In this paper, we present special families of intuitionistic fuzzy sets called fuzzy orthopartitions, which are generalizations of standard fuzzy partitions useful to model situations where both vagueness and uncertainty are involved. Moreover, we de"ne and explain how to compute the lower and upper entropy, which measure the quantity of information contained in a fuzzy orthopartition. The "rst purpose of the present work is to introduce the so-called fuzzy orthopartitions, which are generalized (fuzzy) partitions involving both vagueness and uncertainty. A fuzzy orthopartition is mathematically de"ned as a collection of intuitionistic fuzzy sets satisfying speci"c properties. An intuitionistic fuzzy set A of a universe U is a pair (μA, νA), where μA and νA are functions from U to [0, 1] such that μA(u) + νA(u) ≤ 1 for each u ∈ U , and are respectively called the membership and non-membership functions of A. Moreover, the hesitation margin of an element u of U , given by hA(u) = 1 − (μA(u) + νA(u)), expresses the degree of indeterminacy of u to A [1, 2]. Intuitionistic fuzzy sets can be seen as extensions of both orthopairs and fuzzy sets. More precisely, an orthopair (M, N ) of U , i.e. a pair of disjoint subsets of U [3], is viewed as an intuitionistic fuzzy set (μ, ν) such that μ(u) = 1 if u ∈ M , and μ(u) = 0 otherwise, while ν(u) = 1 if u ∈ N , and ν(u) = 0 otherwise. Furthermore, each fuzzy set π on U , i.e. π : U → [0, 1], is identi"ed with an intuitionistic fuzzy set (π, 1 − π) such that (1 − π)(u) = 1 − π(u) for each u ∈ U . Analogously, fuzzy orthopartitions are considered generalizations of both classical orthopartitions and Ruspini (fuzzy) partitions. Orthopartitions are generalized partitions where the membership class of some elements of the initial universe is not precisely known [4]. Mathematically, an orthopartition of U is a family O = {(M1, N1), . . . , (Mn, Nn)} of orthopairs of U satisfying the following properties: they are mutually disjoint, i.e. Mi ∩ Mj = ∅ and Mi ∩ Bj = ∅ for each i 6= j, where Bj = U \ (Mj ∪ Nj ); the union of P1 ∪ . . . ∪ Pn and B1 ∪ . . . Bn covers U ; for each u ∈ U , if u ∈ Bi then there exits
j 6= i such that u ∈ Bj ; the number of orthopairs of O is less than or equal to the cardinality of U 1.
Ruspini partitions are generalized partitions where elements of the initial universe belong to the blocks with a membership degree of [0,1] [5]. Mathematically, a Ruspini partition of U is a family π = {π1, . . . , πn} of fuzzy sets on U such that π1(u) + . . . + πn(u) = 1 for each u ∈ U .
The second purpose of this work is to present the concepts of logical entropy of a Ruspini partition, and lower and upper entropy of a fuzzy orthopartition. All of them are extensions of the logical entropy of a classical partition introduced in [6].
The logical entropy measures the quantity of information associated with a partition of a "nite universe, and its de"nition is based on the notion of distinction (or dit) considered the atom of information. Given a partition π of U , a distinction is a pair of elements of U that belong to di!erent blocks of π. Therefore, the logical entropy of a partition is mathematically de"ned as the number of distinctions normalized by the number of ordered pairs of U .
We point out that our approach is di!erent from that shown in [7]. More precisely, our measure of entropy in the intuitionistic fuzzy case is based on the generalized notion of distinction, while the authors in [7] provide the logical entropy de"nition of special collections of intuitionistic fuzzy sets representing the results of a random experiment and so, characterized by a probability distribution.
Finally, we notice that we decided to generalize the logical entropy instead of the Shannon one, since already in the crisp orthopartition case, calculating the Shannon entropy is computationally intractable [4].
In the sequel, we will provide all concepts and notions by assuming that our universe is "nite.
This section principally presents the notion of fuzzy orthopartitions. A fuzzy orthopartition is a generalized partition, where an element u of the initial universe belongs to a block represented by the intuitionistic fuzzy set (μi, νi) with a degree of the interval [μi(u), μi(u) + hi(u)] that is not precisely known.
De!nition 1. Let O = {(μ1, ν1), . . . , (μn, νn)} be a "nite family of intuitionistic fuzzy sets of U . Then, O is a fuzzy orthopartition of U if and only if the followings hold for each u ∈ U : (a) Pin=1 μi(u) ≤ 1, (b) μi(u) + hj (u) ≤ 1, for each i 6= j, (c) Pin=1 μi(u) + hi(u) ≥ 1, (d) for each i ∈ {1, . . . , n} with hi(u) > 0, there exists j ∈ {1, . . . , n}\{i} such that hj (u) > 0.
Axioms (a) and (b) capture the idea that intuitionistic fuzzy sets of O must represent mutually disjoints blocks of U , and Axiom (c) corresponds to the covering condition.
The following theorem shows that Axiom (d) allows fuzzy orthopartitions to be extensions of crisp orthopartitions.
