. It is proven that the suggested entropy measure is invariant under isomorphism of product MV-algebra dynamical systems. The last section contains a brief summary.

2. Preliminaries

We begin with recalling the definitions of the basic notions and some known results used in the paper. For defining the notion of MV-algebra, various (but of course equivalent) axiom systems were used (see e.g., [28,50,51]). In this paper, the definition of MV-algebra given by Riečan in Reference [52] is used, which is based on the Mundici representation theorem [23,24]. In view of the Mundici theorem, any MV-algebra may be considered to be an interval of an abelian lattice ordered group. Recall that by an abelian lattice ordered group [53], we mean a triplet where is an abelian group, is a partially ordered set being a lattice and, for every

([52]). An MV-algebra is an algebraic structure

there exists an abelian lattice ordered group 0 is the neutral element of

and are binary operations on A satisfying the following identities:

([27]). A state on an MV-algebra

ifsuch thatthen

([28]). A product MV-algebra is an algebraic structure where is an MV-algebra and ⋅ is an associative and abelian binary operation on A with the following properties:

for every

ifsuch thatthenand

In the following text, we will briefly write instead of A relevant probability theory for the product MV-algebras was developed by Riečan in Reference [54]; the entropy theory of Shannon and Kolmogorov-Sinai type for the product MV-algebras was proposed in References [31,32]. The logical entropy theory for the product MV-algebras was proposed in Reference [8]. We present the main idea and some results of these theories that will be used in the following text.

By a partition in a product MV-algebra we mean a k-tuple of (not necessarily different) members of A satisfying the condition Let and be two partitions in We say that is a refinement of and we write if there exists a partition of the set such that for Further, we define the join of and as an r-tuple (where consisting of the members Since the r-tuple is a partition in It represents an experiment consisting of the realization of and

Consider a probability space and define where stands for the indicator function of the set The family A is closed under the product of indicator functions and it is a special case of product MV-algebras. The map defined, for every of by is a state on the considered product MV-algebra Evidently, if is a measurable partition of then the k-tuple is a partition in the product MV-algebra

Consider a probability space and the family A of all -measurable functions so called full tribe of fuzzy sets (cf., e.g., [ defined, for every by the formula is a state on the product MV-algebra The concept of a partition in the product MV-algebra Tcoincides with the notion of a fuzzy partition (cf. [

The definition of entropy of Shannon type of a partition in a product MV-algebra was introduced in [31] as follows.

Let be any partition in a product MV-algebra and be a state. Then the entropy of with respect to is defined by:

If and are two partitions in then the conditional entropy of given is defined by: where:

The conditional entropy of given is defined by:

It is used the standard convention that if . The basis of the logarithm may be any positive real number, but as a rule logarithms to the basis 2 are taken; the entropy is then expressed in bits. If the natural logarithms are taken in the definition, then the entropy is expressed in nats. The entropy of partitions in a product MV-algebra possesses properties corresponding to properties of Shannon’s entropy of measurable partitions; more details can be found in Reference [31].

The definition of logical entropy of a partition in a product MV-algebra was introduced in Reference [8] as follows.

Let be a partition in a product MV-algebra and be a state. Then the logical entropy of with respect to is defined by: where is the logical entropy function defined by Equation (2). If and are two partitions in then the conditional logical entropy of given is defined by:

3. The Tsallis Entropy of Partitions in a Product MV-Algebra

We begin this section with the definition of Tsallis entropy of a partition in a product MV-algebra and then we will examine its properties. In the following, we will suppose that is a state.

Letbe a partition in a product MV-algebraThen we define the Tsallis entropy of orderwhereof the partitionwith respect to by:

Let us consider the function defined by Equation (4). Since the formula (11) can be expressed in the following form:

Evidently, if we put the logical entropy is obtained. It is possible to verify that the function is, for every a non-negative function. Namely, if then we have for every hence for every On the other hand, for we have for every hence for every

Let be any product MV-algebra. Let us consider the partition representing an experiment resulting in a certain event. Then for every partition X in and

Let be any partition in a product MV-algebra Then: The equality holds if and only if the state s is uniform over X, i.e., if and only if for

The inequality follows from the non-negativity of function so it is sufficient to prove the second assertion. We will use the Jensen inequality. It is easy to verify that the function is concave, therefore, applying the Jensen inequality, we have:with the equality if and only if Since it follows that: The equality holds if and only if i.e., if and only if for  ☐

The following propositions will be needed for the proofs of our results.

