Generalization of the Dempster-Shafer theory: a fuzzy-valuedmeasure

The Dempster-Shafer theory (DST) may be considered as a generalization of the probability theory, which assigns mass values to the subsets of the referential set and suggests an interval-valued probability measure. There have been several attempts for fuzzy generalization of the DST by assigning mass (probability) values to the fuzzy subsets of the referential set. The interval-valued probability measures thus obtained are not equivalent to the original fuzzy body of evidence. In this paper, a new generalization of the DST is put forward that gives a fuzzy-valued definition for the belief, plausibility, and probability functions over a finite referential set. These functions are all equivalent to one another and to the original fuzzy body of evidence. The advantage of the proposed model is shown in three application examples. It can be seen that the proposed generalization is capable of modeling the uncertainties in the real world and eliminate the need for extra preassumptions and preprocessing     

信念修正的完全和可操作的方法

给出了命题逻辑上信念修正的两种可操作的完全方法.首先对R-演算的规则进行了修改,使得对任何一个极大协调的子集都通过这组规则得到.然后,给出了求得所有的极小不协调子集的一组规则.最后,给出一个过程,该过程能求得所有的极大协调子集.因为这两种方法都能求得所有的极大协调子集,所以把它们称为完全的.     

A new belief rule base knowledge representation scheme and inference methodology using the evidential reasoning rule for evidence combination

In this paper, we extend the original belief rule-base inference methodology using the evidential reasoning approach by i) introducing generalised belief rules as knowledge representation scheme, and ii) using the evidential reasoning rule for evidence combination in the rule-base inference methodology instead of the evidential reasoning approach. The result is a new rule-base inference methodology which is able to handle a combination of various types of uncertainty.Generalised belief rules are an extension of traditional rules where each consequent of a generalised belief rule is a belief distribution defined on the power set of propositions, or possible outcomes, that are assumed to be collectively exhaustive and mutually exclusive. This novel extension allows any combination of certain, uncertain, interval, partial or incomplete judgements to be represented as rule-based knowledge. It is shown that traditional IF-THEN rules, probabilistic IF-THEN rules, and interval rules are all special cases of the new generalised belief rules.The rule-base inference methodology has been updated to enable inference within generalised belief rule bases. The evidential reasoning rule for evidence combination is used for the aggregation of belief distributions of rule consequents.     

维护的证明论系统

给定一阶语言及该语言的一个理论，假设需要在理论中添加一个与理论不和谐的语句，并要求保持理论的扩张是和谐的，就必须删除理论内的某些语句．删除理论中尽可能少的语句，即保留理论与需要添加语句和谐的一个极大子集，是构造理论扩张的一种方法．本文构造出了针对上述理论扩张的证明论．该证明论的可靠性及完全性刻划了一般形式的理论扩张与典型的理论扩张间的联系．对于命题逻辑，本文还给出了判定理论扩张的一种方法．     

An Operational Approach to Belief Revision

A deduction system,called RE-proof system,is constructed for generating the revisions of first order belief sets.When a belief set is rejected by a given fact,all maximal subsets of the belief set consistent with the fact can be deduced from the proof system.The soundness and completeness of the RE-proof system are proved,which imply that there exists a resolution method to decide whether a revision retains a maximal subset of a belief set.     

Associativity in combination of belief functions; a derivation of minC combination

 The nature of a contradiction (conflict) between two belief functions is investigated. Alternative ways of distributing the contradiction among nonempty subsets of frame of discernment are studied. The paper employes a new approach to understanding contradictions and introduces an original notion of potential contradiction. A method of an associative combination of generalized belief functions – minC combination and its derivation – is presented as part of the new approach. A proportionalization of generalized results is suggested as well. RID="*" ID="*" Support by Grant No. 1030803 of the GA AV ČR is acknowledged. I am grateful to Philippe Smets for a fruitful discussion on the topic.     

A survey of multiple contractions

The AGM theory of belief contraction is extended tomultiple contraction, i.e. to contraction by a set of sentences rather than by a single sentence. There are two major variants: Inpackage contraction all the sentences must be removed from the belief set, whereas inchoice contraction it is sufficient that at least one of them is removed. Constructions of both types of multiple contraction are offered and axiomatically characterized. Neither package nor choice contraction can in general be reduced to contractions by single sentences; in the finite case choice contraction allows for reduction.     

