Mathematical Problems in Engineering

Volume 2015, Article ID 134950, 7 pages

http://dx.doi.org/10.1155/2015/134950

## A Max-Term Counting Based Knowledge Inconsistency Checking Strategy and Inconsistency Measure Calculation of Fuzzy Knowledge Based Systems

School of Computer Science and Technology, Henan Polytechnic University, No. 2001, ShiJi Avenue, Jiaozuo 454000, China

Received 27 April 2015; Revised 21 July 2015; Accepted 4 August 2015

Academic Editor: Miguel A. Salido

Copyright © 2015 Hui-lai Zhi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The task of finding all the minimal inconsistent subsets plays a vital role in many theoretical works especially in large knowledge bases and it has been proved to be a NP-complete problem. In this work, at first we propose a max-term counting based knowledge inconsistency checking strategy. And, then, we put forward an algorithm for finding all minimal inconsistent subsets, in which we establish a Boolean lattice to organize the subsets of the given knowledge base and use leaf pruning to optimize the algorithm efficiency. Comparative experiments and analysis also show the algorithm’s improvement over past approaches. Finally, we give an application for inconsistency measure calculation of fuzzy knowledge based systems.

#### 1. Introduction

A large knowledge system operating for a long time almost inevitably becomes polluted by wrong data that make the system inconsistent. Despite this fact, a sizeable part of the system remains unpolluted and retains useful information. It is widely adopted that a maximal consistent subset of a system contains a significant portion of unpolluted data [1]. So, simply characterizing a knowledge base as either consistent or inconsistent is of little practical value, and thus ensuring the consistency becomes an important issue [2–4].

In practice, there are two types of methods: one method is based on minimal inconsistent subsets, where every strict subset is consistent and the other is directly based on maximal consistent subsets. Actually, the relationship between minimal inconsistent subsets and maximal consistent subsets was discovered separately in [1, 5, 6], which is known as the hitting subset problem [7].

As finding minimal inconsistent subsets or maximal consistent subsets is NP-complete, the most efficient algorithm is not known yet, and there are a number of heuristic optimizations that can be used to substantially reduce the size of the search space. In practice, heuristic information [8, 9], optimization [10, 11], and hybrid techniques [12, 13] are recognized to reduce time complexity. In the latest research, McAreavey et al. presented a computational approach to finding and measuring inconsistency in arbitrary knowledge bases [14], while Mu et al. gave a method for measuring the significance of inconsistency in the viewpoints framework [15]. In all the abovementioned works, effectively finding minimal inconsistent subsets is the critical step which has a great impact on the applications especially for large knowledge bases. Apparently, its computational complexity depends on the underlying strategies used for checking the consistency of subsets of the knowledge base, but till now this important issue has not gotten a satisfying solution.

In this paper, we first propose an efficient strategy to check the consistency of a given knowledge base. And, then, we put forward an algorithm to find all of the minimal inconsistent subsets of the given knowledge base. Thereafter, to illustrate the algorithm’s improvement, we conduct thorough comparative experiments and analysis with respect to one of the latest proposed algorithm MARCO [16] and give a discussion on the relative algorithms DAA [17] and PDDS [18]. Finally, we give an application for inconsistency measure calculation of fuzzy knowledge based systems.

#### 2. Theoretical Basis

Let denote the propositional language built from a finite set of variables using logical connectives and logical constants . Every variable is called an atomic formula or an atom. A literal is an atom or its negation. A clause is a formula restricted to a disjunction of literals and let denote the set of variables in a clause . A knowledge base is a finite set of arbitrary formulae.

As every formula can be converted into an equivalent conjunction normal form (CNF) formula, knowledge base can be normalized in such a way that every formula contained in it is a clause. For a given normalized knowledge base, if there are no redundant clauses, we say it is an optimized knowledge base.

By the syntactic approach in proof theory, if both and can be derived from a knowledge base , then we say is inconsistent. With the semantic approach in model theory, an interpretation or world is a function from to the set of Boolean values . Let denote the set of worlds of . A world is a model of , denoted as , iff is true under in the classical truth-functional manner. Let denote the set of models of ; that is, . We say that is satisfiable iff there exists a model of . Conversely, is unsatisfiable iff there are no models of . These two approaches coincide in propositional logic; that is, a knowledge base is consistent iff is satisfiable.

In the following discussion, let the Greek lower case letters be formulae from and English lower case letters variables from .

*Definition 1. *For a Boolean function of variables , a sum term in which each of the variables appears once (in either its complemented or uncomplemented form) is called a max-term.

Proposition 2. *Let be variables and the different max-terms built on these variables. Then .*

For example, let , , , and be formulas built on . Then it is trivial to show that , which means no assignments to and satisfy , , , and simultaneously.

*Definition 3. *Let be a formula built on a set of variables . Then we call the extension of ; that is,and, for each , , .

For example, let be a formula built on . Then we have and . Actually, a simple manipulation leads to .

*Remark 4. *It is easy to see that carrying extension of does not change the original meaning of .

Theorem 5. *Let and be two formulas built on a set of variables , in which , . Then the following propositions hold:*(i)*If there exists a variable such that one of and appears in and the other one appears in , then .*(ii)*Otherwise, .*

*Proof. *(i) As there exists a variable such that one of and appears in and the other one appears in , then one of and must appear in and the other one must appear in , which makes differ from . Therefore we have .

(ii) If there does not exist a variable such that one of and appears in and the other one appears in , then there are two situations that need to be surveyed, respectively. *Situation *. If , then there exists only one common formula of and , that is, . Hence, we have , which is equivalent to *Situation *. If , then the common formula of and is where . Hence, we haveTherefore, combining the above two situations, the theorem is proved.

*Corollary 6. Let be a set of formulas built on a set of variables . Then the following propositions hold:(1)If there exist a variable and two formulas and such that one of and appears in and the other one appears in , then .(2)Otherwise, .*

*Theorem 7. Let be a set of formulas built on a set of variables . Then after carrying extensions for , respectively, and using to denote the number of different formulas obtained, we have Moreover, if , then is inconsistent; otherwise is consistent.*

*Proof. *According to inclusion-exclusion principle and in light of Proposition 2 the proof is trivial.

*Let be a set of formulas defined on . According to Theorem 7, we have and thus is consistent.*

*3. An Algorithm for Finding All Minimal Inconsistent Subsets*

*3. An Algorithm for Finding All Minimal Inconsistent Subsets*

*In this section, at first we propose an algorithm for finding all nominal inconsistent subsets via Boolean lattice. And, then, we give an illustrative example and a thorough comparative study with algorithm MARCO by using the number of visited subsets as the benchmark. Besides this, we also give a discussion on relative algorithms DAA and PDDS.*

*3.1. Algorithm*

*3.1. Algorithm*

*An algorithm to find the minimal inconsistent subsets of a given knowledge system must check each of its subsets for inconsistency. One way to proceed is to construct a Boolean lattice of subsets of the given knowledge system, which is initially used by Bird and Hinze in the process of finding the maximal consistent subsets [19].*

*A lattice is called a Boolean lattice if(i) is distributive,(ii) has 0 and 1,(iii)each has a complement .*

*Figure 1 sketches a three-variable Boolean lattice, where all the labels of the nodes consist of the power set of set .*