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Fuzzy Ontology Representation using OWL 2

Fernando Bobillo, Umberto Straccia

arXiv:1009.3391v3cs.LO

TL;DR

Real-world knowledge often contains vagueness that classical ontology languages cannot represent adequately, while existing fuzzy extensions lack a standard and sufficient expressive scope. The paper encodes fuzzy SROIQ(D) ontologies using OWL 2 annotation properties, reports prototype support, and illustrates the approach through matchmaking and decision-making examples.

  • Problem

    Classical ontology languages do not adequately handle vague knowledge, while existing fuzzy reasoners use different representation languages and current fuzzy extensions are considered insufficiently expressive.

  • Method

    The paper identifies five categories of syntactic differences and represents a general fuzzy SROIQ(D) extension using OWL 2 annotation properties.

  • Results

    The methodology supports prototype implementations and application examples including an optimal matchmaking degree of 0.7625 and selection of alternative a2 with satisfiability degree 0.32.

  • Takeaways & Limitations

    OWL 2 annotation properties enable fuzzy ontology representation with current OWL 2 editors while OWL 2 reasoners discard the fuzzy part and retain crisp reasoning results.

  • Takeaways & Limitations

    A full reasoning algorithm for the presented logic is not yet known, so implementations cover only fragments supported by existing reasoners.

Abstract

from arXiv · show

The need to deal with vague information in Semantic Web languages is rising in importance and, thus, calls for a standard way to represent such information. We may address this issue by either extending current Semantic Web languages to cope with vagueness, or by providing a procedure to represent such information within current standard languages and tools. In this work, we follow the latter approach, by identifying the syntactic differences that a fuzzy ontology language has to cope with, and by proposing a concrete methodology to represent fuzzy ontologies using OWL 2 annotation properties. We also report on the prototypical implementations.

1. Introduction

Classical ontology languages do not adequately represent the vagueness and imprecision common in real-world knowledge, motivating a standard representation for fuzzy ontologies. The paper proposes encoding fuzzy SROIQ(D) ontologies with OWL 2 annotation properties while retaining compatibility with current OWL 2 tools.

  • Classical ontology languages are unsuitable for representing vagueness or imprecision, which commonly occurs in real-world applications.
  • OWL 2 is equivalent to the description logic SROIQ(D), providing the logical basis for the paper’s fuzzy ontology representation.
  • Existing fuzzy DL reasoners use different fuzzy DL languages, creating a need for a standard representation of fuzzy ontologies.
  • Current fuzzy OWL extensions are not expected to become W3C standards soon and are considered insufficiently expressive because they modify only the ABox syntax.
  • The proposed methodology uses OWL 2 annotation properties to encode fuzzy SROIQ(D) ontologies and identify the syntactic differences fuzzy ontology languages must handle.
  • Annotation properties allow current OWL 2 editors to represent fuzzy ontologies while OWL 2 reasoners discard fuzzy annotations and produce the corresponding crisp results.

2. Fuzzy Logic

Fuzzy logic represents statements and set relationships by degrees rather than only true or false membership. The section introduces fuzzy connectives, common fuzzy logics, fuzzy relations, and model-based semantic reasoning.

  • Fuzzy statements express that a statement’s degree of truth is at least α or at most β, with α and β in [0, 1].
  • For example, ripeTomato ≥0.9 states that the tomato’s degree of ripeness is at least 0.9.
  • Fuzzy intersection, union, complement, and implication use t-norm, t-conorm, negation, and implication functions, respectively.
  • The four fuzzy logics presented are Zadeh, Łukasiewicz, Gödel, and Product; continuous t-norms can be obtained from combinations of the latter three.
  • A fuzzy relation maps pairs from two classical sets to [0, 1], with inverse, composition, and transitivity defined through fuzzy operations.
  • A fuzzy interpretation satisfies φ ≥l when I(φ) ≥l and satisfies φ ≤u when I(φ) ≤u; tight logical consequence uses the greatest entailed lower bound.

