 # Modeling Uncertainty and Imprecision in Nonmonotonic Reasoning using Fuzzy Numbers

To deal with uncertainty in reasoning, interval-valued logic has been developed. But uniform intervals cannot capture the difference in degrees of belief for different values in the interval. To salvage the problem triangular and trapezoidal fuzzy numbers are used as the set of truth values along with traditional intervals. Preorder-based truth and knowledge ordering are defined over the set of fuzzy numbers defined over [0,1]. Based on this enhanced set of epistemic states, an answer set framework is developed, with properly defined logical connectives. This type of framework is efficient in knowledge representation and reasoning with vague and uncertain information under nonmonotonic environment where rules may posses exceptions.

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## 1 Introduction

Modern applications of artificial intelligence in decision support systems, plan generation systems require reasoning with imprecise and uncertain information. Logical frameworks based on bivalent reasoning are not suitable for such applications, because the set

cannot capture the vagueness or uncertainty of underlying proposition. Though fuzzy logic-based systems can represent imprecise linguistic information by ascribing membership values to attributes (or truth values to propositions) taken from the interval [0,1], but this graded valuation becomes inadequate if the precise membership can not be determined due to some underlying uncertainty. This uncertainty may arise from lack of complete information or from lack of reliability of source of information or lack of unanimity among rational agents in a multi-agent reasoning system or from many other reasons. This uncertainty with respect to the assignment of membership degrees is captured by assigning a range of possible membership values, i.e. by assigning an interval. In other words by replacing the crisp {0,1} set by the set of sub-intervals of [0,1]. The intuition of such interval-valued system is that the actual degree, though still unknown, would be some value within the assigned interval and all the values in the interval are equally-likely to be the actual one.

However, there may be situations where all the values of an interval are not equally likely, rather, the information in hand suggests that some values are more plaussible. For instance, consider the motivating example presented in . It states that, ”if the tumor suppressing genes(TSG) are lost due to mutation during cell division and chromosomal instability (CIN) is activated, a reasonably large tumor will grow”. In the proposed approach, this single information is represented by four rules and the resultant valuation assigned to the fact tumor is given by . This representation is very inefficient and the number of rules and number of elements in the valuation would grow proportionately to the number of truth degrees considered within

. This example denotes that in real-world applications assignment of uniform intervals is inadequate. Instead, if arbitrary distributions over the interval [0,1] are allowed for truth values of propositions that would hugely increase the expressibility of the system and reduce the number of necessary rules in the logic program. Therefore, instead of assigning a sub-interval of [0,1] as the epistemic state to some vague, uncertain proposition, a

fuzzy number defined over [0,1] would be a better choice, since, fuzzy numbers precisely allow to specify a membership distribution over [0,1].

Specifying the set of epistemic states is not enough, there has to be some underlying algebraic structure for ordering the values with respect to their degree of truth and degree of certainty(or uncertainty). For uniform interval-valued case Bilattice-based triangle structure were proposed . However later it is demonstrated  that bilattice-based ordering is not suitable for belief revision in nonmonotonic reasoning and a preorder-based algebraic structure was constructed. Similar type of ordering has to be extended over the fuzzy numbers defined on .

The main contributions of this work are as follows:

The set of fuzzy numbers defined on [0,1] is considered as the set truth values for nonmonotonic reasoning with vague and uncertain information. In this work uniform, triangular and trapezoidal fuzzy numbers are considered only.

Truth ordering and knowledge ordering over the set are defined (section 3) to construct the underlying preorder-based algebraic structure (section 4).

This approach is used for answer set programming (section 5) to demonstrate the advantage.

## 2 Fuzzy Numbers

This section provides necessary preliminary concepts.

Definition 1:

A fuzzy set over some is called a fuzzy number if

1. is convex, i.e.,

where, and .

2. is normalised, i.e. .

3. There is some such that .

4. is piecewise continuous.

### 2.1 Triangular and Trapezoidal Fuzzy Number

The membership function of a triangular fuzzy number for and is specified as:

The membership function of a trapezoidal fuzzy number for and is specified as:

The uniform interval is a special case of when , i.e., can be thought of as an interval so that all the values within the range has membership value 1. In this work an interval will be denoted as to keep parity with the other two notations.

### 2.2 α-cut decomposition of fuzzy numbers

Another way of specifying a fuzzy number is by computing -cuts for . For any fuzzy number and any specific value of , the -cut produces an interval of the form , where and are the intersection values with the left and right segment of . The -cuts for a specific for a TFN and TrFN are shown in Figure 1. for , will be referred to as base-range of ().

