Uncertainty measure in rough logic: A probabilistic approach

Abstract

Uncertainty measure is one of the key issues in the study of rough set theory, however, the existing studies on uncertainty measure are restricted to set-theoretic rough set model(crisp or fuzzy). This paper extends the uncertainty measure of formulae in rough logic to probabilistic environments. By employing the probability measure theory, a new notion of probabilistic rough truth degree (P-rough truth degree for short) is proposed. This notion is demonstrated to be adequate for measuring the extent to which any formula is roughly true in probabilistic environments. Then based upon the fundamental notion, the notions of P-rough similarity degree, P-accuracy degree and P-roughness degree of formulae in rough logic are also proposed. The properties of these concepts are investigated in detail. Moreover, the notion of P-rough similarity degree can also induce, in a natural way, three kinds of pseudo-metrics on the set of rough formulae, which can be used to develop a kind of approximate reasoning in rough logic.

Keywords

Uncertainty measure rough truth degree rough similarity degree approximate reasoning

1 Introduction

Rough set theory [1, 2] is proposed to account for the definability of a concept in terms of some elementary ones in an approximation space. It captures and formalizes the basic phenomenon of information granulation. The finer the granulation is, the more concepts are defined in it. For those concepts not definable in an approximation space, the lower and upper approximation (See Definition 2.2) can be defined. As an effective tool in extracting knowledge from data tables, rough set theory has been widely applied in intelligent data analysis, decision making, machine learning and other related fields [3 –5].

As one of the most important issues in rough set theory, uncertainty of a set has been widely studied. Pawlak presented several numerical measures in [2], which are accuracy and roughness of a set and approximation accuracy of a rough classification. Although these measures are effective, they have some restrictions. The applications of rough set theory in some fields are hence limited. To solve this problem, an improved accuracy measure for rough sets was given in [6], which calculates the imprecision of a set using an excess entropy. However, this measure has a complex mathematical form. To overcome the shortcomings of the existing measures, Liang address the issues of uncertainty of a set in an information system and approximation accuracy of a rough classification in a decision table in [7]. There are, of course, some other research works along this research line, for instance, Mi [8] extended the uncertainty measure of Pawlak’s rough set to partition-based fuzzy rough sets, Hu [9] studied uncertainty measure of covering generated rough Set, etc. However, all the above mentioned studies are restricted to set-theoretic rough set model (crisp or fuzzy). The notion of uncertainty measure is seldom explored from the viewpoint of rough logic theory, which, however, will be helpful in knowledge discovery and data-based reasoning in the framework of rough logic. In [10], She et al. proposed the notion of rough truth degree for formulae in rough logic for the first time, owing to the main tool he uses, i.e., even probability measures, the rough (upper, lower) truth degrees of each atomic proposition are all the same.

Motivated by such a consideration, we proceed to study the uncertainty measure of formulae in rough logic in a more reasonable probabilistic environment. Specifically, by employing the probability measure theory, a new notion of P-rough truth degree is proposed. This notion is demonstrated to be adequate for measuring the extent to which any formula is roughly true. Then, based upon the fundamental notion, the notions of P-rough similarity degree, P-accuracy degree and P-roughness degree are also proposed.

2 Pre-rough algebra and Pre-rough logic

Let’s briefly review the basic notions of rough set theory initially proposed by Pawlak [1, 2].

Definition 2.1. An approximation space is a tuple AS = (U, R), where U is a non-empty set, also called the universe of discourse, R is an equivalence relation on U, representing indiscernibility at the object level.

Definition 2.2. Let AS = (U, R) be an approximation space defined as above. For any set X ⊆ U, if X is a union of some equivalence classes produced by R, then we call X a definable set, and otherwise, a rough set. As for any rough set X, two definable sets are employed to approximate it from above and from below, respectively. They are $\underline{R} (X) = {x \in U | [x] \subseteq X},$ (1) $\bar{R} (X) = {x \in U | [x] \cap X \neq \emptyset},$ (2) where [x] denotes the equivalence block containing x.

Then we call $\bar{R} (X) (\underline{R} (X))$ the rough upper(lower) approximation of X. Note that X is a definable set if and only if $\bar{R} (X) = \underline{R} (X)$ , and therefore, we also treat definable sets as special cases of rough sets.

Definition 2.3. [11, 12] A structure $P = (P, \leq, ⊓, ⊔, ⇁, L, \to, 0, 1)$ is a pre-rough algebra, if and only if

(P, ≤ , ⊓ , ⊔ , ⇁ , L, → , 0, 1) is a bounded distributive lattice with least element 0 and largest element 1,

⇁⇁ a = a,

⇁ (a ⊔ b) = ⇁ a ⊓ ⇁ b,

La ≤ a,

L (a ⊓ b) = La ⊓ Lb,

LLa = La,

L1 = 1,

MLa = La,

⇁La ⊔ La = 1,

L (a ⊔ b) = La ⊔ Lb,

La ≤ Lb and Ma ≤ Mb imply a ≤ b,

a → b = (⇁ La ⊔ Lb) ⊓ (⇁ Ma ⊔ Mb),

where ∀a ∈ P, Ma = ⇁ L ⇁ a.

Example 2.4. [11, 12] Let $3 = ({0, \frac{1}{2}, 1}, \leq, ⊓, ⊔, ⇁, L, \to, 0, 1)$ , where ≤ is the usual order on real numbers, i.e., $0 \leq \frac{1}{2} \leq 1$ , ⊓ and ⊔ are maximum and minimum, respectively. In addition, ⇁0 =1, $⇁ \frac{1}{2} = \frac{1}{2}$ , ⇁1 =0, $L 0 = L \frac{1}{2} = 0$ , L1 = 1. Then it can be easily checked that 3 is a pre-rough algebra, and also the smallest non-trival pre-rough algebra.

