Revision over Partial Pre-orders: A Postulational Study

Belief revision is the process that incorporates, in a consistent way,
a new piece of information, called input, into a belief base. When both belief
bases and inputs are propositional formulas, a set of natural and rational properties, known as AGM postulates, have been proposed to define genuine revision operations. This paper addresses the following important issue : How to revise a partially pre-ordered information (representing initial beliefs) with a new partially pre-ordered information (representing inputs) while preserving AGM postulates? We first provide a particular representation of partial pre-orders (called units) using the concept of closed sets of units. Then we restate AGM postulates in this framework by defining counterparts of the notions of logical entailment and logical consistency. In the second part of the paper, we provide some examples of revision operations that respect our set of postulates. We also prove that our revision methods extend well-known lexicographic revision and natural revision for both cases where the input is either a single propositional formula or a total pre-order.

A Syntax-based approach to measuring the degree of inconsistency for belief bases

Measuring the degree of inconsistency of a belief base is an important issue in many real world applications. It has been increasingly recognized that deriving syntax sensitive inconsistency measures for a belief base from its minimal inconsistent subsets is a natural way forward. Most of the current proposals along this line do not take the impact of the size of each minimal inconsistent subset into account. However, as illustrated by the well-known Lottery Paradox, as the size of a minimal inconsistent subset increases, the degree of its inconsistency decreases. Another lack in current studies in this area is about the role of free formulas of a belief base in measuring the degree of inconsistency. This has not yet been characterized well. Adding free formulas to a belief base can enlarge the set of consistent subsets of that base. However, consistent subsets of a belief base also have an impact on the syntax sensitive normalized measures of the degree of inconsistency, the reason for this is that each consistent subset can be considered as a distinctive plausible perspective reflected by that belief base,whilst eachminimal inconsistent subset projects a distinctive viewof the inconsistency. To address these two issues,we propose a normalized framework formeasuring the degree of inconsistency of a belief base which unifies the impact of both consistent subsets and minimal inconsistent subsets. We also show that this normalized framework satisfies all the properties deemed necessary by common consent to characterize an intuitively satisfactory measure of the degree of inconsistency for belief bases. Finally, we use a simple but explanatory example in equirements engineering to illustrate the application of the normalized framework.

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 MIVC, 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 MIVC. 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.

Measuring conflict and agreement between two prioritized knowledge bases in possibilistic logic.

In this paper we investigate the relationship between two prioritized knowledge bases by measuring both the conflict and the agreement between them.First of all, a quantity of conflict and two quantities of agreement are defined. The former is shown to be a generalization of the well-known Dalal distance which is the hamming distance between two interpretations. The latter are, respectively, a quantity of strong agreement which measures the amount ofinformation on which two belief bases “totally” agree, and a quantity of weak agreement which measures the amount of information that is believed by onesource but is unknown to the other. All three quantity measures are based on the weighted prime implicant, which represents beliefs in a prioritized belief base. We then define a degree of conflict and two degrees of agreement based on our quantity of conflict and quantities of agreement. We also consider the impact of these measures on belief merging and information source ordering.

Context-dependent Combination of Sensor Information in Dempster-Shafer Theory for BDI

There has been much interest in the belief–desire–intention (BDI) agent-based model for developing scalable intelligent systems, e.g. using the AgentSpeak framework. However, reasoning from sensor information in these large-scale systems remains a significant challenge. For example, agents may be faced with information from heterogeneous sources which is uncertain and incomplete, while the sources themselves may be unreliable or conflicting. In order to derive meaningful conclusions, it is important that such information be correctly modelled and combined. In this paper, we choose to model uncertain sensor information in Dempster–Shafer (DS) theory. Unfortunately, as in other uncertainty theories, simple combination strategies in DS theory are often too restrictive (losing valuable information) or too permissive (resulting in ignorance). For this reason, we investigate how a context-dependent strategy originally defined for possibility theory can be adapted to DS theory. In particular, we use the notion of largely partially maximal consistent subsets (LPMCSes) to characterise the context for when to use Dempster’s original rule of combination and for when to resort to an alternative. To guide this process, we identify existing measures of similarity and conflict for finding LPMCSes along with quality of information heuristics to ensure that LPMCSes are formed around high-quality information. We then propose an intelligent sensor model for integrating this information into the AgentSpeak framework which is responsible for applying evidence propagation to construct compatible information, for performing context-dependent combination and for deriving beliefs for revising an agent’s belief base. Finally, we present a power grid scenario inspired by a real-world case study to demonstrate our work.

A belief revision framework for revising epistemic states with partial epistemic states

Belief revision performs belief change on an agent’s beliefs when new evidence (either of the form of a propositional formula or of the form of a total pre-order on a set of interpretations) is received. Jeffrey’s rule is commonly used for revising probabilistic epistemic states when new information is probabilistically uncertain. In this paper, we propose a general epistemic revision framework where new evidence is of the form of a partial epistemic state. Our framework extends Jeffrey’s rule with uncertain inputs and covers well-known existing frameworks such as ordinal conditional function (OCF) or possibility theory. We then define a set of postulates that such revision operators shall satisfy and establish representation theorems to characterize those postulates. We show that these postulates reveal common characteristics of various existing revision strategies and are satisfied by OCF conditionalization, Jeffrey’s rule of conditioning and possibility conditionalization. Furthermore, when reducing to the belief revision situation, our postulates can induce Darwiche and Pearl’s postulates C1 and C2.

