Syntax Splitting = Relevance + Independence: New Postulates for Nonmonotonic Reasoning From Conditional Belief Bases

2020 ◽  
Author(s):  
Gabriele Kern-Isberner ◽  
Christoph Beierle ◽  
Gerhard Brewka
Keyword(s):  
Knowledge Base ◽  
Belief Revision ◽  
Inductive Inference ◽  
Nonmonotonic Reasoning ◽  
Irrelevant Information ◽  
Knowledge Bases ◽  
P System ◽  
Conditional Belief ◽  
Conditional Knowledge ◽  
System P

Syntax splitting, first introduced by Parikh in 1999, is a natural and desirable property of KR systems. Syntax splitting combines two aspects: it requires that the outcome of a certain epistemic operation should only depend on relevant parts of the underlying knowledge base, where relevance is given a syntactic interpretation (relevance). It also requires that strengthening antecedents by irrelevant information should have no influence on the obtained conclusions (independence). In the context of belief revision the study of syntax splitting already proved useful and led to numerous new insights. In this paper we analyse syntax splitting in a different setting, namely nonmonotonic reasoning based on conditional knowledge bases. More precisely, we analyse inductive inference operators which, like system P, system Z, or the more recent c-inference, generate an inference relation from a conditional knowledge base. We axiomatize the two aforementioned aspects of syntax splitting, relevance and independence, as properties of such inductive inference operators. Our main results show that system P and system Z, whilst satisfying relevance, fail to satisfy independence. C-inference, in contrast, turns out to satisfy both relevance and independence and thus fully complies with syntax splitting.

scholarly journals Normal forms of conditional knowledge bases respecting system P-entailments and signature renamings

2021 ◽  
Author(s):  
Christoph Beierle ◽  
Jonas Haldimann
Keyword(s):  
Normal Form ◽  
Knowledge Base ◽  
Normal Forms ◽  
Nonmonotonic Reasoning ◽  
Knowledge Bases ◽  
The Other ◽  
Conditional Knowledge ◽  
System P ◽  
The One ◽  
Model Equivalence

AbstractConditionals are defeasible rules of the form If A then usually B, and they play a central role in many approaches to nonmonotonic reasoning. Normal forms of conditional knowledge bases consisting of a set of such conditionals are useful to create, process, and compare the knowledge represented by them. In this article, we propose several new normal forms for conditional knowledge bases. Compared to the previously introduced antecedent normal form, the reduced antecedent normal form (RANF) represents conditional knowledge with significantly fewer conditionals by taking nonmonotonic entailments licenced by system P into account. The renaming normal form(ρNF) addresses equivalences among conditional knowledge bases induced by renamings of the underlying signature. Combining the concept of renaming normal form with other normal forms yields the renaming antecedent normal form (ρ ANF) and the renaming reduced antecedent normal form (ρ RANF). For all newly introduced normal forms, we show their key properties regarding, existence, uniqueness, model equivalence, and inferential equivalence, and we develop algorithms transforming every conditional knowledge base into an equivalent knowledge base being in the respective normal form. For the most succinct normal form, the ρ RANF, we present an algorithm KBρra systematically generating knowledge bases over a given signature in ρ RANF. We show that the generated knowledge bases are consistent, pairwise not antecedentwise equivalent, and pairwise not equivalent under signature renaming. Furthermore, the algorithm is complete in the sense that, when taking signature renamings and model equivalence into account, every consistent knowledge base is generated. Observing that normalizing the set of all knowledge bases over a signature Σ to ρ RANF yields exactly the same result as KBρra (Σ), highlights the interrelationship between normal form transformations on the one hand and systematically generating knowledge bases in normal form on the other hand.

scholarly journals Nonmonotonic reasoning from conditional knowledge bases with system W

2021 ◽  
Author(s):  
Christian Komo ◽  
Christoph Beierle
Keyword(s):  
Upper Bound ◽  
Constraint Satisfaction Problem ◽  
Nonmonotonic Reasoning ◽  
Knowledge Bases ◽  
Upper And Lower Bounds ◽  
Major Objective ◽  
Ranking Models ◽  
Conditional Knowledge ◽  
System P ◽  
Account System