Example 2.5. [11, 12]. Assume that AS = (U, R) is an approximation space, and $P = {(\underline{X}, \bar{X}) | X \subseteq U}$ . Define a binary relation ≤ on P by $(\underline{X}, \bar{X}) \leq (\underline{Y}, \bar{Y}) \Leftrightarrow \underline{X} \subseteq \underline{Y}$ , $\bar{X} \subseteq \bar{Y}$ . Moreover, define operations ⊔, ⊓, ⇁, L and → on P as follows: $\forall (\underline{X}, \bar{X}), (\underline{Y}, \bar{Y}) \in P$ , $(\underline{X}, \bar{X}) ⊔ (\underline{Y}, \bar{Y}) = (\underline{X} \cup \underline{Y}, \bar{X} \cup \bar{Y}),$ (3) $(\underline{X}, \bar{X}) ⊓ (\underline{Y}, \bar{Y}) = (\underline{X} \cap \underline{Y}, \bar{X} \cap \bar{Y}),$ (4) $\begin{matrix} ⇁ (\underline{X}, \bar{X}) & = & (⇁ \bar{X}, ⇁ \underline{X}), L (\underline{X}, \bar{X}) \\ = & (\underline{X}, \underline{X}) \end{matrix}$ (5) $\begin{matrix} (\underline{X}, \bar{X}) \to (\underline{Y}, \bar{Y}) & = & ((⇁ \underline{X} \cup \underline{Y}) \cap (⇁ \bar{X} \cup \bar{Y}), \\ (⇁ \underline{X} \cup \underline{Y}) \cap (⇁ \bar{X} \cup \bar{Y})) . \end{matrix}$ (6)

It can be easily checked that P is closed under the above operations, and moreover, (P, ≤ , ⊔ , ⊓ , ⇁ , → L, (∅ , ∅) , (U, U)) forms a pre-rough algebra.

3 P-rough truth degree of logic formulae in PRL

In this section, by employing product probability measure on the set of valuations Ω₃, we introduce the notion of probabilistic truth degree (P-rough truth degree) for rough formulae in PRL, with the intention of measuring the extent to which formulae in PRL are roughly true. As shown below, we adopt the integrated approach, that is, the valuations are taken collectively but not individually.

Definition 3.1. Let Ω₃ be the set of all valuations of the form v : F (S) → 3. For any formula A = A (p₁, ⋯ , p_n) ∈ F (S), define a mapping $φ_{A} : Ω \to {0, \frac{1}{2}, 1}$ by $φ_{A} (v) = v (A) .$ (7)

Then φ_A is called the A-induced function.

Some fundamental results concerning the infinite dimensional product of measure spaces are stated int he following theorem.

Definition 3.2. [13] Let ${X_{k}, A_{k}, μ_{k}}_{k = 1}^{\infty}$ be a sequence of probabilistic measure spaces, i.e., ∀k ∈ {1, 2, ⋯}, X_k is a nonempty set, $A_{k}$ is the σ-algebra on X_k and μ_k is the probability measure on $A_{k}$ . Then we call the smallest σ-algebra containing A₁× ⋯ × A_n × X_n+1 × X_n+2 × ⋯ $(A_{k} \in A_{k}, k = 1, \dots, n, n = 1, 2, \dots)$ the product σ-algebra generated by $A_{1}, A_{2}, \dots$ and denote it by $A$ . In addition, by taking $Ω = \prod_{k = 1}^{\infty} X_{k}$ , we call $(Ω, A)$ a product space of ${X_{k}, A_{k}, μ_{k}}_{k = 1}^{\infty}$ .

Theorem 3.3. [13] Let $(X, A)$ be a product space of probability measure spaces ${X_{k}, A_{k}, μ_{k}}_{k = 1}^{\infty}$ . Then there exists a unique probabilistic measure μ on the σ-algebra $A$ such that for each measurable set G of the form E× X_n+1 × X_n+2 × ⋯, where E ⊆ X₁ × ⋯ × X_n, $μ (G) = (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E) .$

Especially, in case E = A₁ × ⋯ × A_n, $\begin{matrix} μ (E \times X_{n + 1} \times X_{n + 2} \times \dots) \\ = (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E) \\ = μ_{1} (A_{1}) \times μ_{2} (A_{2}) \times \dots \times μ_{n} (A_{n}) . \end{matrix}$

In the subsequent discussion, unless stated otherwise, for any probability measure space $(X_{k}, A_{k}, μ_{k})$ , $k = 1, 2, \dots, A_{k}$ will always denote the largest σ-algebra on X_k and μ_k denotes the general probability measure on X_k.

We are now to define the notion of P-rough truth degree for rough formulae in PRL, before we go, it is necessary to show that each formula-induced function is integrable in the product probability measure space $(X, A, μ)$ .

Note that for each valuation v : F (S) → 3, it is a homomorphism of (⊓ , L, ⇁) type, and hence is completely determined by the infinite sequence $(v (p_{1}), v (p_{2}), \dots) \in {0, \frac{1}{2}, 1}^{ω}$ (i.e., the infinite Cartesian product of ${0, \frac{1}{2}, 1}$ . Conversely, given any such infinite sequence in ${0, \frac{1}{2}, 1}^{ω}$ , we can construct a valuation v : F (S) → 3 such that (v (p₁) , v (p₂) , ⋯) coincides with the sequence. And hence, we can conclude that there exists a one-to-one correspondence between Ω₃ and ${0, \frac{1}{2}, 1}^{ω}$ . Due to this, we don’t distinguish Ω₃ and ${0, \frac{1}{2}, 1}^{ω}$ (i.e., the infinite Cartesian product of ${0, \frac{1}{2}, 1}$ ) in the subsequent discussion.