Operational and complete approaches to belief revision

Two operational approaches to belief revision are presented in this paper.The rules of R-calculus are modified in order to deduce all the maximal consistent subsets.Another set of given in order to deduce all the minimal inconsistent subsets.Then a procedure,which can generate all the maximal consistent subsets,is presented.They are complete approaches,since all the maximal consistent subsets can be deduced or generated.In this paper,only the case of propositional logic is considered.     

Infobase Change: A First Approximation

Generalisations of theory change involving operations on arbitrary sets ofwffs instead of on belief sets (i.e., sets closed under a consequencerelation), have become known as base change. In one view, a base should bethought of as providing more structure to its generated belief set, whichmeans that it can be employed to determine the theory contraction operationassociated with a base contraction operation. In this paper we follow suchan approach as the first step in defining infobase change. We think of an infobase as a finite set of wffs consisting of independently obtainedbits of information. Taking AGM theory change (Alchourrón et al. 1985) as the general framework, we present a method that uses the structure of aninfobase B to obtain an AGM theory contraction operation for contractingthe belief set Cn(B). Both the infobase and the obtained theory contraction operation then play a role in constructing a unique infobasecontraction operation. Infobase revision is defined in terms of an analogueof the Levi Identity, and it is shown that the associated theory revisionoperation satisfies the AGM postulates for revision. Because every infobaseis associated with a unique infobase contraction and revision operation, the method also allows for iterated base change.     

Changes of disjunctively closed bases

An operator of contraction for a belief set (a theory) can be obtained by assigning to it a belief base and an operator of partial meet contraction for that base. It is argued that closure of the base under disjunction is an intuitively reasonable condition. Axiomatic characterizations are given of the contractions of belief sets that can be generated by (various types of) partial meet contraction on disjunctively closed bases. The corresponding revision operators are also characterized. Finally, some results are reported on operations on bases that are closed under material implication.I would like to thank Hans Rott and two anonymous referees for valuable comments and the Swedish Council for Research in the Humanities and Social Sciences for financial support.     

A general framework for measuring inconsistency through minimal inconsistent sets

Hunter and Konieczny explored the relationships between measures of inconsistency for a belief base and the minimal inconsistent subsets of that belief base in several of their papers. In particular, an inconsistency value termed MIV _C , defined from minimal inconsistent subsets, can be considered as a Shapley Inconsistency Value. Moreover, it can be axiomatized completely in terms of five simple axioms. MinInc, one of the five axioms, states that each minimal inconsistent set has the same amount of conflict. However, it conflicts with the intuition illustrated by the lottery paradox, which states that as the size of a minimal inconsistent belief base increases, the degree of inconsistency of that belief base becomes smaller. To address this, we present two kinds of revised inconsistency measures for a belief base from its minimal inconsistent subsets. Each of these measures considers the size of each minimal inconsistent subset as well as the number of minimal inconsistent subsets of a belief base. More specifically, we first present a vectorial measure to capture the inconsistency for a belief base, which is more discriminative than MIV _C . Then we present a family of weighted inconsistency measures based on the vectorial inconsistency measure, which allow us to capture the inconsistency for a belief base in terms of a single numerical value as usual. We also show that each of the two kinds of revised inconsistency measures can be considered as a particular Shapley Inconsistency Value, and can be axiomatically characterized by the corresponding revised axioms presented in this paper.     

A sound and complete R-calculi with respect to contraction and minimal change

AGM postulates are for belief revision (revision by a single belief), and DP postulates are for iterated revision (revision by a finite sequence of beliefs). R-calculus is given for R-configurations Δ|Γ, where Δ is a set of atomic formulas or the negations of atomic formulas, and Γ is a finite set of formulas. We shall give two R-calculi C and M (sets of deduction rules) such that for any finite consistent sets Γ, Δ of formulas in the propositional logic, there is a consistent set Θ ? Γ of formulas such that Δ|Γ ? Δ, Θ is provable and Θ is a contraction of Γ by Δ or a minimal change of Γ by Δ; and prove that C and M are sound and complete with respect to the contraction and the minimal change, respectively.     