Analytically the -cut for the fuzzy numbers can be specified as follows:

For ; ;

For ; ;

For ; .

Since, is a special case of can be obtained from by setting and . Similarly if the condition is imposed on a is obtained. Hence both are special cases of . Therefore, in later sections some concepts will be explained in terms of s only because same will be applicable for and by imposing the aforementioned conditions.

## 3 Fuzzy numbers as truth assignment and their truth and knowledge ordering:

It is already demonstrated by means of an example that specifying an interval of real numbers from [0,1] is not sufficient to express the epistemic state of propositions in real life reasoning with vague and uncertain information. Now, general fuzzy numbers can be used as truth assignment of a proposition to capture various degrees of belief over the range of . However, just specifying fuzzy numbers as the set of epistemic states is not enough, there must be some ordering to order two such epistemic states with respect to the degree of truth (truth ordering) and degree of certainty (knowledge ordering). Instead of considering any general type of fuzzy numbers, here, only the three types, that are specified in Section 2 (i.e., IFN, TFN and TrFN), are considered as truth assignments.

Definition 2:

A is said to be restricted if , i.e., the base-range . . Similarly restricted versions of and are defined.

A is semi-restricted if and or both . A is semi-restricted if and any or both of and .

### 3.1 Construction of the Set of Epistemic States:

In this section the set of truth assignments is constructed so that any element from can be assigned to some proposition to express its degree of belief. is constructed from following conditions:

1. All restricted , and are member of .

2. For a semi-restricted its truncated version confined within [0,1] is included in . Thus,

Here some intuitive aspects are explained to justify the necessity of by means of examples.

Example 1: The case described in the introduction section can be re-considered. The epistemic state of the fact that was specified by can be approximately represented by . This assignment(shown in Figure 2a) is more compact representation.

Example 2: Suppose a group of agents with different degree of expertise is asserting their degree of belief about some proposition under uncertainty. They all agree that is not false and has moderate to high degree of truth. The most reliable experts tend to ascribe very high degree of truth, which shows that they believe will be true. This scenario can be expressed by using a trapezoidal fuzzy number , as shown in Figure 2b.

Example 3: If nothing is known about a proposition then is assigned. If a proposition is known to be True, with absolute certainty, then is assigned.

Bimodal or Multi-modal distributions can not be expressed using .

### 3.2 Truth ordering and knowledge ordering of restricted TrFNs and restricted Tfns:

Now that the set of epistemic states is specified and intuitively justified, elements of are to be ordered with respect to their degree of truth and certainty. These orderings play crucial role in revising beliefs during nonmonotonic reasoning. For instance, suppose, based on available knowledge the truth status of certain proposition has been determined. Now, some additional information becomes available and based on the new enhanced information set, the proposition is re-evaluated. In such a scenario, it becomes important to compare the two new assignment with the previous one with respect to degree of truth and degree of certainty. If some contradiction arises, some previously known facts or rules are to be withdrawn and this withdrawal procedure mandates ordering various rules or facts with respect to their degree of certainty. It is demonstrated in , for s preorder-based ordering is more intuitive and suitable for performing nonmonotonic reasoning with imprecise and uncertain information.

Definition 3: For any two , and the truth ordering() and knowledge ordering(), defined in , are as follows:

.

.

The truth ordering () and the knowledge ordering () are preorders and combined they give rise to a preorder-based triangle. These definitions are generalized for s and s in the next subsections.

#### 3.2.1 Truth-ordering

The intuition of assigning a fuzzy number for the epistemic state of a proposition, is that, due to uncertainty the actual truth assignment for (say, ) is unknown, and hence is approximated by the assigned fuzzy number. If is approximated by , then every value within the interval

is equally probable to be

. If is assigned to , then it signifies, in the range , has a higher chance of being the actual truth status () of . Assignment of a can be interpreted similarly.

If we perform a random experiment, where an agent guesses the actual truth value of proposition , then

can be thought of as a random variable, which follows a probability distribution. Now given the information in hand, assigning an epistemic state for

is same as assigning an equivalent probability distribution over the random variable . So, for any restricted fuzzy number in , an equivalent probability distribution can be defined (as shown in Table 1).

For two propositions and with truth assignments and from , the intuition for ordering the truth assignments, with respect to the degree of truth is 

iff

where, and stands for the actual (yet unknown) truth status of propositions and respectively.