Theorem 3.4. Suppose that $X_{k} = {0, \frac{1}{2}, 1}$ , Ω₃ = (X_k) ^ω (k = 1, 2, ⋯), $A$ is the σ-algebra generated by $A_{1}, A_{2}, \dots$ , and μ is the infinite product measure of probability measures on $X_{k}^{'} s$ . Then for any logic formula A∈ PRL, the A-induced function φ_A is measurable and its integral exists.

Proof. Assume that A contains n atomic formulae p₁, p₂, ⋯ , p_n then we have from (7) that φ_A is a function from Ω₃ to ${0, \frac{1}{2}, 1}$ , and for any v = (v (p₁) , v (p₂) , ⋯) ∈ Ω₃, the value of φ_A (v) is determined by the segment v (n) = (v (p₁) , ⋯ , v (p_m)). Hence $φ_{A}^{- 1} (\frac{i}{2}) (i = 0, 1, 2)$ enjoy the following form: $E_{n} \times \prod {X_{k} : k = n + 1, n + 2, \dots},$ (8) where E_n ⊆ X₁ × ⋯ × X_n. Moreover, it is clear that $φ_{A}^{- 1} (0)$ , $φ_{A}^{- 1} (\frac{1}{2})$ and $φ_{A}^{- 1} (1)$ are pairwisely distinct measurable sets, which shows that φ_A is a staircase function, and hence φ_A is integrable. Lastly, it follows from the fact φ_A is bounded that $φ_{A}^{'} s$ integral exists.

We are now ready to define the notion of randomized truth degree for rough formula in PRL.

Definition 3.5. Suppose that $X_{k} = {0, \frac{1}{2}, 1}$ , $X = {Ω_{3}, A, μ}$ , where Ω₃ = (X_k) ^ω (k = 1, 2, ⋯), $A$ is the σ-algebra generated by $A_{1}, A_{2}, \dots$ and μ is the infinite product measure of probability measures on $X_{k}^{'} s$ . Then for any formula A ∈ F (S), define $τ (A) = \int_{Ω} φ_{A} d μ,$ (9) $\bar{τ} (A) = \int_{Ω} φ_{MA} d μ,$ (10) $\underline{τ} (A) = \int_{Ω} φ_{LA} d μ,$ (11)

Then we call $τ (A) (\bar{τ} (A), \underline{τ} (A))$ the randomized rough (upper, lower) truth degree of A, respectively, and call $τ (\bar{τ}, \underline{τ})$ the randomized rough (upper, lower) truth degree mapping, respectively.

Remark 3.6. (1) Since φ_A is a staircase function, it follows that $τ (A) = \sum {\frac{i}{2} μ (φ_{A}^{- 1} (\frac{i}{2})) : i = 0, 1, 2} .$ (12)

Also, since the value of A under each valuation v is uniquely determined by its segment v (n) = (v (p₁) , ⋯ , v (p_n)), $φ_{A}^{- 1} (\frac{i}{2}) (i = 0, 1, 2)$ enjoy the form of (8). Denote

$\begin{matrix} φ_{A}^{- 1} (\frac{i}{2}) \\ = E_{A, n}^{i} \times \prod {X_{k} : k = n + 1, n + 2, \dots}, \\ i = 0, 1, 2 . \end{matrix}$ (13)

Then it follows from the fact μ is the infinite product measure of the probability measures $μ_{k}^{'} s (k = 1, 2, \dots)$ that

$\begin{matrix} τ (A) & = & \sum {\frac{i}{2} (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{A, n}^{i}) : \\ i = 0, 1, 2,} . \end{matrix}$ (14)

For $\bar{τ} (A)$ , since ∀v ∈ Ω₃, v (MA) ∈ {0, 1} and v (MA) =1 if and only if $v (A) \in {\frac{1}{2}, 1}$ , we have that $\bar{τ} (A) = (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{A, n}^{1} \cup E_{A, n}^{2}) .$ (15)

Similarly, we can obtain $\underline{τ} (A) = (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{A, n}^{2}) .$ (16)

(2) In case $μ_{k}^{'} s (k = 1, 2, \dots)$ are even probability measures on X_k, then (14 – 16) reduce to the following form: $\begin{matrix} τ (A) & = & \sum {\frac{i}{2} (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{A, n}^{i}) : \\ i = 0, 1, 2} \\ = & \frac{1}{2} \times \sum {μ_{1} (v (p_{1})) \times \dots \\ \times μ_{n} (v (p_{n})) | (v (p_{1}), \dots, v (p_{n})) \in E_{A, n}^{1}} \\ + 1 \times \sum {μ_{1} (v (p_{1})) \times \dots \\ \times μ_{n} (v (p_{n})) | (v (p_{1}), \dots, v (p_{n})) \in E_{A, n}^{2}} \\ = & \frac{1}{2} \times \frac{1}{3^{n}} | E_{A, n}^{1} | + 1 \times \frac{1}{3^{n}} | E_{A, n}^{2} | \\ = & \frac{1}{3^{n}} (\frac{1}{2} | E_{A, n}^{1} | + | E_{A, n}^{2} |) \\ = & \frac{1}{3^{n}} (\frac{1}{2} \times | f_{A}^{- 1} (\frac{1}{2}) | + 1 | f_{A}^{- 1} (1) |) \end{matrix}$

Similarly, we can obtain that $\begin{matrix} \bar{τ} (A) & = & \frac{1}{3^{n}} (| f_{A}^{- 1} (\frac{1}{2}) | + | f_{A}^{- 1} (1) |) \\ \bar{τ} (A) & = & \frac{1}{3^{n}} (| f_{A}^{- 1} (1) |) . \end{matrix}$