AbstractFor nonmonotonic reasoning in the context of a knowledge base $\mathcal {R}$ R containing conditionals of the form If A then usually B, system P provides generally accepted axioms. Inference solely based on system P, however, is inherently skeptical because it coincides with reasoning that takes all ranking models of $\mathcal {R}$ R into account. System Z uses only the unique minimal ranking model of $\mathcal {R}$ R , and c-inference, realized via a complex constraint satisfaction problem, takes all c-representations of $\mathcal {R}$ R into account. C-representations constitute the subset of all ranking models of $\mathcal {R}$ R that are obtained by assigning non-negative integer impacts to each conditional in $\mathcal {R}$ R and summing up, for every world, the impacts of all conditionals falsified by that world. While system Z and c-inference license in general different sets of desirable entailments, the first major objective of this article is to present system W. System W fully captures and strictly extends both system Z and c-inference. Moreover, system W can be represented by a single strict partial order on the worlds over the signature of $\mathcal {R}$ R . We show that system W exhibits further inference properties worthwhile for nonmonotonic reasoning, like satisfying the axioms of system P, respecting conditional indifference, and avoiding the drowning problem. The other main goal of this article is to provide results on our investigations, underlying the development of system W, of upper and lower bounds that can be used to restrict the set of c-representations that have to be taken into account for realizing c-inference. We show that the upper bound of n − 1 is sufficient for capturing c-inference with respect to $\mathcal {R}$ R having n conditionals if there is at least one world verifying all conditionals in $\mathcal {R}$ R . In contrast to the previous conjecture that the number of conditionals in $\mathcal {R}$ R is always sufficient, we prove that there are knowledge bases requiring an upper bound of 2n− 1, implying that there is no polynomial upper bound of the impacts assigned to the conditionals in $\mathcal {R}$ R for fully capturing c-inference.

scholarly journals A Complete Map of Conditional Knowledge Bases in Different Normal Forms and Their Induced System P Inference Relations Over Small Signatures

2021 ◽  
Vol 34 (1) ◽  
Author(s):  
Christoph Beierle ◽  
Jonas Haldimann ◽  
Steven Kutsch
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Knowledge Bases ◽  
Predominant Role ◽  
New Normal ◽  
Conditional Knowledge ◽  
System P

Conditional knowledge bases consisting of qualitativeconditionals play a predominant role in knowledge representationand reasoning. In this paper, we develop a full map of allconsistent conditional knowledge bases over a small signature indifferent normal forms. We introduce two new normal formsthat take the induced system P inference relation into account,the system P normal form (SPNF) and the renaming SPNF(ρSPNF) considering additionally renamings of theunderlying signature. For a two-element signature, we systematicallygenerate and compare all consistent knowledge bases in ANF,RANF, SPNF, and their renaming counterparts, as well as allcomplete system P inference relations induced by conditionalknowledge bases.

scholarly journals InfOCF-Web: An Online Tool for Nonmonotonic Reasoning with Conditionals and Ranking Functions

2021 ◽  
Author(s):  
Steven Kutsch ◽  
Christoph Beierle
Keyword(s):  
Nonmonotonic Reasoning ◽  
Minimal Models ◽  
Ranking Functions ◽  
Online Tool ◽  
Ranking Models ◽  
Conditional Knowledge ◽  
System P ◽  
Nonmonotonic Inference

InfOCF-Web provides implementations of system P and system Z inference, and of inference relations based on c-representation with respect to various inference modes and different classes of minimal models. It has an easy-to-use online interface for computing ranking models of a conditional knowledge R, and for answering queries and comparing inference results of nonmonotonic inference relations induced by R.

A Probabilistic Approach to Ordering Formulas in a Possibilistic Knowledge Base

1998 ◽  
Vol 06 (03) ◽  
pp. 241-256
Author(s):  
P. H. Giang ◽  
D. Dubois ◽  
H. Prade
Keyword(s):  
Probability Distribution ◽  
Knowledge Base ◽  
Probabilistic Approach ◽  
Nonmonotonic Reasoning ◽  
Careful Analysis ◽  
Knowledge Bases ◽  
Possibilistic Logic ◽  
Many Sources ◽  
Special Case ◽  
Principle Of Maximum

In this paper, a careful analysis of interval-valued possibilistic knowledge bases indicates that there exists a natural probability distribution over the set of orderings of formulae compatible with the weights given in the knowledge base. We propose a new view, by which a possibilistic knowledge base can be considered in term of such probability distribution. It reveals some interconnections between probabilistic and possibilistic logics. We show that the principle of minimum specificity, widely used in possibilistic logic, is a special case of the principle of maximum likehood (at least, from the standpoint of nonmonotonic reasoning). We propose a formula to calculate probability for a defeasible conclusion. Moreover, the proposed view seems to be useful for other practical purposes. As an example, we apply it to a traditional problem of fusion of possibilistic knowledge from many sources and derive a new solution.

