Modeling Uncertainty and Imprecision in Nonmonotonic Reasoning using Fuzzy Numbers

Abstract

To deal with uncertainty in reasoning, interval-valued logic has been developed. But uniform intervals cannot capture difference in degrees of belief for different values in the interval. To salvage the problem triangular and trapezoidal fuzzy numbers are used as 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.Keywords Fuzzy numbers, Interval valued fuzzy sets, Preorderbased triangle, Answer Set Programming.[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 subintervals 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 [4]. 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 {tumor 0.8:0.4 , tumor 0.6:0.6 , tumor 0.4:0.8 , tumor 0.2:1 }. 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 [0, 1]. This example denotes that in realworld 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].Specif...