Let be partitions in a product MV-algebra Then:

implies

For the proof, see Reference [8]. ☐

Let be partitions in a product MV-algebra such that and Then

Let Then there exists a partition of the set such that for and there exists a partition of the set such that for Put for We get:for what means that  ☐

Let be a partition in a product MV-algebra and be a state. Then:

for every

for every with

For the proof of the claim (i), see [8]. If with then using the previous equality, we obtain:

Let be partitions in a product MV-algebra such that Then

Suppose that Then there exists a partition of the set such that for Hence for Consider the case when Then:for Summing both sides of the above inequality over i, we get:

In this case we have hence: The case of can be obtained in the same way. ☐

As an immediate consequence of the previous theorem and Proposition 1, we obtain the following result.

For every partitions in a product MV-algebra it holds:

Let and be partitions in a product MV-algebra Then, for it holds:

Applying the Jensen inequality, we have:for and consequently: The assumption that implies the inequality for The function is non-negative, therefore, for we get:and consequently: The last inequality combined with (13) yields the claim. ☐

Let be partitions in a product MV-algebra Then, for it holds:

Suppose that Let us calculate: In the last step we used Proposition 4. ☐

Let us consider the family A of all Borel measurable functions and define in A the operation as the natural product of fuzzy sets. Then the system is a product MV-algebra. In addition, we define a state by the formula for every and consider the pairs where for every Evidently, X and Y are partitions in the product MV-algebra By elementary calculations we get that they have the state values and of the corresponding elements, respectively. The partition has the state values of the corresponding elements. We want to find out whether the statement of the previous theorem is true in the case under consideration. Using the formula (11), it can be computed that It holds and which is consistent with the assertion of Theorem 3. Put We obtain: It can be seen that The result means that the Tsallis entropy of order does not have the property of sub-additivity.

One of the most important properties of Shannon entropy is additivity. In the following theorem it is shown that the Tsallis entropy does not have the property of additivity; it satisfies the following weaker property of pseudo-additivity.

If partitions in a product MV-algebra are statistically independent with respect to i.e., for every and then:

Suppose that Let us calculate:

In the last part of this section, we will prove the concavity of Tsallis entropy on the family of all states defined on a given product MV-algebra

Let be two states defined on a common product MV-algebra Then, for every real number the map is a state on

The proof is simple, so it is omitted. ☐

Let be two states defined on a common product MV-algebra Then, for every partition in a product MV-algebra and for every real number the following inequality holds:

Assume that The function is concave, therefore, for every real number we get:

As a consequence of Theorem 5, we get the concavity of the logical entropy as a function of s. The result of the previous theorem is illustrated in the following example.

Consider the product MV-algebra from Example 4 and the real functions defined by the equalities for every real number We define two states by the formulas for every of Further, we consider the partition in By simple calculation we get that it has the -state values of the corresponding elements, and the -state values of the corresponding elements. In the previous theorem we put We will show that, for the chosen the following inequality holds:

Put We calculated that and One can easily check that in this case:

For the case of i.e., for the logical entropy, we get: and for the case of we obtain: One can easily check that in both cases the inequality (14) holds.

4. The Conditional Tsallis Entropy of Partitions in a Product MV-Algebra

In this section we introduce and study the concept of conditional Tsallis entropy of partitions in a product MV-algebra

Let and be partitions in a product MV-algebra We define the conditional Tsallis entropy of order where of X given Y as the number:

Evidently, if we put

At the value of is undefined because it gives the shape In the following theorem it is shown that for the conditional Tsallis entropy tends to the conditional Shannon entropy defined by the formula (8), when the natural logarithm is taken in this formula.