1In this paper, we deal with more general orthopartitions, where the last property can be unsatis"ed. Theorem 1. Let O = {(μ1, ν1), . . . , (μn, νn)} be a fuzzy orthopartition of U such that μi(u), νi(u) ∈ {0, 1}, for each u ∈ U and i ∈ {1, . . . , n}. Then, {(M1, N1), . . . , (Mn, Nn)} is an orthopartition of U , where
Mi = {u ∈ U | μi(u) = 1} and Ni = {u ∈ U | νi(u) = 1}, for each i ∈ {1, . . . , n}.
Proof. Let i, j ∈ {1, . . . , n} such that i 6= j. If u ∈ Mi, then u cannot belong neither to Mj from Axiom (a) or to U \ (Mj ∪ Nj ) from Axiom (b). Therefore, (Mi, Ni) and (Mj , Nj ) are disjoint.
Suppose that there exists u ∈ U such that u ∈ Ni for each i ∈ {1, . . . , n}. Then, since (μ1, ν1), . . . , (μn, νn) are intuitionistic fuzzy sets, we get Pin=1 μi(u) + hi(u) = 0. But, the latter contradicts Axiom (c). Consequently, we can conclude Sin=1(Mi ∪ Bi) = U .
Lastly, given u ∈ U and i ∈ {1, . . . , n}, hi(u) > 0 if and only if hi(u) = 1 (i.e. u ∈ Bi), from hypothesis. Thus, Axiom (d) is equivalent to the third property of classical orthopartitions.
Additionally, Ruspini partitions can be viewed as special fuzzy orthopartitions. More precisely, the following theorem holds.
Theorem 2. Let π1, . . . , πn be some fuzzy sets on U . Then, {π1, . . . , πn} is a Ruspini partition if and only if {(π1, 1 − π1), . . . , (πn, 1 − πn)} is a fuzzy orthopartition.
Proof. The thesis clearly follows from De"nition 1.
By the previous theorem, each Ruspini partition is identi"ed with a fuzzy orthopartition. Vice-versa, a fuzzy orthopartition corresponds to a collection of Ruspini partitions. Their de"nition is formulated by thinking that fuzzy orthopartitions approximate Ruspini partitions when the truth degree of elements of the initial universe is not exactly known. De!nition 2. A Ruspini partition π = {π1, . . . , πn} of U is compatible with a fuzzy orthopartition O = {(μ1, ν1), . . . , (μn, νn)} of U if and only if μi(u) ≤ πi(u) ≤ μi(u) + hi(u) for each u ∈ U and i ∈ {1, . . . , n}.
Moreover, we indicate with ΠO the set of all Ruspini partitions compatible with O. Example 1. Let U = {u}. We consider an orthopartition O = {(μ1, ν1), (μ2, ν2), (μ3, ν3)} of U , where μ1(u) = 0.2, ν1(u) = 0.6, μ2(u) = 0.4, ν2(u) = 0.3, μ3(u) = 0.2, and ν3(u) = 0.3. Then, π = {π1, π2, π3} such that π1(u) = 0.2, π2(u) = 0.4 and π3(u) = 0.4 is an example of Ruspini partition compatible with O.
This section aims to generalize the logical entropy of standard partitions to Ruspini partitions and fuzzy orthopartitions.
In this subsection, we provide the notion of logical entropy of a Ruspini partition.
Given a Ruspini partition π of U , since the total membership degree of each element of U is distributed among all blocks of π, a pair of elements (u, u′) of U × U forms a distinction of π with a certain degree in [0, 1]. Therefore, we can generalize the concept of distinction to the case of Ruspini partitions by de"ning a fuzzy relation that assigns to each pair (u, u′) ∈ U × U the maximum of the distances between u and u′ w.r.t. all blocks of π.
De!nition 3. Let π = {π1, . . . , πn} be a Ruspini partition of U . Then, we consider a fuzzy
relation ditπ : U × U → [0, 1] such that let u, u′ ∈ U ,
ditπ(u, u′) = max{|πi(u) − πi(u′)| such that i ∈ {1, . . . , n}}.
(
A Ruspini partition π = {π1, . . . , πn} of U such that πi(u) ∈ {0, 1} for each u ∈ U and
i ∈ {1, . . . , n}, corresponds to a crisp partition. Thus, it is easy to prove that a pair of elements
is a distinction in the classical sense, i.e. it is made by elements of di!erent blocks, if and only if
the degree of distinction given by (
Proposition 1. Let π = {π1, . . . , πn} be Ruspini partition of U such that πi(u) ∈ {0, 1} for each u ∈ U and i ∈ {1, . . . , n}. Then, ditπ(u, u′) = 1 if and only if there exist i, j ∈ {1, . . . , n} with i 6= j such that πi(u) = 1 and πj (u′) = 1.
Proof. The thesis clearly follows from De"nition 3.
We now de"ne the logical entropy of a Ruspini partition as the normalized sum of the degrees of distinctions of all pairs of the initial universe.