Now following this intuition we intend to extend the truth ordering from Definition 3 to ordering s and s.

As explained above, for two propositions and , and can be thought of as two random variables. In order to calculate or another random variable is defined as:

.

Then, and . Moreover the expectations(or means) of the random variables are related by .

Now, if probability distributions of and are chosen so that , then . This makes, . Thus, . Since, the truth ordering is a total preorder, this would signify that propositions and have same degree of truth. This occurs irrespective of the chosen probability distribution of and .

Definition 4: For any member , its equivalent-interval is any restricted so that mean value of the equivalent probability distribution of is equal to , i.e., the expected value of a random variable that follows the probability density function . Therefore, any centered around the value is an equivalent-interval to .

The truth ordering defined over s (from Definition 3) can be extended for ordering restricted s and s using their equivalent-intervals.

Theorem 1: For any members ,

iff .

where, and are random variables following probability density functions equivalent to and (as specified in Table 1) respectively.

Proof: If and are s then the theorem directly follows from Definition 3, as .

Suppose, and are respectively and , and their corresponding equivalent-intervals are and respectively. Following the aforementioned rationale and have same degree of truth and same holds for and . The two s, and can be ordered with respect to following Definiton 3.Thus,

iff .

In other words,

iff ,

iff

Since, following Definition 4, and .

s being special cases of s the theorem can similarly be proved if and are s, or if is an and is a or a as well. (Q.E.D)

Theorem 1 essentially gives the definition of preorder-based truth ordering () of restricted fuzzy numbers of . Therefore, for any restricted fuzzy numbers ;

;

iff .

;

iff .

;

iff .

;

iff .

Example 4: This example analytically validates Theorem 1. Consider two truth assignments and , with and being their actual truth values approximated by , respectively. The actual truth status and are independent random variables, that follow a uniform and a trapezoidal probability density functions and respectively. So, and . The joint probability density function .

,

,

,

,

,

.

Now,

,

,

,

,

As a special case, having in gives . Putting this condition in the above derivation gives,

.

Consider three propositions , and , ascribed with , and .

, , . It can be seen, and . Also, and .

So, and .

#### 3.2.2 Knowledge-ordering

As evident from Definition 3, the knowledge ordering is based on the length of s, i.e., more is the length more is the underlying uncertainty. Therefore, the length of an identifies its level of uncertainty.

Uncertainty degree of TFN:

The concept of length is not so obvious for as it is for s. To do so, the -cut decomposition of is used.

For the -cut for any any value of is an given as . Now being an , the degree of its uncertainty can be evaluated to be:

.

varies with different values of in [0,1]. Hence the average uncertainty(or length) is obtained as:

Thus for two s in , namely and ; it can be said,

Uncertainty degree of TrFN:

For and for some in ; and . Therefore,

.

Hence, for and ,

.

In a nutshell,

uncertainty degree of , ;

uncertainty degree of , ;

uncertainty degree of , .

For any restricted fuzzy numbers ,

iff .

Example 5: Consider , , and , and .

Now, , i.e., . Therefore,

.

This is intuitive, since in case of all values in are equally probable, whereas for , is more likely than any other value in ; which means provides more information about the truth status of the underlying proposition than . lies in between.

Note: One notable point is that the uncertainty degrees of a fuzzy number as calculated is actually equal to the underlying area of the membership function of the fuzzy number. When there is no uncertainty and a specific membership value is assigned then the uncertainty degree is zero and so is the area under the curve of the form , for some . This can be utilised for calculating the uncertainty degree of semi-restricted s and s.

### 3.3 Truth ordering and knowledge ordering of truncated semi-restricted TrFNs and Tfns in T:

The notion of truth and knowledge ordering, as defined over restricted fuzzy numbers of , can be extended to every pair of members of .

#### 3.3.1 Uncertainty degree and knowledge ordering:

For semi-restricted fuzzy numbers their truncated versions within the interval [0,1] are considered. Therefore, the expressions for uncertainty degree as specified in section 3.2.2 is no longer valid if the base-range of the fuzzy number exceeds [0,1]. However, from the notion developed in previous subsection, the uncertainty degree can be easily calculated from evaluating the area underlying the curve in the interval [0,1]. The more general expressions for uncertainty degree of elements of are presented here.

For ,

For ,

In general, for any in , the knowledge degree can be specified as:

.