(3) Note that the idea of applying probability method to mathematical logic has already been introduced in some literatures. The idea was initiated by Adams E.’s influential work “probability logic” [14]. In probability logic, the probability p (A) of a proposition A, which is supposed to measure the credibility of A, is given by means of a probability distribution p on a set of finite state-descriptions. This idea was further brought to great height of development by Wang [15], who initiated a kind of “quantitative logic” by combining mathematical logic with probability computation. The key notion of quantitative logic is the truth degree of a proposition in many- valued propositional logic systems. Note that the notion of truth degree is thoroughly different from the probability of a proposition in probability logic. The probability of a proposition in probability logic depends on a probability distribution on a finite set, and hence is arbitrarily given. However, in quantitative logic, the truth degree assigned to each proposition A is not arbitrarily given, but is completely determined by the logic structure of A. Hence it is intrinsic but not arbitrary. However, the above analysis are all restricted to some commonly used propositional logic systems, they are seldom explored from the viewpoint of rough logic theory, consequently, the proposed notions such as the probability of a proposition in probability logic, the notion of truth degree in quantitative logic can not reflect the idea of rough approximations, however, the combination of probability and rough set theory will be helpful in the knowledge discovery, as stated in [15].

Example 3.7. Let $(X_{k}, A_{k}, μ_{k}) (k = 1, 2)$ be two probability measure spaces, where $X_{k} = {0, \frac{1}{2}, 1}$ and $μ_{k}^{'} s (k = 1, 2)$ are defined as follows: $\begin{matrix} μ_{1} (0) = 0.2, μ_{1} (\frac{1}{2}) = 0.3, μ_{1} (1) = 0.5, \\ μ_{2} (0) = 0.3, μ_{2} (\frac{1}{2}) = 0.4, μ_{2} (1) = 0.3 . \end{matrix}$

Compute the P-rough truth degrees of A = p₁, B = p₁ ⊓ p₂.

Solution. Easy calculation shows that $φ_{p_{1}}^{- 1} (1) = {1} \times \prod {X_{k} : k = 2, 3, \dots}$ and $φ_{p_{1}}^{- 1} (\frac{1}{2}) = {\frac{1}{2}} \times \prod {X_{k} : k = 2, 3, \dots}$ , then it follows from (14-16) that $\begin{matrix} τ (A) & = & \sum {\frac{i}{2} (μ_{1}) ({i}) : i = 0, 1, 2} \\ = & \frac{1}{2} \times 0.3 + 1 \times 0.5 = 0.65, \\ \bar{τ} (A) & = & μ_{1} ({\frac{1}{2}, 1}) = 0.3 + 0.5 = 0.8, \end{matrix}$ and $\underline{τ} (A) = μ_{1} ({1}) = 0.5$

Similarly, we have that $\begin{matrix} τ (p_{1} ⊓ p_{2}) & = & \frac{1}{2} (μ_{1} \times μ_{2}) {(\frac{1}{2}, \frac{1}{2}), (\frac{1}{2}, 1), (1, \frac{1}{2})} \\ + (μ_{1} \times μ_{2}) {(1, 1)} = 0.355, \\ \bar{τ} (p_{1} ⊓ p_{2}) & = & (μ_{1} \times μ_{2}) {(\frac{1}{2}, \frac{1}{2}), \\ (\frac{1}{2}, 1), (1, \frac{1}{2}), (1, 1)} = 0.56, \end{matrix}$ and $\bar{τ} (p_{1} ⊓ p_{2}) = (μ_{1} \times μ_{2}) {(1, 1)} = 0.5 \times 0.3 = 0.15 .$

The notions of P-rough (upper, lower) truth degrees obey the following properties.

Proposition 3.8. Let $τ (\bar{τ}, \underline{τ})$ be the P-rough (upper, lower) truth degree mappings as defined in (9), (10) and (11). Then

$0 \leq \underline{τ} (A) \leq τ (A) \leq \bar{τ} (A) \leq 1$ ,

$\bar{τ} (A) = τ (MA)$ , $\underline{τ} (A) = τ (LA)$ ,

If ⊢A, i.e., A is a theorem in PRL, then $τ (A) = \bar{τ} (A) = \underline{τ} (A) = 1$ ,

If ⊢MA, i.e., MA is a theorem in PRL, then $\bar{τ} (A) = 1$ ,

If ⊢A, i.e., LA is a theorem in PRL, then $\underline{τ} (A) = 1$ ,

If ⊢MA → MB, then $\bar{τ} (A) \leq \bar{τ} (B)$ ;

If ⊢LA → LB, then $\underline{τ} (A) \leq \underline{τ} (B)$ ;

If ⊢A → B, then τ (A) ≤ τ (B), $\bar{τ} (A) \leq \bar{τ} (B)$ and $\underline{τ} (A) \leq \underline{τ} (B)$ ,

If A and B are logically equivalent, i.e., both ⊢A → B and ⊢B → A are theorems in PRL, then τ (A) = τ (B),

If A and B are rough-upperly equivalent, i.e., ⊢MA ↔ MB, then $\bar{τ} (A) = \bar{τ} (B)$ ,

If A and B are rough-lowerly equivalent, i.e., ⊢LA ↔ LB, then $\underline{τ} (A) = \underline{τ} (B)$ ,

τ (⇁ A) =1 - τ (A), $\bar{τ} (⇁ A) = 1 - \underline{τ} (A)$ , $\underline{τ} (⇁ A) = 1 - \bar{τ} (A)$ ,

τ (A ⊔ B) = τ (A) + τ (B) - τ (A ⊓ B);

$\bar{τ} (A ⊔ B) = \bar{τ} (A) + \bar{τ} (B) - \bar{τ} (A ⊓ B)$ ;

$\underline{τ} (A ⊔ B) = \underline{τ} (A) + \underline{τ} (B) - \underline{τ} (A ⊓ B)$ .

