On the measure of conflicts: Shapley Inconsistency Values

There are relatively few proposals for inconsistency measures for propositional belief bases. However inconsistency measures are potentially as important as information measures for artificial intelligence, and more generally for computer science. In particular, they can be useful to define various operators for belief revision, belief merging, and negotiation. The measures that have been proposed so far can be split into two classes. The first class of measures takes into account the number of formulae required to produce an inconsistency: the more formulae required to produce an inconsistency, the less inconsistent the base. The second class takes into account the proportion of the language that is affected by the inconsistency: the more propositional variables affected, the more inconsistent the base. Both approaches are sensible, but there is no proposal for combining them. We address this need in this paper: our proposal takes into account both the number of variables affected by the inconsistency and the distribution of the inconsistency among the formulae of the base. Our idea is to use existing inconsistency measures in order to define a game in coalitional form, and then to use the Shapley value to obtain an inconsistency measure that indicates the responsibility/contribution of each formula to the overall inconsistency in the base. This allows us to provide a more reliable image of the belief base and of the inconsistency in it.     

Relevance sensitive belief structures

We propose a new relevance sensitive model for representing and revising belief structures, which relies on a notion of partial language splitting and tolerates some amount of inconsistency while retaining classical logic. The model preserves an agent's ability to answer queries in a coherent way using Belnap's four‐valued logic. Axioms analogous to the AGM axioms hold for this new model. The distinction between implicit and explicit beliefs is represented and psychologically plausible, computationally tractable procedures for query answering and belief base revision are obtained. This revised version was published online in June 2006 with corrections to the Cover Date.     

An AGM-style belief revision mechanism for probabilistic spatio-temporal logics

There is now extensive interest in reasoning about moving objects. A probabilistic spatio-temporal (PST) knowledge base (KB) contains atomic statements of the form “Object o is/was/will be in region r at time t with probability in the interval [?,u]”. In this paper, we study mechanisms for belief revision in PST KBs. We propose multiple methods for revising PST KBs. These methods involve finding maximally consistent subsets and maximal cardinality consistent subsets. In addition, there may be applications where the user has doubts about the accuracy of the spatial information, or the temporal aspects, or about the ability to recognize objects in such statements. We study belief revision mechanisms that allow changes to the KB in each of these three components. Finally, there may be doubts about the assignment of probabilities in the KB. Allowing changes to the probability of statements in the KB yields another belief revision mechanism. Each of these belief revision methods may be epistemically desirable for some applications, but not for others. We show that some of these approaches cannot satisfy AGM-style axioms for belief revision under certain conditions. We also perform a detailed complexity analysis of each of these approaches. Simply put, all belief revision methods proposed that satisfy AGM-style axioms turn out to be intractable with the exception of the method that revises beliefs by changing the probabilities (minimally) in the KB. We also propose two hybrids of these basic approaches to revision and analyze the complexity of these hybrid methods.     

Internal structure of coalitions in competitive and altruistic graphical coalitional games

This paper introduces a certain graphical coalitional game where the internal topology of the coalition depends on a prescribed communication graph structure among the agents. The game Value Function is required to satisfy four Axioms of Value. These axioms make it possible to provide a refined study of coalition structures on graphs by defining a formal graphical game and by assigning a Positional Advantage, based on the Shapley value, to each agent in a coalition based on its connectivity properties within the graph. Using the Axioms of Value the graphical coalitional game can be shown to satisfy properties such as convexity, fairness, cohesiveness, and full cooperativeness. Three measures of the contributions of agents to a coalition are introduced: marginal contribution, competitive contribution, and altruistic contribution. The mathematical framework given here is used to establish results regarding the dependence of these three types of contributions on the graph topology, and changes in these contributions due to changes in graph topology. Based on these different contributions, three online sequential decision games are defined on top of the graphical coalitional game, and the stable graphs under each of these sequential decision games are studied. It is shown that the stable graphs under the objective of maximizing the marginal contribution are any connected graph. The stable graphs under the objective of maximizing the competitive contribution are the complete graph. The stable graphs under the objective of maximizing the altruistic contribution are any tree.     