3. The Fuzzy DL SROIQ(D)

Fuzzy DL SROIQ(D) extends the OWL 2 foundation with graded concepts, roles, datatypes, modifiers, axioms, and reasoning tasks for representing vague knowledge.

  • Fuzzy SROIQ(D) interprets concepts as fuzzy sets, roles as fuzzy binary relations, and axioms as statements that may hold to a degree.
  • Fuzzy modifiers: Fuzzy modifiers map [0, 1] to [0, 1] and alter membership functions using linear hedges or triangular functions.The paper gives very as the example modifier linear(0.8).
  • Fuzzy concrete domains: Fuzzy concrete domains use trapezoidal, triangular, left-shoulder, and right-shoulder membership functions over rational intervals, with modified datatypes also allowed.Modified datatypes result from applying a fuzzy modifier to a fuzzy concrete-domain interpretation.
  • Syntax: The syntax adds modified, weighted, and weighted-sum concepts, while fuzzy knowledge bases organize axioms into ABox, TBox, and RBox groups.All listed fuzzy axioms have counterparts in crisp SROIQ(D).
  • Semantics: Interpretations assign membership functions to concepts, abstract roles, concrete roles, concrete predicates, and modifiers, and satisfaction compares axiom truth degrees with thresholds.
  • Reasoning tasks: Reasoning tasks include satisfiability, entailment, subsumption, best degree bounds, and maximal concept satisfiability degree, but these belong to querying rather than ontology representation.

4. Representation of Fuzzy Ontologies in OWL 2

The paper represents fuzzy ontologies within OWL 2 by preserving ordinary OWL structures and attaching fuzzy information through annotation properties.

  • OWL 2 annotation properties encode fuzzy ontology features that OWL 2 cannot directly represent.
  • The method first represents the crisp OWL 2 assertion and then annotates it with the fuzzy degree or constructor information.For example, paul: Tall is annotated with ≥0.5.
  • The methodology addresses six syntactic cases, including fuzzy modifiers, datatypes, concepts, roles, degree-qualified axioms, and fuzzy logic annotations.
  • The fuzzyLabel annotation property is used instead of OWL 2 defaults, with at most one such annotation allowed per ontology element.
  • Each annotation is enclosed by FuzzyOwl2 tags, whose fuzzyType attribute identifies the fuzzy element being represented.
  • Fuzzy modifiers: Fuzzy modifiers are encoded with modifier-specific parameters, using xsd:double and omitting explicit 0 and 1 bounds when they are assumed.

4.4. Fuzzy datatypes

Fuzzy datatypes are represented as OWL 2 datatype definitions plus annotations specifying their fuzzy shape, parameters, and optional modifications.

  • The datatype cases cover left-shoulder, right-shoulder, triangular, trapezoidal, and modified fuzzy datatypes.
  • Fuzzy datatype parameters include k1, k2, a, b, c, and d, with c and d appearing only in later datatype cases.
  • The OWL 2 datatype declaration uses xsd:integer or xsd:double as its base and can specify xsd:minInclusive and xsd:maxInclusive bounds.If bounds are omitted, the minimum and maximum of the datatype parameters are assumed.
  • A fuzzy datatype annotation identifies the datatype shape and its parameters inside a fuzzyOwl2 element with fuzzyType="datatype".
  • YoungAge is represented as an xsd:integer datatype with range [0, 200], then annotated as a left-shoulder datatype with parameters 10 and 30.
  • A modified datatype records a previously defined fuzzy modifier and base datatype, and the parser checks that both already exist.VeryYoungAge applies very to YoungAge.

4.5. Fuzzy concepts

Fuzzy concepts are represented by annotating OWL 2 concept declarations with constructors for modification, weighting, weighted sums, and fuzzy nominals.