Proof. Both (1) and (2) are clear from Definition 3.5.

(3) Suppose that A contains n atomic propositions p₁, ⋯ , p_n, then it follows from Theorem 2.8 that A is a valid formula, which in turn entails that $E_{A, n}^{2} = {0, \frac{1}{2}, 1}^{n}$ . Hence, we have $τ (A) = \sum {\frac{i}{2} (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{n}^{i}) : i = 0, 1, 2} = (μ_{1} \times μ_{2} \times \dots \times μ_{n}) ({0, \frac{1}{2}, 1}^{n}) = 1$ , $\underline{τ} (A) = \sum {\frac{i}{2} (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{n}^{2})} = (μ_{1} \times μ_{2} \times \dots \times μ_{n}) ({0, \frac{1}{2}, 1}^{n}) = 1$ . Moreover, $\bar{τ} (A) = 1$ follows immediately from Proposition 3.8(1).

It can be shown in a similar way that if ⊢MA, then $\bar{τ} (A) = 1$ , and if ⊢A, then $\underline{τ} (A) = 1$ .

(4) We assume, without any loss of generality, that both A and B contain the same atomic formulae p₁, ⋯ , p_n. If ⊢A → B, v (A → B) =1 for any valuation v ∈ Ω₃, which immediately entails $E_{A, n}^{2} \subseteq E_{B, n}^{2}$ , $E_{A, n}^{1} \subseteq E_{B, n}^{1} \cup E_{B, n}^{2}$ . Hence $τ (A) = \frac{1}{2} (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{A, n}^{1}) + (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{A, n}^{2}) \leq \frac{1}{2} (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{B, n}^{1}) + (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{B, n}^{2}) = τ (B)$ .

If ⊢MA → MB, then it can be proved in a similar manner as above (by letting A = MA, B = MB in the above argument) that τ (MA) ≤ τ (MB), applying Proposition 3.8 (2) here, we obtain immediately that $\bar{τ} (A) \leq \bar{τ} (B)$ .

Similarly, we can conclude $\underline{τ} (A) \leq \underline{τ} (B)$ under the premise of ⊢LA → LB.

(5) It follows immediately from Proposition 3.8 (4).

(6) It follows from v (⇁ A) =1 - v (A) that $\begin{matrix} E_{⇁ A, n}^{0} & = & E_{A, n}^{2}, E_{⇁ A, n}^{1} = E_{A, n}^{1}, \\ E_{⇁ A, n}^{2} & = & E_{A, n}^{0} . \end{matrix}$

Then, we have from (14) that $\begin{matrix} τ (⇁ A) & = & \sum {\frac{i}{2} (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{⇁ A, n}^{i}) : \\ i = 0, 1, 2} \\ = & \sum {\frac{1}{2} (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{⇁ A, n}^{1}) \\ + (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{⇁ A, n}^{2})} \\ = & \sum {\frac{1}{2} (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{A, n}^{1}) \\ + (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{A, n}^{0})} \\ = & 1 - \sum {\frac{1}{2} (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{A, n}^{1}) \\ + (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{A, n}^{2})} \\ = & 1 - τ (A) . \end{matrix}$

It can be easily checked that M (⇁ A) is provably equivalent to ⇁LA, applying τ (⇁ A) =1 - τ (A), Propositions 3.8(2) and 3.8(5) here, we conclude immediately that $\bar{τ} (⇁ A) = τ (M (⇁ A)) = τ (⇁ LA) = 1 - τ (LA) = 1 - \underline{τ} (A)$ .

Similarly, we can show that $\underline{τ} (⇁ A) = 1 - \bar{τ} (A)$ .

(7) Assume that both A and B contain n atomic formulae p₁, ⋯ , p_n. Then it is clear that $\begin{matrix} E_{A ⊔ B, n}^{1} = E_{A, n}^{1} \cup E_{B, n}^{1} - E_{A ⊓ B, n}^{1}, \\ E_{A ⊔ B, n}^{2} = E_{A, n}^{2} \cup E_{B, n}^{2} - E_{A ⊓ B, n}^{2}, \end{matrix}$

Hence by (14), we have that $\begin{matrix} τ (A ⊔ B) & = & \sum {\frac{i}{2} (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{A ⊔ B, n}^{i}) : \\ i = 0, 1, 2} \\ = & \frac{1}{2} (μ_{1} \times μ_{2} \times \dots \times μ_{n}) \\ (E_{A, n}^{1} \cup E_{B, n}^{1} - E_{A ⊓ B, n}^{1}) \\ + (μ_{1} \times μ_{2} \times \dots \times μ_{n}) \\ (E_{A, n}^{2} \cup E_{B, n}^{2} - E_{A ⊓ B, n}^{2}) \\ = & (\frac{1}{2} (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{A, n}^{1}) \\ + (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{A, n}^{2})) \\ + (\frac{1}{2} (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{B, n}^{1}) \\ + (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{B, n}^{2})) \\ - (\frac{1}{2} (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{A ⊓ B, n}^{1}) \\ + (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{A ⊓ B, n}^{2})) \\ = & τ (A) + τ (B) - τ (A ⊓ B) . \end{matrix}$

The other two equations can be proved in an analogous way and hence are omitted here.

