Morphologic for knowledge dynamics: revision, fusion and abduction

AbstractSeveral tasks in artificial intelligence require the ability to find models about knowledge dynamics. They include belief revision, fusion and belief merging, and abduction. In this paper, we exploit the algebraic framework of mathematical morphology in the context of propositional logic and define operations such as dilation or erosion of a set of formulas. We derive concrete operators, based on a semantic approach, that have an intuitive interpretation and that are formally well behaved, to perform revision, fusion and abduction. Computation and tractability are addressed, and simple examples illustrate the main results.KEYWORDS: Mathematical morphologymorphologicknowledge representationknowledge dynamicsbelief revisionfusionabduction Disclosure statementNo potential conflict of interest was reported by the author(s).Notes1 Although mathematical morphology has also been extended to complete semi-lattices and general posets (Keshet, Citation2000), based on the notion of adjunction, in this paper we only consider the case of complete lattices.2 Let d be a discrete metric on a set M. We say that d has the betweenness property if for all x,y∈M and all k∈{0,1,…,d(x,y)} there exists z∈M such that δ(x,z)=k and δ(z,y)=d(x,y)−k. The Hamming distance has this property.3 Note that, in contrast to the Hausdorff distance, the minimum distance is improperly called distance since it does not satisfy all the properties of a true metric.4 In this paper, we do not assume any closure constraint on the theory, which is simply a set of formulas Σ={φi,i∈I}, where I is an index set. To identify a theory with a formula, we will also use Σ to denote ∧i∈Iφi.5 It is easy to see that we can define an AGM operator ⋆ starting from °, by letting K⋆ψ=Cn(φ∘ψ) where φ satisfies K=Cn(φ).6 The notation min(A,≤) where ≤ is a total pre-order, stands for {ω∈A|∀ω′∈A, ω≤ω′}.7 This story is inspired by a famous example in Cognitive Psychology of an experiment by Tversky and Kahneman (Citation1983).8 In knowledge dynamics the fusion of pieces of information having a logical representation is usually called belief merging (Konieczny & Pino Pérez, Citation1998, Citation2002; Konieczny & Pino Pérez, Citation2011).9 Actually the sources can have different reliabilities, but we will focus on the case where all the sources have the same reliability; there is already a lot to say in this case.10 The triangle inequality is not required.11 Strictly, they are called Δd,Σ, Δd,Gmax and Δd,max, respectively, to emphasise the chosen distance d.12 Often in this work we will identify a finite set of formulas Σ with the conjunction of all its formulas and, by abuse of language, we continue to call this formula Σ. Thus, for instance, we denote the conjunction of formulas of Σ∪{α} by Σ∧α.13 E−W−CM in fact was already considered by Flach (Citation1996) but he did not provide any example for it not satisfying already the stronger version E-CM.Additional informationFundingThe first author would like to thank the Universidad de Los Andes who has given partial funding during a sabbatical visit. This work was also partly supported by her chair in Artificial Intelligence (Sorbonne Université and SCAI), and a part of it was done while she was with LTCI, Télécom Paris. The second author thanks the ‘Investissements d'avenir’ program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA Institute). The third author thanks Télécom Paris who has given partial funding during several sabbatical visits. The third author has benefited from the support of the AI Chair BE4musIA of the French National Research Agency (ANR-20-CHIA-0028) and also thanks the CDCHT-ULA who has given partial funding through the Project N∘ C-1855-13-05-A. A part of this work was done while the third and fourth authors were at the Universidad de Los Andes, Mérida, Venezuela.