Let and be partitions in a product MV-algebra Then:

For every we have:where and are continuous functions defined, for every by the equalities: The functions and are differentiable and evidently, Also, it can easily be verified that Indeed, by Proposition 3, we get:

Using L’Hôpital’s rule, it follows that under the assumption that the right hand side exists. It holds and:

It follows that:

Let be any partition in a product MV-algebra and Then:

Let be any partition in a product MV-algebra Then:

The statement is an immediate consequence of the previous theorem; it suffices to put  ☐

For arbitrary partitions in a product MV-algebra it holds:

Let

By Proposition 3, it holds for hence, we can write:

Suppose that For we have which implies that for Since for it follows that On the other hand, for it holds for In this case hence

By direct calculations, we have:

The statement is an immediate consequence of the previous property; it suffices to put  ☐

By combining the property (iii) from Theorem 8 with Theorem 4, we obtain the following property of conditional Tsallis entropy .

If partitions in a product MV-algebra are statistically independent with respect to then:

Let be partitions in a product MV-algebra Then, for it holds:

Let Then by the use of the property (iii) from Theorem 8 and Theorem 3, we get:

To illustrate the result of previous theorem, we provide the following example, which is a continuation of Example 4.

Consider the product MV-algebra the state and the partitions from Example 4. We have calculated that By easy calculations we get that It can be seen that and which is consistent with the assertion of Theorem 10. On the other hand, we have and The result means that the conditional Tsallis entropy of order does not have the property of monotonicity.

5. The Tsallis Entropy of Dynamical Systems in a Product MV-Algebra

In this section, we introduce and study the concept of the Tsallis entropy of a dynamical system in a product MV-algebra

([32]). By a dynamical system in a product MV-algebra we understand a system where is a state, and is a map such that and, for every the following conditions are satisfied:

ifthenand

We say also briefly a product MV-algebra dynamical system instead of a dynamical system in a product MV-algebra.

Let be a classical dynamical system. Let us consider the product MV-algebra and the state from Example 1. In addition, let us define the mapping by the equality for every Then the system is a dynamical system in the considered product MV-algebra

Let be a classical dynamical system. Let us consider the product MV-algebra and the state from Example 2. If we define the mapping by the equality for every then it is easy to verify that the system is a dynamical system in the considered product MV-algebra

Let be a dynamical system in a product MV-algebra and be a partition in Put Since according to Definition 8, we have what means that the k-tuple is a partition in

Let be a dynamical system in a product MV-algebra and be partitions in Then

implies

The property (i) follows from the condition (ii) of Definition 8. Suppose that Then there exists a partition of the set such that for Therefore, by the condition (i) from Definition 8, we have: However, this means that  ☐

Define and put for where is the identical mapping. It is obvious that the mapping possesses the properties from Definition 8. Hence, for any non-negative integer the system is a dynamical system in a product MV-algebra

Let be a dynamical system in a product MV-algebra and be partitions in Then, for any non-negative integer the following equalities hold:

Suppose that

Since for any non-negative integer and it holds we obtain:

Based on the same argument, we get:

Let be a dynamical system in a product MV-algebra and be a partition in Then, for the following equality holds:

We use proof by mathematical induction on starting with For the claim holds by the property (iii) of Theorem 8. We suppose that the claim holds for a given integer and we will prove that it holds for By the property (i) of Theorem 11, we get: Therefore, using the property (iii) of Theorem 8 and our inductive hypothesis, we obtain: In conclusion, the claim is obtained by the principle of mathematical induction. ☐

In the following, we will define the Tsallis entropy of a dynamical system First, we define the Tsallis entropy of relative to a partition in Then we remove the dependence on to get the Tsallis entropy of a dynamical system The following proposition will be needed.

