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Fuzzy Ontologies over Lattices with T-norms

Stefan Borgwardt, Rafael Peñaloza
2011 International Workshop on Description Logics  
In this paper we introduce the lattice-based fuzzy DL ALC L , where L is a complete De Morgan lattice equipped with a t-norm operator.  ...  In fuzzy logics, conjunctions and disjunctions are interpreted with the help of t-norms and t-conorms.  ... 
dblp:conf/dlog/BorgwardtP11 fatcat:hnaxacew7bad7nefg4wkargou4

Consistency reasoning in lattice-based fuzzy Description Logics

Stefan Borgwardt, Rafael Peñaloza
2014 International Journal of Approximate Reasoning  
On the other hand, we identify decidable and undecidable classes of fuzzy Description Logics over infinite lattices. For all the decidable classes, we also provide tight complexity bounds.  ...  Fuzzy Description Logics have been widely studied as a formalism for representing and reasoning with vague knowledge.  ...  In other words, the Gödel t-norms is the largest possible t-norm over a given lattice L.  ... 
doi:10.1016/j.ijar.2013.07.006 fatcat:fwhh3nszw5bsvlnghbzkoo2oxa

Fuzzy DLs over Finite Lattices with Nominals

Stefan Borgwardt
2014 International Workshop on Description Logics  
The complexity of reasoning in fuzzy description logics (DLs) over a finite lattice L usually does not exceed that of the underlying classical DLs.  ...  In this paper, this approach is modified to deal with nominals and constants in L-ISCHOI.  ...  Acknowledgments The author is indebted to Rafael Peñaloza for many discussions on the topics of (fuzzy) DLs in general and automata-based reasoning procedures in particular.  ... 
dblp:conf/dlog/Borgwardt14 fatcat:gv2y336a35bbdjaj5dwaozinny

A Tableau Algorithm for Fuzzy Description Logics over Residuated De Morgan Lattices [chapter]

Stefan Borgwardt, Rafael Peñaloza
2012 Lecture Notes in Computer Science  
This paper analyses the consistency and satisfiability problems in the description logic SHI with semantics based on a complete residuated De Morgan lattice.  ...  Fuzzy description logics can be used to model vague knowledge in application domains.  ...  However, the most important task is to reduce the search space created by the choice of lattice values in most of the rules. We plan to study these optimizations in the future.  ... 
doi:10.1007/978-3-642-33203-6_3 fatcat:hvvvyybm3fdcnfabduq5estdhe

Decidability and Complexity of Fuzzy Description Logics

Franz Baader, Stefan Borgwardt, Rafael Peñaloza
2016 Künstliche Intelligenz  
As a result, we have provided an almost complete classification of the decidability and complexity of FDLs with GCIs.  ...  Fuzzy description logics (FDLs) have been introduced to represent concepts for which membership cannot be determined in a precise way, i.e., where instead of providing a strict border between being a member  ...  ≥-assertions and any t-norm except the Gödel t-norm, and -ELC with crisp ontologies and the Product t-norm.  ... 
doi:10.1007/s13218-016-0459-3 fatcat:kvlgugali5ebdey55yep3zqzcq

The Complexity of Lattice-Based Fuzzy Description Logics

Stefan Borgwardt, Rafael Peñaloza
2012 Journal on Data Semantics  
We study the complexity of reasoning in fuzzy description logics with semantics based on finite residuated lattices.  ...  For the logic SHI, we show that deciding satisfiability and subsumption of concepts, with or without a TBox, are ExpTime-complete problems.  ...  Similar results have been shown for several fuzzy DLs based on ALC with t-norms over the unit interval [3, 4, 16, 18] .  ... 
doi:10.1007/s13740-012-0013-x fatcat:4o7th7isk5d3rlrnt4works4fy

Fuzzy and Bipolar Mathematical Morphology, Applications in Spatial Reasoning [chapter]

Isabelle Bloch
2009 Lecture Notes in Computer Science  
Mathematical morphology is based on the algebraic framework of complete lattices and adjunctions, which endows it with strong properties and allows for multiple extensions.  ...  Another, more recent, extension, concerns bipolar fuzzy sets. These extensions have numerous applications, two of each being presented here.  ...  Two types of t-norms and t-conorms are considered in [29] : operators called t-representable t-norms and t-conorms, which can be expressed using usual t-norms t and t-conorms T , and Lukasiewicz operators  ... 
doi:10.1007/978-3-642-02906-6_1 fatcat:gbyqydvw2ncvvbtr3vuulupn4e

Reasoning in Fuzzy Description Logics using Automata

Stefan Borgwardt, Rafael Peñaloza
2016 Fuzzy sets and systems (Print)  
Very recently, the ideas behind these automata-based approaches were adapted for reasoning also in fuzzy extensions of DLs, with semantics based either on finitely many truth degrees or the Gödel t-norm  ...  over the interval [0, 1].  ...  Peñaloza was affiliated with TU Dresden, and the Center for Advancing Electronics Dresden, Germany.  ... 
doi:10.1016/j.fss.2015.07.013 fatcat:zepnuj6iazcvrndlaradjyafzy

