Localising iceberg inconsistencies

Bona, Glauber De; Hunter, Anthony

doi:10.1016/j.artint.2017.02.005

Cited by 16 publications

(14 citation statements)

References 34 publications

(81 reference statements)

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“…We cannot just use the results from the propositional case for two reasons: first, our definitions for the postulates are different and second, the transformation of a general information space allows only formulas in propositional logic of a certain form. For example, iceberg inconsistencies [13] cannot occur.…”

Section: Postulate Satisfaction For General Information Spacesmentioning

confidence: 99%

General information spaces: measuring inconsistency, rationality postulates, and complexity

Grant

Parisi

2021

Ann Math Artif Intell

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AI systems often need to deal with inconsistent information. For this reason, since the early 2000s, some AI researchers have developed ways to measure the amount of inconsistency in a knowledge base. By now there is a substantial amount of research about various aspects of inconsistency measuring. The problem is that most of this work applies only to knowledge bases formulated as sets of formulas in propositional logic. Hence this work is not really applicable to the way that information is actually stored. The purpose of this paper is to extend inconsistency measuring to real world information. We first define the concept of general information space which encompasses various types of databases and scenarios in AI systems. Then, we show how to transform any general information space to an inconsistency equivalent propositional knowledge base, and finally apply propositional inconsistency measures to find the inconsistency of the general information space. Our method allows for the direct comparison of the inconsistency of different information spaces, even though the data is presented in different ways. We demonstrate the transformation on four general information spaces: a relational database, a graph database, a spatio-temporal database, and a Blocks world scenario, where we apply several inconsistency measures after performing the transformation. Then we review so-called rationality postulates that have been developed for propositional knowledge bases as a way to judge the intuitive properties of these measures. We show that although general information spaces may be nonmonotonic, there is a way to transform the postulates so they can be applied to general information spaces and we show which of the measures satisfy which of the postulates. Finally, we discuss the complexity of inconsistency measures for general information spaces.

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Section: Postulate Satisfaction For General Information Spacesmentioning

confidence: 99%

General information spaces: measuring inconsistency, rationality postulates, and complexity

Grant

Parisi

2021

Ann Math Artif Intell

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“…In contrast, the absence of strong rationality postulates leave a vacuum for rather meaningless proposals to arise, such as I e and I e . To address this dilemma, more or less demanding properties, as those proposed in De Bona and Hunter (2017), could be employed to describe and classify inconsistency measures, instead of being required in their definition. Augmented inconsistency graphs, or simply inconsistency graphs, can be useful for the formulation of properties of this kind.…”

Section: From Inconsistency Measures To the Inconsistency Graphmentioning

confidence: 99%

“…So IG(K 1 ) = IG(K 2 ); however, the augmented inconsistency graph has an extra vertex in U . The formula ¬b ∧ d is called an iceberg inconsistency (De Bona & Hunter, 2017) as it is free in this set but if the first formula were broken into two formulas, as a ∧ ¬a and b ∧ ¬c, where the two formulas together are logically equivalent to the original formula, ¬b ∧ d would no longer be free. I 1 (K 2 ) = I 1 (K 1 ) = 3 as the atom d in the free formula is not counted.…”

Section: A Hierarchy Of Non-syntactic Measuresmentioning

confidence: 99%

Classifying Inconsistency Measures Using Graphs

Bona¹,

Grant²,

Hunter³

et al. 2019

jair

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The aim of measuring inconsistency is to obtain an evaluation of the imperfections in a set of formulas, and this evaluation may then be used to help decide on some course of action (such as rejecting some of the formulas, resolving the inconsistency, seeking better sources of information, etc). A number of proposals have been made to define measures of inconsistency. Each has its rationale. But to date, it is not clear how to delineate the space of options for measures, nor is it clear how we can classify measures systematically. To address these problems, we introduce a general framework for comparing syntactic measures of inconsistency. It is based on the notion of an inconsistency graph for each knowledgebase (a bipartite graph with a set of vertices representing formulas in the knowledgebase, a set of vertices representing minimal inconsistent subsets of the knowledgebase, and edges representing that a formula belongs to a minimal inconsistent subset). We then show that various measures can be computed using the inconsistency graph. Then we introduce abstractions of the inconsistency graph and use them to construct a hierarchy of syntactic inconsistency measures. Furthermore, we extend the inconsistency graph concept with a labeling that extends the hierarchy to include some other types of inconsistency measures.

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“…If two inconsistent pairs satisfy (x, y) ≤ p (x ′ , y ′ ), the latter can be viewed at least as inconsistent as the former. In a more general context, researches have been trying to address questions like "where is the inconsistency", "how severer is it", and how to make changes to an inconsistent theory (see, e.g., (Bona and Hunter 2017)). A deeper understanding of inconsistencies in the context of AFT presents an interesting future direction.…”

Section: Examplementioning

confidence: 99%

Alternating Fixpoint Operator for Hybrid MKNF Knowledge Bases as an Approximator of AFT

Liu¹,

You²

2019

Rules and Reasoning

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Approximation fixpoint theory (AFT) provides an algebraic framework for the study of fixpoints of operators on bilattices and has found its applications in characterizing semantics for various classes of logic programs and nonmonotonic languages. In this paper, we show one more application of this kind: the alternating fixpoint operator by Knorr et al. for the study of the well-founded semantics for hybrid MKNF knowledge bases is in fact an approximator of AFT in disguise, which, thanks to the power of abstraction of AFT, characterizes not only the well-founded semantics but also two-valued as well as three-valued semantics for hybrid MKNF knowledge bases. Furthermore, we show an improved approximator for these knowledge bases, of which the least stable fixpoint is information richer than the one formulated from Knorr et al.'s construction. This leads to an improved computation for the well-founded semantics. This work is built on an extension of AFT that supports consistent as well as inconsistent pairs in the induced product bilattice, to deal with inconsistencies that arise in the context of hybrid MKNF knowledge bases. This part of the work can be considered generalizing the original AFT from symmetric approximators to arbitrary approximators.

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Localising iceberg inconsistencies

Cited by 16 publications

References 34 publications

General information spaces: measuring inconsistency, rationality postulates, and complexity

General information spaces: measuring inconsistency, rationality postulates, and complexity

Classifying Inconsistency Measures Using Graphs

Alternating Fixpoint Operator for Hybrid MKNF Knowledge Bases as an Approximator of AFT

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