Normal forms of conditional knowledge bases respecting system P-entailments and signature renamings
type: Item
instance of: scholarly article
title: Normal forms of conditional knowledge bases respecting system P-entailments and signature renamings
author: C. Beierle
author: Jonas Haldimann
published in: Annals of Mathematics and Artificial Intelligence
publication date: 2022-02-14T00:00:00Z
review text: The paper revises and extends the work presented in [the authors, Lect. Notes Comput. Sci. 12012, 22--41 (2020; Zbl 1456.68184)]. The setting is that of conditional logic, built upon a finite set \(\Sigma\) of propositional letters, with formulas (conditionals) of the form \((B|A)\) with \(A\) and \(B\) propositional formulas over \(\Sigma\) meant to express that \textit{if \(A\) then usually \(B\)}; for instance, \((e\overline{f}|e)\) can formalise that ``e-cars usually are e-cars that do not need fossil fuel''. The semantics is defined on the basis of \textit{ordinal conditional functions} (OCF), that map possible worlds to plausibilities, in the form of natural numbers, lower numbers corresponding to ``less surprise''. An OCF \(\kappa\) then accepts a conditional \((B|A)\), written \(\kappa\models (B|A)\) if \(\kappa(AB)<\kappa(A\overline{B})\) (verification of the conditional is less surprising than its falsification). Given a knowledge base \(\mathcal R\), all models of \(\mathcal R\) accept a conditional \((B|A)\) iff \((B|A)\) can be derived from \(\mathcal R\) using the 6 rules of system P [\textit{D. Lehmann} and \textit{M. Magidor}, Artif. Intell. 55, No. 1, 1--60 (1992; Zbl 0762.68057)]. The paper proposes: \begin{itemize} \item a notion of unique renaming normal form (\(\rho\)NF) for knowledge bases thanks to which signatures can be renamed; \item a reduced antecedent normal form (RANF) for knowledge bases, taking system P inferences into account; \item a renaming antecedent normal form (\(\rho\)ANF) and a renaming reduced antecedent normal form (\(\rho\)RANF) that combine \(\rho\)NF with ANF and RANF, respectively; \item algorithms for transforming a knowledge base into any of the introduced normal forms; \item an algorithm to systematically generate all knowledge bases in \(\rho\)RANF normal form over a given signature that are consistent, pairwise not antecedentwise equivalent and pairwise not equivalent under signature renaming. \end{itemize}
reviewed by: Éric Martin
Mathematics Subject Classification ID: 68T27
Mathematics Subject Classification ID: 68T30
zbMATH DE Number: 7473198
zbMATH Keywords: conditional knowledge base
zbMATH Keywords: signature renaming
zbMATH Keywords: inferential equivalence
zbMATH Keywords: renaming normal form
zbMATH Keywords: antecedent normal form
zbMATH Keywords: reduced antecedent normal form
zbMATH Keywords: knowledge base generation
MaRDI profile type: Publication
full work available at URL: s10472-021-09745-3
OpenAlex ID: W3195130889
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DOI: 10.1007/S10472-021-09745-3
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description: scientific article; zbMATH DE number 7473198