A Generalization of Approximation Fixpoint Theory and Application

Bi, Yi; You, Jia-Huai; Feng, Zhiyong

doi:10.1007/978-3-319-11113-1_4

Cited by 10 publications

(7 citation statements)

References 13 publications

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“…Consider for instance the set of AICs {¬a ⊃ +a, a ⊃ −a}; intuitively, we expect a semantic operator to derive an inconsistency from any partial action set; in standard AFT this is not possible. However, extensions of AFT that accommodate this have been defined (Bi et al, 2014); it would be interesting to see how AICs fit in this general theory. Another AFT-based topic of interest could be to study what safe inductions yield in the context of AICs and whether they can fix problems with the well-founded semantics.…”

Section: Resultsmentioning

confidence: 99%

“…More recently, Strass (2013) showed that AFT can also be used to characterize the major semantics of Dung's argumentation frameworks (Dung, 1995) and abstract dialectical frameworks (Brewka and Woltran, 2010). Other recent applications of AFT include: defining extensions of logic programming (Antic et al, 2013), defining new logics (Bogaerts et al, 2014), integrating different formalisms (Bi et al, 2014), studying complexity (Strass and Wallner, 2014), and studying modularity and predicate introduction in a uniform way (Vennekens et al, 2006(Vennekens et al, , 2007a.…”

Section: Introductionmentioning

confidence: 99%

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Fixpoint semantics for active integrity constraints

Bogaerts

Cruz-Filipe

2018

Artificial Intelligence

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Active integrity constraints (AICs) constitute a formalism to associate with a database not just the constraints it should adhere to, but also how to fix the database in case one or more of these constraints are violated. The intuitions regarding which repairs are "good" given such a description are closely related to intuitions that live in various areas of non-monotonic reasoning, such as logic programming and autoepistemic logic. In this paper, we apply approximation fixpoint theory, an abstract, algebraic framework designed to unify semantics of non-monotonic logics, to the field of AICs. This results in a new family of semantics for AICs. We study properties of our new semantics and relationships to existing semantics. In particular, we argue that two of the newly defined semantics stand out. Grounded repairs have a simple definition that is purely based on semantic principles that semantics for AICs should adhere to. And, as we show, they coincide with the intended interpretation of AICs on many examples. The second semantics of interest is the AFT-well-founded semantics: it is a computationally cheap semantics that provides upper-and lower bounds for many other classes of repairs.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fixpoint semantics for active integrity constraints

Bogaerts

Cruz-Filipe

2018

Artificial Intelligence

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“…They identified approximating operators for default logic and autoepistemic logic and showed that AFT induces all main and some new semantics in these fields (Denecker, Marek, and Truszczyński 2003). AFT has been applied to various other research domains, including active integrity constraints (Bogaerts and Cruz-Filipe 2018), and extensions of logic programming (Antic, Eiter, and Fink 2013;Bi, You, and Feng 2014;Charalambidis, Rondogiannis, and Symeonidou 2018). Most importantly for this paper, showed that many semantics for Dung's argumentation frameworks and abstract dialectical frameworks can be retrieved from AFT.…”

mentioning

confidence: 78%

Weighted Abstract Dialectical Frameworks through the Lens of Approximation Fixpoint Theory

Bogaerts

2019

AAAI

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Weighted abstract dialectical frameworks (wADFs) were recently introduced, extending abstract dialectical frameworks to incorporate degrees of acceptance. In this paper, we propose a different view on wADFs: we develop semantics for wADFs based on approximation fixpoint theory, an abstract algebraic theory designed to capture semantics of various non-monotonic reasoning formalisms. Our formalism deviates from the original definition on some basic assumptions, the most fundamental is that we assume an ordering on acceptance degrees. We discuss the impact of the differences, the relationship between the two versions of the formalism, and the advantages each of the approaches offers. We furthermore study complexity of various semantics.

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“…The only other work that treats inconsistency in AFT explicitly is Bi et al . (2014), where in case of inconsistency, instead of computing (lfp(A(•, v) 1 ), lfp(A(u, •) 2 )) on the respective domains [⊥, v] and [u, ], one computes (lfp(A(•, v) 1 ), A(u, v) 2 ) because lfp(A(u, •) 2 ) may no longer be defined on [u, ].…”

Section: Propositionmentioning

confidence: 99%

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

Liu¹,

You²

2021

Theory and Practice of Logic Programming

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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 minimal knowledge and negation as failure (MKNF) knowledge bases is in fact an approximator of AFT in disguise, which, thanks to the abstraction power 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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A Generalization of Approximation Fixpoint Theory and Application

Cited by 10 publications

References 13 publications

Fixpoint semantics for active integrity constraints

Fixpoint semantics for active integrity constraints

Weighted Abstract Dialectical Frameworks through the Lens of Approximation Fixpoint Theory

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

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