Remark 3.9. The converse direction in Proposition 3.8(3) is not true, that is, we can not conclude from τ (A) =1 that A is a theorem in PRL. Please see the following counterexample:

Let μ be the infinite product measure of probability measures $μ_{k}^{'} s (k = 1, 2, \dots)$ , where each μ_k satisfies $μ_{k} (0) = μ_{k} (\frac{1}{2}) = 0$ , μ_k (1) =1. Then for each atomic proposition p_k; $τ (p_{k}) = \frac{1}{2} μ_{k} ({\frac{1}{2}}) + μ_{k} ({1}) = 1,$ however, p_k is not a theorem in PRL.

Nonetheless, we can obtain a weak result as follows.

Proposition 3.10. Let μ be the infinite product measure of probability measures $μ_{k}^{'} s (k = 1, 2, \dots)$ , where each μ_k satisfies the condition that $μ_{k} (\frac{i}{2}) > 0$ , i = 0, 1, 2, k = 1, 2, ⋯. Then τ (A) =1 if and only if A is a valid formula in PRL.

Proof. We only need to show the sufficient condition below. Suppose that A (p₁, ⋯ , p_n) is not valid in PRL, there is a valuation v : F (S) → 3 such that v (A) <1, that is, $φ_{A} (v) \in {0, \frac{1}{2}}$ . Then by definition, we have that $(v (p_{1}, \dots, v (p_{n})) \in E_{A, n}^{1} \cup E_{A, n}^{0}$ . Assume, without any loss of generality, that $(v (p_{1}, \dots, v (p_{n})) \in E_{A, n}^{1}$ , then we conclude immediately that $τ (A) = \frac{1}{2} (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{A, n}^{1}) + (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (E_{A, n}^{2}) < 1$ , a contradiction!

4 P-accuracy degree and P-roughness degree of formulae in PRL

In this section, based upon the proposed notion of P-rough truth degree, we extent the uncertain measure for set-theoretic rough set model to rough logic theory for the first time. Concretely, we introduce the notions of P-accuracy degree and P-roughness degree, with the aim of measuring the degree to which any rough formula in PRL is accurate and rough, respectively.

Definition 4.1. Let A ∈ F (S), define $\begin{matrix} Acc (A) & = & τ (MA \to LA), \\ Rou (A) & = & 1 - τ (MA \to LA) . \end{matrix}$

Then we call Acc(A), Rou(A) the accuracy degree and the roughness degree of A, respectively.

Example 4.2. Let μ be the infinite product measure of probability measures $μ_{k}^{'} s$ , where μ₁ {0} =0.2, $μ_{1} ({\frac{1}{2}}) = 0.3$ , μ₁ {1} =0.5 and μ₂ {0} =0.3, $μ_{2} ({\frac{1}{2}}) = 0.4$ , μ₁ {1} =0.3. Compute the P-accuracy degree and the P-roughness degree of p₁ and p₁ ⊔ p₂.

Solution. (1) Easy calculation shows that $τ (Mp \to Lp) = \frac{1}{3}$ , then by definition, Acc(p₁) = $\frac{1}{3}$ and Rou( $p_{1}) = 1 - τ (MA \to LA) = \frac{2}{3}$ .

(2) Similarly, $τ (M (p_{1} ⊔ p_{2}) \to L (p_{1} ⊔ p_{2})) = \frac{6}{3^{2}}$ , then by definition, Acc( $p_{1} ⊔ p_{2}) = \frac{1}{3}$ and Rou( $p_{1} ⊔ p_{2}) = 1 - τ (MA \to LA) = \frac{2}{3}$ .

Proposition 4.3. Let A, B ∈ F (S), then $Acc (A) = 1 - \bar{τ} (A) + \underline{τ} (A), Rou (A) = \bar{τ} (A) - \underline{τ} (A) .$

Proof. We have from Proposition 3.8(2), (6) and (7) that Acc( $A) = τ (MA \to LA) = τ (⇁ MA ⊔ LA) = τ (⇁ MA) + τ (LA) - τ (⇁ MA ⊓ LA) = (1 - τ (MA)) + τ (LA) - 0 = 1 - \bar{τ} (A) + \underline{τ} (A)$ , and Rou(A) =1- Acc( $A) = 1 - (1 - \bar{τ} (A) + \underline{τ} (A)) = \bar{τ} (A) - \underline{τ} (A)$ .

The notions of accuracy degree and roughness degree obey the following properties.

Proposition 4.4. Let A, B ∈ F (S), then

Acc(LA) = Acc(MA) = 1, Rou(LA) = Rou(MA) = 0;

If A ∼ MA, or equivalently, A ∼ LA, then Acc(A) = 1 and Rou(A) = 0.

A ∼ B implies that Acc(A) = Acc(B) and Rou(A) = Rou(B),

Acc(A) = 1 if and only if Rou(A) = 0, if and only if $\bar{τ} (A) = \underline{τ} (A)$ ;

For a fixed $\bar{τ} (A)$ , Acc(A) strictly monotonically increase with $\underline{τ} (A)$ ;

For a fixed $\underline{τ} (A)$ , Acc(A) strictly monotonically decrease with $\bar{τ} (A)$ .

Proof. (1) It can be verified that ⊢MLA → LLA ↔ LA → LA and ⊢MMA ↔ LMA, then we have from Proposition 3.8(3), (5) that Acc(MA) = τ (MMA → LMA) = τ (MA → MA) =1 and Acc(LA) = τ (MLA → LLA) = τ (LA → LA) =1, which directly entails that Rou(LA) = Rou(MA) = 0.

(2) It is clear that A ∼ MA if and only if A ∼ LA. If one of them holds, then by Proposition 3.8(5), we have Acc(A) = τ (MA → LA) = τ (A → A) =1, Rou(A) = 1 - 1 =0.

(3) If A ∼ B, then it can be easily checked that MA → LA is logically equivalent to MB → LB. Hence, by Proposition 3.8(5), τ (MA → LA) = τ (MB → LB), then by Definition 4.1, Acc(A) = Acc(B) and Rou(A) = Rou(B) immediately follow.