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.     

On the cardinality index and attitudinal character of fuzzy measures

We briefly introduce the fuzzy measure and then discuss its use in representing information about uncertain variables. A relationship between the fuzzy measure and the Dempster-Shafer belief structure is discussed and a method for generating the family of fuzzy measures associated with a belief structure is described. We discuss the use of the Shapley index as a means for introducing an extension of the concept of entropy to fuzzy measures called the Shapley entropy. It is shown that all fuzzy measures generated from a given Dempster-Shafer belief structure have the same value for their Shapley entropy. We introduce the cardinality index of a fuzzy measure and use it to define the attitudinal character of a fuzzy measure. A semantics for this attitudinal character in the framework of using fuzzy measures to represent information about uncertain variables is suggested.     

Method for Restoring Consistency in Probabilistic Knowledge Bases

ABSTRACT

Ensuring consistency of knowledge systems is always one of the essential requirements because, without it, most of these systems become useless. Because of the importance, many studies have involved the restoration of consistency in knowledge systems. However, these approaches are only implemented on knowledge systems that are represented by logic or probabilistic logic, thus when we apply them to probabilistic knowledge systems, there are many inadequacies. To overcome these drawbacks, in this paper, we put forward a new model for restoring the consistency of a probabilistic knowledge base by focusing on changing the probabilities in this knowledge base via several inconsistency measures. To this end, a set of inconsistency measures is presented and a family of consistency restoring operators for probabilistic knowledge bases is introduced. Next, an axiomatic model consists of a set of axioms is built to characterize the desirable properties of the consistency restoring operators. Finally, the properties of each consistency restoring operator in the introduced family are investigated and discussed.     

A programmable approach to maintenance of a finite knowledge base

In this paper,we present a programmable method of revising a finite clause set.We first present a procedure whose formal parameters are a consistent clause set Γand a clause A and whose output is a set of minimal subsets of Γwhich are inconsistent with A.The maximal consistent subsets can be generated from all minimal inconsistent subsets.We develop a prototype system based on the above procedure,and discuss the implementation of knowledge base maintenance.At last,we compare the approach presented in this paper with other related approaches,The main characteristic of the approach is that it can be implemented by a computer program.     

Decentralized checking of context inconsistency in pervasive computing environments

Contexts are often inconsistent in pervasive computing environments, owing to many heterogeneous devices with limited processing capabilities, imperfect measurement techniques, and user movement. A variety of schemes have been proposed to check context inconsistency. However, they implicitly require central control. This requirement inhibits their effectiveness in some pervasive computing environments (e.g., transport systems) where all nodes are resource-constrained and cannot act as a centralized node. To this end, we propose in this paper DCCI—a scheme of Decentralized Checking of Context Inconsistency in pervasive computing environments. DCCI exploits a simple, yet efficient, preference-based locality that denotes nodes requiring that the same context can check the inconsistency on this type of contexts. According to this locality, DCCI constructs a preference-based shortcut structure such that it checks context inconsistency within the shortcut structure. Extensive experiments show that DCCI can accurately and efficiently check context inconsistency in the presence of node churns and heterogeneity.     

基于量测噪声和观测次数的EKF-SLAM一致性分析

Inconsistency is a fundamental problem in simultaneous localization and mapping (SLAM). Previous works from predecessors have studied the inconsistent problem of extended Kalman filter (EKF) SLAM algorithm focusing on the linearization errors. In this paper, we studied the inconsistency issue of EKF SLAM in theory based on measurement noise and observation time. In a simplified situation, we deduced some useful theorems of estimated covariance matrix. Then, we made use of them to investigate the inconsistency issue. We showed that the measurement noise and the observation times can drive the EKF SLAM out of consistency. Moreover, we demonstrated the explicit effects of measurement noise and observation times on inconsistency of the EKF SLAM. Our simulation experiments verified the results.     