All About Fuzzy Description Logics and Applications [chapter]

Umberto Straccia
2015 Lecture Notes in Computer Science  
The aim of this talk is to present a detailed, self-contained and comprehensive account of the state of the art in representing and reasoning with structured fuzzy knowledge.  ...  Fuzzy knowledge comes into play whenever one has to deal with concepts for which membership is a matter of degree (e.g., the degree of illness is a function of, among others, the body temperature).  ...  ALC with GCIs over finite lattices being EXPTIMEcomplete, as for the crisp variant, while [33] shows that the KB satisfiability problem for ALC with GCIs over non finite lattices is undecidable.  ... 
doi:10.1007/978-3-319-21768-0_1 fatcat:dpgq6ttx6zbohpbrnzjesuul5m

Reasoning With Weighted Ontologies

Rafael Peñaloza
2009 International Workshop on Description Logics  
We study the problem of reasoning over weighted ontologies.  ...  We assume that every axiom is labeled with an element of a distributive lattice (called its weight) and try to compute its so-called boundary, with respect to a given property.  ...  Such a generalization could be helpful for reasoning with respect to other t-norms, or the variants of possibilistic logic.  ... 
dblp:conf/dlog/Penaloza09 fatcat:2f3lwhmgcfgpblagmpro5gtayy

Refining Semantic Matching for Job Recruitment: An Application of Formal Concept Analysis [chapter]

Gábor Rácz, Attila Sali, Klaus-Dieter Schewe
2018 Lecture Notes in Computer Science  
We also give sharp estimates how the sizes of the original ontology lattice and our new constructions relate.  ...  Skills are organized into ontologies that form a lattice by the specialization relation. Matching functions were defined based on filters of the lattice generated by the profiles.  ...  For two fuzzy sets f, g of S and for a skill s ∈ S let (f ∩ g)(s) := min{f (s), g(s)}, and ||f || := (v,γv)∈f γ v , i.e. || · || denotes sigma cardinality and intersection is defined as the min t-norm.  ... 
doi:10.1007/978-3-319-90050-6_18 fatcat:hxltorqwozgjnadiyytbfh7rwq

Towards a Fuzzy Context Logic [chapter]

Hedda Schmidtke
2021 Fuzzy Systems [Working Title]  
First, a proof is presented that context logic with a lattice semantics can be understood as an extension of fuzzy logic. Second, a fuzzification for context logic is proposed.  ...  The resulting language, which can be understood as a two-layered fuzzy logic or as a fuzzified context logic, expands both fields in a novel manner.  ...  However, fuzzy logic can be given a generalized t-norm semantics based on residuated lattices, i.e., other lattice structures L, ⪯ ðÞ instead of 0, 1 ½ , ≤ ðÞ .  ... 
doi:10.5772/intechopen.95624 fatcat:hao6gv6f7jaqjgrmbfcm3i7rl4

Type-2 fuzzy description logic

Ruixuan Li, Kunmei Wen, Xiwu Gu, Yuhua Li, Xiaolin Sun, Bing Li
2011 Frontiers of Computer Science in China  
We also propose type-2 fuzzy web ontology language (OWL) to build a fuzzy ontology based on type-2 fuzzy ALC and analyze the soundness, completeness, and complexity of the reasoning algorithms.  ...  In this paper, we present a type-2 fuzzy attributive concept language with complements (ALC) and provide its knowledge representation and reasoning algorithms.  ...  A triangular norm is named T-norm if T (a, 1) = a, (a 2 0,1 ½ ), while a triangular norm is called S-norm if T (0, a) = a, (a 2 0,1 ½ ).  ... 
doi:10.1007/s11704-011-0109-8 fatcat:552zdscmqrdbtkvlgcnkcnczdi

Finite Lattices Do Not Make Reasoning in ALCI Harder

Stefan Borgwardt, Rafael Peñaloza
2011 International Semantic Web Conference  
We consider the fuzzy logic ALCI with semantics based on a finite residuated lattice.  ...  Over the unit interval, the product and Lukasiewicz t-norms cause undecidability w.r.t. witnessed models [3, 11] ; for arbitrary models decidability is unknown in these cases.  ...  Satisfiability w.r.t. general TBoxes and residuated total orders has been shown to be undecidable [9] , but it remains open to find subclasses of infinite lattices and t-norms for which the problem is  ... 
dblp:conf/semweb/BorgwardtP11 fatcat:7kyzev74wrawdmnjtcxsgnytsa

Fuzzy set and semantic similarity in ontology alignment

Valerie Cross, Xueheng Hu
2012 2012 IEEE International Conference on Fuzzy Systems  
Many semantic similarity measures are derivative of fuzzy set similarity measures.  ...  Ontology alignment (OA) addresses this challenge by identifying correspondences between entities in different ontologies.  ...  The set intersection uses a t-norm, typically minimum, and the set union uses a t-co-norm, typically maximum.  ... 
doi:10.1109/fuzz-ieee.2012.6251265 dblp:conf/fuzzIEEE/CrossH12 fatcat:j5gyuzim7zbrnnle4sm4rezrbm
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