  • A concept annotation records the constructor type and its parameters, while recursive definitions of a concept in terms of itself are disallowed.
  • Modified concepts: Modified concepts specify a fuzzy modifier and a base fuzzy concept, with the parser checking that the modifier exists and the base differs from the annotated concept.The paper illustrates this with VeryC as a modification of C by very.
  • Weighted concepts: Weighted concepts specify a value in (0, 1] and the fuzzy concept being weighted.The example represents (0.8 C) using the value 0.8 and base C.
  • Weighted sums: Weighted-sum concepts contain one or more weighted concepts, and their component concept names must differ from the annotated concept.The paper represents (0.8 A + 0.2 B) with two weighted components.
  • Fuzzy nominals: Fuzzy nominal concepts assign a value in (0, 1] to an individual, using a nominal constructor with value and individual attributes.The example represents {0.75/ind} as ind075.

4.6. Fuzzy roles

Fuzzy roles are represented in OWL 2 by annotating object or data properties with role-constructor metadata. The syntax supports modified roles, subject to nonrecursive construction and naming restrictions.

  • Role construction is nonrecursive, and the paper supports fuzzy modified roles.The role name must differ from the annotated concept, and the modifier must already be defined.
  • A fuzzy role annotation applies to an OWL 2 object or data property declaration.Its annotation domain is an OWL 2 property declaration.
  • The role syntax records the constructor type, modifier, and base role through fuzzyOwl2 and a modified Role tag.The modifier and base identify how the new role is formed.
  • Example 13 represents very(R) by defining the atomic property VeryR with modifier very and base R.The annotation uses fuzzyType role and a modified Role element.

4.7. Fuzzy axioms

Fuzzy axioms are encoded by annotating crisp OWL 2 axioms with a degree of truth and, where applicable, a selected fuzzy logic. The ontology-level syntax currently supports Lukasiewicz and Zadeh logics.

  • Supported OWL 2 axiom types can receive a truth degree through an axiom annotation.The domain includes concept assertions, role assertions, GCIs, and RIAs.
  • If the Degree value is omitted, the axiom’s assumed degree is 1.Explicit degree values must lie in (0, 1].
  • Inequality directions can be reflected, but certain lower- or upper-bound forms lack equivalent expressions in classical description logics.The restriction applies to axioms involving the specified operator in axioms (A8), (A12), and (A13).
  • An axiom with degree 0.5 is encoded using fuzzyType axiom and a Degree element with value 0.5.The paper illustrates this encoding after creating a crisp OWL 2 concept assertion.
  • An ontology annotation specifies the default fuzzy logic through a FuzzyLogic element.The current implementation allows only Lukasiewicz and Zadeh fuzzy logics.

5. Some Applications of Fuzzy Ontologies

The paper applies fuzzy ontologies to matchmaking and multi-criteria decision making by combining fuzzy requirements and weighted preferences. The examples identify optimal agreements or alternatives by maximizing satisfiability degrees.

  • 5.1. Matchmaking: The matchmaking example models buyer and seller preferences, vehicle knowledge, and reservation constraints in a fuzzy ontology.The domain includes sedan vehicles, car features, insurance packages, warranties, and price requirements.
  • 5.1. Matchmaking: Fuzzy datatypes encode gradual preferences such as price and warranty ranges, while equal shoulder parameters produce a crisp concept.The examples define ls22000-24000 and a crisp leq26000 datatype.
  • 5.1. Matchmaking: 0.7625 is the optimal matchmaking satisfiability degree under Lukasiewicz fuzzy logic.The corresponding agreement uses a price of 22500, a 100000 kilometer warranty, and a 60 month warranty.
  • 5.2. Multi-criteria Decision Making: The MCDM application represents performances and local and global values as fuzzy concepts combined through weighted sums.The final ranking orders alternatives by descending global value and selects the maximizer.
  • 5.2. Multi-criteria Decision Making: 0.32 versus 0.26 yields alternative a2 as the optimal MCDM choice.The values are the satisfiability degrees of GlobalValue-2 and GlobalValue-1, respectively.