Strong theorems on the form of probability distributions of Euclidean distances between an ordered sets of points in the plane upon random rotations or reflections of their subsets

This paper presents theorems about the form of probability distributions of Euclidean distances between ordered sets of points in the plane for random transformations of their subsets. Four options are considered for the change: a random rotation as a whole, a random reflection as a whole, simultaneous independent random rotations of two ordered disjoint subsets that make up the original ordered set, and simultaneous independent random reflection of two ordered disjoint subsets that make up the original ordered set. The theorems contain expressions to calculate the probability density function to calculate the initial moments.     

一个在Horn子句中求解极大缩减的算法

在信念修正理论中,一个核心问题是求解一个公式集合关于事实集合的所有极大协调子集,即极大缩减.本文尝试从算法的角度来解决这一问题,研究在Horn子句中求解所有极大缩减的算法.首先,本文指出并证明了公式集合和事实集合并集的极小不协调子集与公式集合关于事实集合的极大缩减之间的转化关系.其次,给出并证明了Horn子句集合极小不协调的一个必要条件.然后,基于上述两个结论,本文提出了一个在Horn子句中枚举公式集合和事实集合并集的极小不协调子集的交互式算法和一个通过这些极小不协调子集计算所有极大缩减的算法.最后,综合这两个算法,提出了一个在Horn子句中求解所有极大缩减的交互式算法.     

An algorithm to compute maximal contractions for Horn clauses

In the theory of belief revision, the computation of all maximal subsets (maximal contractions) of a formula set with respect to a set of facts is one of the key problems. In this paper, we try to solve this problem by studying the algorithm to compute all maximal contractions for Horn clauses. First, we point out and prove the conversion relationship between minimal inconsistent subsets of union of the formula set and the set of facts and maximal contractions of the formula set with respect to th...     

Recovery and Epistemic Residue

Two recent defences of the recovery postulate for contraction of belief sets are analyzed. It is concluded that recovery is defensible as a by-product of a formalization that is idealized in the sense of being simplified for the sake of clarity. However, recovery does not seem to be a required feature of the doxastic behaviour of ideal (perfectly rational) agents. It is reasonable to expect that there should be epistemic residues (remnants of rejected beliefs), but not that these should always suffice to recover all the original beliefs if the discarded belief is reinstated.     

Axiomatic characterization of belief merging by negotiation

Belief merging has been an active research field with many important applications. The major approaches for the belief merging problems, considered as arbitration processes, are based on the construction of the total pre-orders of alternatives using distance functions and aggregation functions. However, these approaches require that all belief bases are provided explicitly and the role of agents, who provide the belief bases, are not adequately considered. Therefore, the results are merely ideal and difficult to apply in the multi-agent systems. In this paper, we approach the merging problems from other point of view. Namely, we treat a belief merging problem as a game, in which rational agents participate in a negotiation process to find out a jointly consistent consensus trying to preserve as many important original beliefs as possible. To this end, a model of negotiation for belief merging is presented, a set of rational and intuitive postulates to characterize the belief merging operators are proposed, and a representation theorem is presented.     

Iterated Belief Change

Most existing formalizations treat belief change as a single-step process, and ignore several problems that become important when a theory, or belief state, is revised over several steps. This paper identifies these problems, and argues for the need to retain all of the multiple possible outcomes of a belief change step, and for a framework in which the effects of a belief change step persist as long as is consistently possible. To demonstrate that such a formalization is indeed possible, we develop a framework, which uses the language of PJ-default logic (Delgrande and Jackson 1991) to represent a belief state, and which enables the effects of a belief change step to persist by propagating belief constraints . Belief change in this framework maps one belief state to another, where each belief state is a collection of theories given by the set of extensions of the PJ-default theory representing that belief state. Belief constraints do not need to be separately recorded; they are encoded as clearly identifiable components of a PJ-default theory. The framework meets the requirements for iterated belief change that we identify and satisfies most of the AGM postulates (Alchourrón, Gärdenfors, and Makinson 1985) as well.