Both (4) and (5) follow immediately from Proposition 4.3.

5 P-rough similarity between formulae in PRL

Recall first that two formulae A and B in PRL are said to be roughly equal if and only if ⊢MA ↔ MB and ⊢LA ↔ LB, they are said to be rough-upperly equal if and only if ⊢MA ↔ MB and rough-lowerly equal if and only if ⊢LA ↔ LB, these notions can be graded to introduce the notions of rough similarity degree, rough upper similarity degree and rough lower similarity degree, respectively. As will be shown below, such notions can be introduced by the notion of rough truth degree in a natural way.

Definition 5.1. Let A, B ∈ F (S), define $ξ (A, B) = τ ((A \to B) ⊓ (B \to A)),$ (17) $\bar{ξ} (A, B) = τ ((MA \to MB) ⊓ (MB \to MA)),$ (18) $\underline{ξ} (A, B) = τ ((LA \to LB) ⊓ (LB \to LA)) .$ (19)

Then we call ξ (A, B) $(\bar{ξ} (A, B), \underline{ξ} (A, B))$ probabilistic rough (upper, lower) similarity degree (P- rough (upper, lower) similarity degree for short) between A and B.

Example 5.2. Let μ be the infinite product measure of probability measures $μ_{k}^{'} s$ , where μ₁ {0} =0.2, $μ_{1} ({\frac{1}{2}}) = 0.3$ , μ₁ {1} =0.5 and μ₂ {0} =0.3, $μ_{2} ({\frac{1}{2}}) = 0.4$ , μ₁ {1} =0.3. Compute ξ (p₁, p₂), $\bar{ξ} (p_{1}, p_{2})$ and $\underline{ξ} (p_{1}, p_{2})$ .

Solution. Note that for any valuation v : F (S) → 3, v ((p₁ → p₂) ⊓ (p₂ → p₁)) ∈ {0, 1}, and it follows from Definition 5.1 that $ξ (p_{1}, p_{2}) = τ ((p_{1} \to p_{2}) ⊓ (p_{1} \to p_{2})) = (μ_{1} \times μ_{2}) (E_{A, 2}^{2}) = \sum {(μ_{1} \times μ_{2}) (\frac{i}{2}, \frac{i}{2}) | i = 0, 1, 2} = 0.2 \times 0.3 + 0.3 \times 0.4 + 0.5 \times 0.3 = 0.33$ , where A = (p₁ → p₂) ⊓ (p₂ → p₁).

$\bar{ξ} (p_{1}, p_{2}) = τ (({Mp}_{1} \to {Mp}_{2}) ⊓ ({Mp}_{2} \to {Mp}_{1})) = (μ_{1} \times μ_{2}) (E_{B, 2}^{2}) = (μ_{1} \times μ_{2}) {(0, 0), (\frac{1}{2}, \frac{1}{2}), (\frac{1}{2}, 1), (1, \frac{1}{2}), (1, 1)} = (μ_{1} \times μ_{2}) ({(0, 0)}) + (μ_{1} \times μ_{2}) ({(\frac{1}{2}, \frac{1}{2})}) + (μ_{1} \times μ_{2}) ({(\frac{1}{2}, 1)}) + (μ_{1} \times μ_{2}) ({(1, \frac{1}{2})}) + (μ_{1} \times μ_{2}) ({(1, 2)}) = 0.2 \times 0.3 + 0.3 \times 0.4 + 0.5 \times 0.4 + 0.3 \times 0.3 + 0.5 \times 0.3 = 0.52$ , where B = (Mp₁ → Mp₂) ⊓ (Mp₂ → Mp₁).

Similarly, we have that $\underline{ξ} (p_{1}, p_{2}) = 0.5$ .

P-rough (upper, lower) similarity degree enjoys the following properties.

Proposition 5.3. ∀A, B, C ∈ F (S),

$ξ (A, A) = \bar{ξ} (A, A) = \underline{ξ} (A, A) = 1$ ,

ξ (A, B) = ξ (B, A), $\bar{ξ} (A, B) = \bar{ξ} (B, A)$ , $\underline{ξ} (A, B) = \underline{ξ} (B, A)$ ,

$\bar{ξ} (A, B) = ξ (MA, MB)$ , $\underline{ξ} (A, B) = ξ (LA, LB)$ ,

If ⊢A ↔ B, then ξ (A, B) =1;

If ⊢MA ↔ MB, then $\bar{ξ} (A, B) = 1$ ;

If ⊢LA ↔ LB, then $\underline{ξ} (A, B) = 1$ ,

$\bar{ξ} (A, B) + \bar{ξ} (B, C) \leq \bar{ξ} (A, C) + 1$ ;

ξ (A, B) + ξ (B, C) ≤ ξ (A, C) +1;

$\underline{ξ} (A, B) + \underline{ξ} (B, C) \leq \underline{ξ} (A, C) + 1$ ,

If ⊢MA ↔ ⇁ MB, then $\bar{ξ} (A, B) = 0$ ,

if ⊢LA ↔ ⇁ LB, then

\underline{ξ} (A, B) = 0

Proof. (1), (2) and (3) are clear from Definition 5.1.

(4) It follows immediately from (17–19) and Proposition 3.8(3).

(5) Assume that A, B and C contain the same atomic formulae p₁ ⋯ , p_n. In what follows, for the sake of convenience, we will use D, E and F to denote (MA → MB) ⊓ (MB → MA), (MB → MC) ⊓ (MC → MB) and (MA → MC) ⊓ (MC → MA), respectively.