Rational Partial Choice Functions and Their Application to Belief Revision.

Necessary and sufficient conditions for choice functions to be rational have been intensively studied in the past. However, in these attempts, a choice function is completely specified. That is, given any subset of options, called an issue, the best option over that issue is always known, whilst in real-world scenarios, it is very often that only a few choices are known instead of all. In this paper, we study partial choice functions and investigate necessary and sufficient rationality conditions for situations where only a few choices are known. We prove that our necessary and sufficient condition for partial choice functions boils down to the necessary and sufficient conditions for complete choice functions proposed in the literature. Choice functions have been instrumental in belief revision theory. That is, in most approaches to belief revision, the problem studied can simply be described as the choice of possible worlds compatible with the input information, given an agent’s prior belief state. The main effort has been to devise strategies in order to infer the agents revised belief state. Our study considers the converse problem: given a collection of input information items and their corresponding revision results (as provided by an agent), does there exist a rational revision operation used by the agent and a consistent belief state that may explain the observed results?

Managing Software Requirements Changes Based On Negotiation-Style Revision

For any proposed software project, when the software requirements specification has been established, requirements changes may result in not only a modification of the requirements specification but also a series of modifications of all existing artifacts during the development. Then it is necessary to provide effective and flexible requirements changes management. In this paper, we present an approach to managing requirements changes based on Booth’s negotiation-style framework for belief revision. Informally, we consider the current requirements specification as a belief set about the system-to-be. The request of requirements change is viewed as new information about the same system-to-be. Then the process of executing the requirements change is a process of revising beliefs about the system-to-be. We design a family of belief negotiation models appropriate for different processes of requirements revision, including the setting of the request of requirements change being fully accepted, the setting of the current requirements specification being fully preserved, and that of the current specification and the request of requirements change reaching a compromise. In particular, the prioritization of requirements plays an important role in reaching an agreement in each belief negotiation model designed in this paper.

Bridging Jeffrey's Rule, AGM Revision and Dempster Conditioning in the Theory of Evidence

Belief revision characterizes the process of revising an agent’s beliefs when receiving new evidence. In the field of artificial intelligence, revision strategies have been extensively studied in the context of logic-based formalisms and probability kinematics. However, so far there is not much literature on this topic in evidence theory. In contrast, combination rules proposed so far in the theory of evidence, especially Dempster rule, are symmetric. They rely on a basic assumption, that is, pieces of evidence being combined are considered to be on a par, i.e. play the same role. When one source of evidence is less reliable than another, it is possible to discount it and then a symmetric combination operation
is still used. In the case of revision, the idea is to let prior knowledge of an agent be altered by some input information. The change problem is thus intrinsically asymmetric. Assuming the input information is reliable, it should be retained whilst the prior information should be changed minimally to that effect. To deal with this issue, this paper defines the notion of revision for the theory of evidence in such a way as to bring together probabilistic and logical views. Several revision rules previously proposed are reviewed and we advocate one of them as better corresponding to the idea of revision. It is extended to cope with inconsistency between prior and input information. It reduces to Dempster
rule of combination, just like revision in the sense of Alchourron, Gardenfors, and Makinson (AGM) reduces to expansion, when the input is strongly consistent with the prior belief function. Properties of this revision rule are also investigated and it is shown to generalize Jeffrey’s rule of updating, Dempster rule of conditioning and a form of AGM revision.

On the Merit of Selecting Different Belief Merging Operators

Belief merging operators combine multiple belief bases (a profile) into a collective one. When the conjunction of belief bases is consistent, all the operators agree on the result. However, if the conjunction of belief bases is inconsistent, the results vary between operators. There is no formal manner to measure the results and decide on which operator to select. So, in this paper we propose to evaluate the result of merging operators by using three ordering relations (fairness, satisfaction and strength) over operators for a given profile. Moreover, a relation of conformity over operators is introduced in order to classify how well the operator conforms to the definition of a merging operator. By using the four proposed relations we provide a comparison of some classical merging operators and evaluate the results for some specific profiles.

Revising Partial Pre-Orders with Partial Pre-Orders: A Unit-Based Revision Framework

Belief revision studies strategies about how agents revise their belief states when receiving new evidence. Both in classical belief revision and in epistemic revision, a new input is either in the form of a (weighted) propositional formula or a total
pre-order (where the total pre-order is considered as a whole).
However, in some real-world applications, a new input can be a partial pre-order where each unit that constitutes the partial pre-order is important and should be considered individually. To address this issue, in this paper, we study how a partial preorder representing the prior epistemic state can be revised by another partial pre-order (the new input) from a different perspective, where the revision is conducted recursively on the individual units of partial pre-orders. We propose different revision operators (rules), dubbed the extension, match, inner and outer revision operators, from different revision points of view. We also analyze several properties for these operators.