De!nition 4. Let π = {π1, . . . , πn} be a Ruspini partition of U . Then, the logical entropy of π is given by h(π) =
P(u,u′)∈U×U ditπ(u, u′) |U × U | .
Finally, Proposition 1 implies h(π), given by (
In this subsection, we de"ne and explain how to compute the lower and upper entropy of a fuzzy orthopartition.
De!nition 5. Let O be a fuzzy orthopartition of U , the lower and upper entropy are respectively given by
h∗(O) = min{h(π) | π ∈ ΠO} and h∗(O) = max{h(π) | π ∈ ΠO}.2
2Let us recall that ΠO is the set of all Ruspini partitions compatible with O (see De"nition 2).
(
Moreover, it is easy to verify that if π = {π1, . . . , πn} is a Ruspini partition, then h(π) given
by (
We now intend to show how to compute the lower and upper entropy of a fuzzy orthopartition O = {(μ1, ν1), . . . , (μn, νn)} by using De"nition 5. To achieve this goal, we need to "nd the Ruspini partitions π∗ and π∗ of ΠO such that h∗(O) = h(π∗) and h∗(O) = h(π∗).
By De"nitions 3 and 4, π∗ and π∗ are respectively the minimum and the maximum points of the function f : ΠO → R+ such that f (π) =
X (u,u′)∈U×U
max{ |πi(u) − πi(u′)| such that i ∈ {1, . . . , n} }, for each π = {π1, . . . , πn} ∈ ΠO.
Let us notice that π∗ and π∗ exist because ΠO is always non-empty from De"nition 2.
Then, the problem to "nd π∗ and π∗ can be transformed in a constrained optimization problem as follows.
Let {π1, . . . , πn} ∈ ΠO, and let U = {u1, . . . , um}, we put I = {1, . . . , n} and J = {1, . . . , m}. Then, μi(uj) ≤ πi(uj) for each (i, j) ∈ I × J from De"nition 2, and so, we can write
πi(uj) = μi(uj) + xij.
where xij ≥ 0. Consequently, since every μi(uj) is known, each Ruspini partition π of ΠO can
be identi"ed with a matrix (xij) ∈ Rn×m having the elements determined by (
Hence, our aim becomes to "nd the maximum and the minimum points of a new function
g : Rn×m → R+ such that let m ∈ Rn×m,
(
max{|(μi(uh) + xih) − (μi(uk) + xik)| such that i ∈ I} (Pin=1(μi(uj) + xij) = 1 for each j ∈ J,
0 ≤ xij ≤ hi(uj) for each (i, j) ∈ I × J.
We can observe constraint 1 assures us that π, corresponding to m through (
Finally, this problem can be converted into a linear programming problem by introducing !n2 additional variables. More precisely, let X2 = {X ⊆ U such that |X| = 2}, we consider a new variable yX for each X ∈ X2, and we aim to minimize or maximize a new function PX∈X2 yX subject to the previous constraints, plus the followings: (yX ≥ (μi(uh) + xih) − (μi(uk) + xik), for each X ∈ X2, i ∈ I and (h, k) ∈ J × J, yX ≥ (μi(uk) + xik) − (μi(uh) + xih), for each X ∈ X2, i ∈ I and (h, k) ∈ J × J. Thus, we can compute the optimal solutions, i.e. the matrices of Rn×m representing π∗ and π∗, by using one of the standard techniques in linear programming like the Simplex method [8].
Once obtained π∗ and π∗, we can compute h(π∗) and h(π∗) by using (
Example 2. Let U = {u, v, z}, consider a fuzzy orthopartition O = {(μ1, ν1), (μ2, ν2), (μ3, ν3)} according to the de"nition in Table 1.
u v z
We suppose that (μ1, ν1), (μ2, ν2), and (μ3, ν3) express the interest in three topics T1, T2, and T3 of three di#erent groups u, v, and z of users of a social network. For instance, by Table 1, the users of u are interested in the topic T2 with a degree from μ2(u) = 0.4 to 1 − ν2(u) = 0.7. Therefore, in this case, the upper and lower entropy of O measure how much the interests of u, v, and z (valued w.r.t. {T1, T2, T3}) diverge from each other.
Then, in order to "nd the upper and lower entropy of O, we have to maximize and minimize
the function y{u,v} + y{u,z} + y{v,z} subject to the constraints given by equations (
and
Lastly, h(π∗) = 0.31 and h(π∗) = 0.13 from De"nition 4, which are the upper and lower entropy of O, respectively.
In the future, we intend to extend the contents of this paper as follows. Firstly, we want to verify that h∗(O) and h∗(O) given by De"nition 5 respectively coincide with the upper and lower entropy de"ned in [4], when O is a crisp orthopartition. Then, in order to provide a new measure of entropy, given a fuzzy orthopartition O, we will "nd a Ruspini partition π˜ of ΠO such that h(π˜) is the arithmetic mean of h∗(O) and h∗(O). On the long term, we plan to apply our results to the machine learning "eld (similarly as it has been done with orthopartitions [4, 9]) in order to de"ne new algorithms with the ability to handle both uncertainty and vagueness.