6. Discussion

The approach has prototype parsers for selected fuzzy OWL 2 fragments, but full reasoning for the presented logic remains unavailable. Compared with related representations, it emphasizes compatibility with OWL 2 editors and a distinct fuzzy-information annotation property.

  • 6.1. Implementation: The prototype includes a general parser adapted to fuzzyDL and DeLorean, with similar parsers for other fuzzy DL reasoners considered feasible.The parsers translate OWL 2 annotations of type fuzzyLabel into the target reasoner languages.
  • 6.1. Implementation: Table 3 summarizes the fuzzy OWL 2 fragments supported by fuzzyDL and DeLorean rather than comparing the reasoners directly.The table reports support across axioms A1–A25, with fuzzyDL showing partial support for A12 and no support for several axioms that DeLorean supports.
  • 6.1. Implementation: A full reasoning algorithm for the presented fuzzy SROIQ(D) logic is not known, so the parsers cover only fragments currently supported by the available reasoners.This limits the implemented representation-to-reasoner pipeline to supported subsets.
  • 6.2. Related work: The work describes itself as the first effort toward fuzzy ontology representation using OWL 2.The claim concerns representation using OWL 2, not fuzzy ontology reasoning in general.
  • 6.2. Related work: Unlike approaches that encode concepts, roles, and axioms as individuals, this method reuses current ontology editors and avoids that user-unfriendly modeling style.The alternative representation also introduces met logical problems and replaces standard constructors such as intersectionOf with custom encodings.
  • 6.2. Related work: The current syntax is restricted to Łukasiewicz and Zadeh fuzzy logics, although the authors state it can be extended to Gödel or Product logics.The supported logics reflect the capabilities of fuzzyDL and DeLorean.

7. Conclusions and Future Work

The paper proposes a framework for representing expressive fuzzy ontologies with current OWL 2 resources rather than introducing a new standard language. It identifies five syntactic cases, supports extensibility, and provides editor, reasoner, and parser integration.

  • Conclusions: The framework represents fuzzy SROIQ(D) ontologies through OWL 2 annotation properties instead of proposing a new fuzzy ontology language.This enables development with OWL 2 editors while preserving compatibility with current language resources.
  • Conclusions: The authors identify five syntactic cases that fuzzy ontology languages must handle and cover extensions beyond a simple fuzzy ABox.Examples include fuzzy datatypes, fuzzy modifiers, and weighted sum concepts.
  • Future work: The approach is extensible to alternative fuzzy logics, modifier functions, and fuzzy datatypes.The authors state that these additions can be incorporated easily.
  • Methodology: OWL 2 editors can build the ontology core and add fuzzy information through annotation properties, while standard reasoners continue reasoning over the non-fuzzy core.The methodology separates the core ontology from its fuzzy annotations.
  • Implementation: Parsers translate annotated OWL 2 ontologies into languages supported by fuzzyDL and DeLorean, with a general parser adaptable to other fuzzy DL reasoners.A graphical Protégé plug-in was also under development to hide annotation encoding from users.

A. From Fuzzy DLs to Fuzzy OWL

This section relates fuzzy description logics to OWL 2 by describing OWL 2 entities, constructors, properties, and axioms. It also identifies annotation properties as OWL 2 features without direct DL counterparts.

  • OWL 2 entities: OWL 2 contains descriptions of classes, properties, and individuals, with object and datatype properties corresponding to abstract and concrete roles.The section uses DL terminology to explain these correspondences.
  • Additional properties: Annotation properties and ontology properties have no DL counterparts and instead contain ontology meta-properties.These properties are distinct from the class and role constructs represented in SROIQ(D).
  • Axioms: An OWL 2 document combines optional ontology headers with axioms corresponding to the ABox, TBox, and RBox.The axioms cover facts about individuals, class axioms, and property axioms; Table 5 gives their SROIQ(D) equivalences.
  • Constructors: Table 4 lists OWL 2 class and property constructors together with their correspondences in SROIQ(D).The table connects OWL 2 syntax to the DL formalism underlying the paper’s fuzzy extension.
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