Let $\begin{matrix} M_{1} = E_{D, n}^{2}, \\ M_{2} = E_{E, n}^{2}, \\ M_{3} = E_{F, n}^{2} . \end{matrix}$

Then by (14) and (18), $\begin{matrix} \bar{ξ} (A, B) + \bar{ξ} (B, C) & = & (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (M_{1}) \\ + (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (M_{2}), \\ \bar{ξ} (A, C) + 1 & = & (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (M_{3}) . \end{matrix}$

Trivally, $\bar{ξ} (A, B) + \bar{ξ} (B, C) = (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (M_{1} \cup M_{2}) + (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (M_{1} \cap M_{2})$ . Then, it follows immediately from M₁ ∩ M₂ ⊆ M₃ that $(μ_{1} \times μ_{2} \times \dots \times μ_{n}) (M_{1} \cup M_{2}) + (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (M_{1} \cap M_{2}) \leq 1 + (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (M_{3}) = \bar{ξ} (A, C) + 1$ .

The proof of the other inequalities can be given in a similar way, and hence is omitted here.

(6) Assume, without any loss of generality, that A, B and C contain the same atomic formulae p₁, ⋯ p_n. If ⊢MA ↔ ⇁ MB, we have that v (MA ↔ ⇁ MB) =1 for each v ∈ Ω₃, which shows that v (MA) =0, v (MB) =1 or v (MA) =1, v (MB) =0. In either case, one can not obtain v ((MA → MB) ⊓ (MB → MA)) =1, that is, $E_{C, n}^{2} = \emptyset$ , and hence $\bar{ξ} (A, B) = τ ((MA \to MB) ⊓ (MB \to MA)) = (μ_{1} \times μ_{2} \times \dots \times μ_{n}) (\emptyset) = 0$ .

Similarly, we can prove that ⊢LA ↔ ⇁ LB implies $\underline{ξ} (A, B) = 0$ .

Remark 5.4. Easy verification shows that $\bar{ξ} (A, MA) = 1$ and $\underline{ξ} (A, LA) = 1$ , however, the converse direction is not true, that is, we can not derive ⊢MA ↔ MB and ⊢LA ↔ LB from $\bar{ξ} (A, B) = 1$ and $\underline{ξ} (A, B) = 1$ , respectively. Similarly, the opposite direction of Proposition 5.3(6) is also not true. The corresponding counterexamples can be given in a similar manner as that of Remark 3.9.

However, we have the following weak results.

Proposition 5.5. Let μ be the infinite product measure of probability measures μ_k (k = 1, 2, ⋯), where each μ_k satisfies the condition that $μ_{k} (\frac{i}{2}) > 0$ , i = 0, 1, 2, k = 1, 2, ⋯. Then ⊢A ↔ B if and only if ξ (A, B) =1, ⊢MA ↔ MB if and only if $\bar{ξ} (A, B) = 1$ and ⊢LA ↔ LB if and only if $\underline{ξ} (A, B) = 1$ .

Proof. It can be proved in an analogous manner as that of Proposition 3.10 and hence is omitted here.

If we can induce a pseudo-metric on the set F (S) of all formulae in PRL which is compatible with the connectives on F (S), then it is convenient for us to study approximate reasoning problem under the framework of rough logic PRL. As we will see, such pseudo-metrics can be induced by the notions of P-rough similarity degree, P-rough upper similarity degree and P-rough lower similarity degree, respectively.

Definition 5.6. Define three non-negative functions ρ, $\bar{ρ}$ and $\underline{ρ} : F (S) \times F (S) \to [0, 1]$ as follows: ∀A, B ∈ F (S), $ρ (A, B) = 1 - ξ (A, B),$ (20) $\bar{ρ} (A, B) = 1 - \bar{ξ} (A, B),$ (21) $\underline{ρ} (A, B) = 1 - \underline{ξ} (A, B) .$ (22)

The following proposition states that ρ, $\bar{ρ}$ and $\underline{ρ}$ are indeed the pseudo-metrics on the set of rough formulae in PRL.

Proposition 5.7. Let ρ, $\bar{ρ}$ and $\underline{ρ}$ be the non-negative functions as defined in (20–22), then ∀A, B, C ∈ F (S),

$ρ (A, A) = \bar{ρ} (A, A) = \underline{ρ} (A, A) = 0$ ,

ρ (A, B) = ρ (B, A); $\bar{ρ} (A, B) = \bar{ρ} (B, A)$ ; $\underline{ρ} (A, B) = \underline{ρ} (B, A)$ ,

ρ (A, C) ≤ ρ (A, B) + ρ (B, C);

$\bar{ρ} (A, C) \leq \bar{ρ} (A, B) + \bar{ρ} (B, C)$ ;

$\underline{ρ} (A, C) \leq \underline{ρ} (A, B) + \underline{ρ} (B, C)$ .

Proof. Both (1) and (2) are clear by definition, and (3) follows immediately from Proposition 5.3(5).

By virtue of Proposition 5.7, in what follows, we call ρ, $\bar{ρ}$ and $\underline{ρ}$ the rough pseudo-metric, the rough upper pseudo-metric and the rough lower pseudo-metric, respectively.

6 Concluding remarks

In this paper, we combine rough logic and probability theory together by means of the infinite product measure on the valuation set 3 in PRL.We initially introduce the notion of P-rough truth degree for formulae in PRL, then, based upon the fundamental notion of P-rough truth degree, the uncertainty measure such as P-accuracy measure, P-roughness measure and P-rough similarity degrees are also proposed and their properties are investigated in detail. It is important to note that it has been shown in [12] that pre-rough logic is equivalent to 3-valued Lukasiewicz logic, then an interesting topic is to extend the present study to multiple-valued logic liking rough sets, we will report it in our forthcoming papers.

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