scholarly journals Strong Inconsistency in Nonmonotonic Reasoning

2017 ◽  
Author(s):  
Gerhard Brewka ◽  
Matthias Thimm ◽  
Markus Ulbricht
Keyword(s):  
Knowledge Base ◽  
Nonmonotonic Reasoning ◽  
Knowledge Bases ◽  
Generic Algorithm ◽  
Hitting Sets ◽  
Strong Notion ◽  
Classical Duality

Minimal inconsistent subsets of knowledge bases play an important role in classical logics, most notably for repair and inconsistency measurement. It turns out that for nonmonotonic reasoning a stronger notion is needed. In this paper we develop such a notion, called strong inconsistency. We show that—in an arbitrary logic, monotonic or not—minimal strongly inconsistent subsets play the same role as minimal inconsistent subsets in classical reasoning. In particular, we show that the well-known classical duality between hitting sets of minimal inconsistent subsets and maximal consistent subsets generalizes to arbitrary logics if the strong notion of inconsistency is used. We investigate the complexity of various related reasoning problems and present a generic algorithm for computing minimal strongly inconsistent subsets of a knowledge base. We also demonstrate the potential of our new notion for applications, focusing on repair and inconsistency measurement.

R-CALCULUS: A LOGICAL APPROACH FOR KNOWLEDGE BASE MAINTENANCE

1995 ◽  
Vol 04 (01n02) ◽  
pp. 177-200 ◽  
Author(s):  
WEI LI ◽  
NINGCHUAN SHEN ◽  
JU WANG
Keyword(s):  
Logic Programming ◽  
Knowledge Base ◽  
Belief Revision ◽  
Knowledge Bases ◽  
Logical Approach ◽  
Computational Aspects ◽  
Relevant Work

The concepts of maintenance sequences of a knowledge base and their limits are introduced. Some concepts used in maintenance of knowledge bases, such as new laws, user’s rejections, and reconstructions of a base are defined; the related theorems are proved. A maintenance procedure scheme is defined. The maintenance sequences generated by the procedure are convergent, and their limits are the set of true sentences of the model. Some computational aspects of reconstructions are studied; an R-calculus is given to deduce a reconstruction when a knowledge base meets a rejection. Especially, an R-calculus for logic programming is provided and implemented in Prolog. Finally, our research is compared with AGM’s theory of belief revision and other relevant work.

Open Knowledge Base: Resources and Units of Knowledge

2020 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Knowledge Base ◽  
Knowledge Bases

This paper is about highlighting two categories of knowledge bases, one built as a repository of links, and other based on units of knowledge.

scholarly journals Building an OWL ontology with Xper3

2018 ◽  
Vol 2 ◽  
pp. e25614 ◽  
Author(s):  
Florian Pellen ◽  
Sylvain Bouquin ◽  
Isabelle Mougenot ◽  
Régine Vignes-Lebbe
Keyword(s):  
Semantic Web ◽  
Knowledge Base ◽  
Expressive Power ◽  
Knowledge Bases ◽  
User Friendliness ◽  
Standard Format ◽  
Ontology Language ◽  
Closed World ◽  
Web Standards ◽  
Description Framework

Xper3 (Vignes Lebbe et al. 2016) is a collaborative knowledge base publishing platform that, since its launch in november 2013, has been adopted by over 2 thousand users (Pinel et al. 2017). This is mainly due to its user friendly interface and the simplicity of its data model. The data are stored in MySQL Relational DBs, but the exchange format uses the TDWG standard format SDD (Structured Descriptive DataHagedorn et al. 2005). However, each Xper3 knowledge base is a closed world that the author(s) may or may not share with the scientific community or the public via publishing content and/or identification key (Kopfstein 2016). The explicit taxonomic, geographic and phenotypic limits of a knowledge base are not always well defined in the metadata fields. Conversely terminology vocabularies, such as Phenotype and Trait Ontology PATO and the Plant Ontology PO, and software to edit them, such as Protégé and Phenoscape, are essential in the semantic web, but difficult to handle for biologist without computer skills. These ontologies constitute open worlds, and are expressed themselves by RDF triples (Resource Description Framework). Protégé offers vizualisation and reasoning capabilities for these ontologies (Gennari et al. 2003, Musen 2015). Our challenge is to combine the user friendliness of Xper3 with the expressive power of OWL (Web Ontology Language), the W3C standard for building ontologies. We therefore focused on analyzing the representation of the same taxonomic contents under Xper3 and under different models in OWL. After this critical analysis, we chose a description model that allows automatic export of SDD to OWL and can be easily enriched. We will present the results obtained and their validation on two knowledge bases, one on parasitic crustaceans (Sacculina) and the second on current ferns and fossils (Corvez and Grand 2014). The evolution of the Xper3 platform and the perspectives offered by this link with semantic web standards will be discussed.

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