Resource-Bounded Paraconsistent Inference

In this paper, a new framework for reasoning from inconsistent propositional belief bases is presented. A family of resource-bounded paraconsistent inference relations is introduced. Such inference relations are based on S-3 entailment, an inference relation logically weaker than the classical one and parametrized by a set S of propositional variables. The computational complexity of our relations is identified, and their logical properties are analyzed. Among the strong features of our framework is the fact that tractability is ensured each time |S| is bounded and a limited amount of knowledge is taken into account within the belief base. Furthermore, binary connectives , behave in a classical manner. Finally, our framework is general enough to encompass several paraconsistent multi-valued logics (including S-3, J ₃ and its restrictions), the standard coherence-based approach to inconsistency handling (based on the selection of consistent subbases) and some signed systems for paraconsistent reasoning as specific cases.     

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.     

Reasoning about knowledge and belief: a survey

We examine a number of logics of knowledge and belief from the perspective of knowledge-based systems. We are concerned with the beliefs of a knowledge-based system, including both the system's base set of beliefs–those garnered directly from the world–and beliefs that follow from the base set. Three things to consider with such logics are the expressive power of the language of the logic, the correctness and completeness of the inferences sanctioned, and the speed with which it is possible to determine whether a given sentence is believed. The influential possible worlds approach to representing belief has the property of logical omniscience, which makes for inferences that are unacceptable in the context of belief and may take too much time to make. We examine a number of weak logics which attempt to deal with these problems. These logics divide into three categories: those that admit incomplete or inconsistent situations into their semantics, those that posit a number of distinct states for a believer which correspond roughly to frames of mind, and those that incorporate axioms or other syntactic entities directly into the semantics. As to expressive power, we consider whether belief should be represented by a predicate or a sentential operator and examine the boundary between self-referential and inconsistent systems. Finally, we consider logics of believing only , which add the assumption that a system's base set of beliefs are, in a certain sense, all that it believes.     

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...     

A general framework for computing maximal contractions

This paper investigates the problem of computing all maximal contractions of a given formula set Γ with respect to a consistent set Δ of atomic formulas and negations of atomic formulas. We first give a constructive definition of minimal inconsistent subsets and propose an algorithmic framework for computing all minimal inconsistent subsets of any given formula set. Then we present an algorithm to compute all maximal contractions fromminimal inconsistent subsets. Based on the algorithmic framework and the algorithm, we propose a general framework for computing all maximal contractions. The computability of the minimal inconsistent subset and maximal contraction problems are discussed. Finally, we demonstrate the ability of this framework by applying it to the first-order language without variables and design an algorithmfor the computation of all maximal contractions.     

Suppressing detection of inconsistency hazards with pattern learning

Context: Inconsistency detection and resolution is critical for context-aware applications to ensure their normal execution. Contexts, which refer to pieces of environmental information used by applications, are checked against consistency constraints for potential errors. However, not all detected inconsistencies are caused by real context problems. Instead, they might be triggered by improper checking timing. Such inconsistencies are ephemeral and usually harmless. Their detection and resolution is unnecessary, and may even be detrimental. We name them inconsistency hazards.Objective: Inconsistency hazards should be prevented from being detected or resolved, but it is not straightforward since their occurrences resemble real inconsistencies. In this article, we present SHAP, a pattern-learning based approach to suppressing the detection of such hazards automatically.Method: Our key insight is that detection of inconsistency hazards is subject to certain patterns of context changes. Although such patterns can be difficult to specify manually, they may be learned effectively with data mining techniques. With these patterns, we can reasonably schedule inconsistency detections.Results: The experimental results show that SHAP can effectively suppress the detection of most inconsistency hazards (over 90%) with negligible overhead.Conclusions: Comparing with other approaches, our approach can effectively suppress the detection of inconsistency hazards, and at the same time allow real inconsistencies to be detected and resolved timely.