Let be a dynamical system in a product MV-algebra and be a partition in Then, for there exists the following limit:

Put for Then the sequence is a sequence of non-negative real numbers with the property for every natural numbers Indeed, by means of sub-additivity of Tsallis entropy for and the property (i) from Theorem 11, we have: The result guarantees (in view of Theorem 4.9, [55]) the existence of  ☐

Letbe a dynamical system in a product MV-algebraandbe a partition inThen we define, forthe Tsallis entropy ofrelative to by:

Consider any dynamical system in a product MV-algebra and the partition Evidently, and

Let be a dynamical system in a product MV-algebra and be a partition in Then, for and for any non-negative integer the following equality holds:

Using Definition 9, we can write:

Let be a dynamical system in a product MV-algebra and be partitions in such that Then, for it holds

Let By Propositions 2 and 6, we have for Therefore, by Theorem 2, we get: Consequently, dividing by and letting we get  ☐

The Tsallis entropy of a dynamical system in a product MV-algebra is defined, for by:

Let be a dynamical system in a product MV-algebra Then, for and every natural number it holds

Let be a partition in Then, for every natural number we have: Hence, we obtain:

On the other hand, by Proposition 1, we have Hence, by Theorem 14, we obtain:

This implies that:

Two product MV-algebra dynamical systems are called isomorphic, if there exists some one-to-one and onto map such that and, for every the following conditions are satisfied: In this case, is said to be an isomorphism.

if then

Let be isomorphic product MV-algebra dynamical systems, and be an isomorphism between them. Then, for the inverse the following properties are satisfied:

for every

if such that then

for every

for every

The map is bijective, therefore, for every there exist such that and

Let Then we get:

Let such that Then and, therefore, we have:

Let Then

Let Then

Let be isomorphic product MV-algebra dynamical systems, and Then:

Let be an isomorphism between dynamical systems Consider a partition in a product MV-algebra Then and therefore, by the condition (ii) of Definition 11, it holds This means that the k-tuple is a partition in a product MV-algebra Moreover, according to the condition (iii) of Definition 11, we have: Hence, using the conditions (iv), and (i) of Definition 11, we get:

Therefore, dividing by and letting we obtain:

This implies that:and consequently: The converse can be obtained in a similar way; according to Proposition 8, it suffices to consider the inverse  ☐

It trivially follows from Theorem 16 that if then the corresponding dynamical systems are not isomorphic. This means that some product MV-algebra dynamical systems can be distinguished due to their different Tsallis entropies.

6. Conclusions

In this article we dealt with the mathematical modelling of Tsallis entropy in product MV-algebras. Our results are given in Section 3, Section 4 and Section 5. In Section 3 we have introduced the notion of Tsallis entropy of a partition X in a product MV-algebra and we examined properties of this entropy measure. In Section 4 we have defined and studied the conditional Tsallis entropy of partitions in this algebraic structure. It has been shown that the proposed concepts are consistent, in the case of the limit of with the Shannon entropy expressed in nats, defined and studied in Reference [31]. Moreover, putting in the proposed definitions, we obtain the logical entropy of partitions in a product MV-algebra defined and studied in Reference [8].

Section 5 was devoted to the mathematical modelling of Tsallis entropy in product MV-algebra dynamical systems. From Example 8 it follows that the notion of product MV-algebra dynamical system is a generalization of the concept of classical dynamical system. We have shown that the Tsallis entropy is invariant under isomorphism of product MV-algebra dynamical systems.

In the proofs we used L’Hôpital’s rule and the known Jensen inequality. To illustrate the results, we have provided several numerical examples. In Example 2, we have mentioned that the full tribe of fuzzy sets is a special case of product MV-algebras; hence, all the results of this article can be directly applied to this family of fuzzy sets. We remind that a fuzzy subset of a non-empty set is any mapping where the value is interpreted as the degree of belonging of element of to the fuzzy set (cf. [16]). In Reference [56], Atanassov has generalized the Zadeh fuzzy set theory by introducing the idea of an intuitionistic fuzzy set (IF-set), a set that has the degree of belonging as well as the degree of non-belonging with each of its elements. From the point of view of application, it should be noted that for a given class of IF-sets can be created an MV-algebra such that can be inserted to . Also the operation of product on can be defined by such a way that the corresponding MV-algebra is a product MV-algebra. Therefore, the presented results are also applicable to the case of IF-sets.