This paper contains a proposal for a terminological logic. The formalisms for representing knowledge as well as the needed inferences are described

Baader, Franz

Bürckert, Hans-Jürgen

Heinsohn, Jochen

Hollunder, Bernhard

Müller, Jürgen

Nebel, Bernhard

Nutt, Werner

Profitlich, Hans-Jürgen

Acronym

Deutsches Forschungszentrum fOr KOnstliche Inte li lgenz GmbH Technical Memo TM-90-04 Terminological Knowledge Representation: A Proposal for a Terminological Logic F. Baader, H.-J. Burckert, J. Heinsohn, B. Hollunder, J. Muller, B. Nebel, W. Nutt, H.-J. Profitlich December 1990 Deutsches Forschungszentrum fur Kunstliche Intelligenz GmbH Postfach 20 80 D-6750 Kaiserslautern Tel.: (+49631) 205-3211/13 Fax: (+49631) 205-3210 Stuhlsatzenhausweg 3 D-6600 Saarbrilcken 11 Tel.: (+49681) 302-5252 Fax: (+49681) 302-5341 Deutsches Forschungszentrum fur KOnstliche Intelligenz The German Research Center for Artificial Intelligence (Deutsches Forschungszentrum fOr KOnstliche Intelligenz, DFKI) with sites in Kaiserslautern und SaarbrOcken is a non-profit organization which was founded in 1988 by the shareholder companies ADV/Orga, AEG, IBM, Insiders, Fraunhofer Gesellschaft, GMD, Krupp-Atlas, Mannesmann-Kienzle, Philips, Siemens and Siemens-Nixdorf. Research projects conducted at the DFKI are funded by the German Ministry for Research and Technology, by the shareholder companies, or by other industrial contracts. The DFKI conducts application-oriented basic research in the field of artificial intelligence and other related subfields of computer science. The overall goal is to construct systems with technical knowledge and common sense which - by using AI methods - implement a problem solution for a selected application area. Currently, there are the following research areas at the DFKI : Cl Intelligent Engineering Systems Cl Intelligent User Interfaces Cl Intelligent Communication Networks Cl Intelligent Cooperative Systems. The DFKI strives at making its research results available to the scientific community. There exist many contacts to domestic and foreign research institutions, both in academy and industry. The DFKI hosts technology transfer workshops for shareholders and other interested groups in order to inform about the current state of research. From its beginning, the DFKI has provided an attractive working environment for AI researchers from Germany and from all over the world. The goal is to have a staff of about 100 researchers at the end of the building-up phase. Prof. Dr. Gerhard Barth Director Terminological Knowledge Representation: A Proposal for a Terminological Logic Franz Baader, Hans-Jiirgen Biirckert, Jochen Heinsohn, Bernhard Hollunder, Jiirgen Miiller, Bernhard Nebel, Werner Nutt, Hans-Jiirgen Profitlich DFKI-TM-90-04 © Deutsches Forschungszentrum fUr Kunstliche Intelligenz 1990 This work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the follow ing: a notice that such copying is by permission of Deutsches Forschungszentrum fur Kunstliche Intelligenz, Kaiserslautern, Federal Republic of Germany; an acknowledgement of the authors and individual contributors to the work; all applicable portions of this copyright notice. Copying, reproducing, or republish ing for any other purpose shall require a licence with payment of fee to Deutsches Forschungszentrum fUr Kunstliche Intelligenz. Terminological Knowledge Representation: A Proposal for a Terminological Logic Franz Baader, Hans-Jurgen Burckert, Jochen Heinsohn, Bernhard Hollunder, Jurgen Muller, Bernhard Nebel, Werner Nutt, Hans-Jurgen Profitlich Deutsches Forschungszentrum fur Kunstliche Intelligenz (DFKI) Postfach 2080, D-6750 Kaiserslautern Stuhlsatzenhausweg 3, D-6600 Saarbrucken 11 West Germany Abstract This paper contains a proposal for a terminological logic. The formalisms for representing knowledge as well as the needed inferences are described. 1 1 Introduction An important aspect of intelligence is the use of existing knowledge. In order to realize this in AI-Systems we need both adequate methods to represent knowledge and effective procedures to retrieve and reuse the needed knowledge. One of the basic mechanisms of human knowledge representation and processing is the division of the world into classes or concepts ("find the right pigeonhole") which usually are given with a hierarchical structure. Let us consider some knowledge base about families and relationships. We have to deal with persons which are of sex male or female. We have parents, mothers, fathers etc. A verbal description of this knowledge might be as follows: • Persons are of sex Male or Female. • Woman is defined as Person with sex Female. • Man is defined as Person with sex Male. • Nobody can be both Man and Woman. • Parents are defined as Persons which have some child (which is also a P erson) . • Mothers are defined to be Parents with sex Female. • Fathers are defined to be Parents with sex Male. • Mother_with_many_children is defined as Mother with at least three children. We also have individuals (or objects) which are instances of concepts. For example, • John is a Father. • Tom is a child of John. • Mary is a Woman. Now every knowledge representation system should offer a couple of services that allow to arrange, manage, modify or retrieve information of the above kind . It should be able to answer the following questions: • Is an introduced concept defined in a meaningful way at all (or does it denote the empty concept in all worlds) ? (satisfiability) • Is a concept more general than another one? (subsumption) • Where exactly is the concept situated in a concept hierarchy? (classification) • Is the represented knowledge consistent? (consistency) • What facts are deducible from the knowledge? (instantiation) • Which are the concepts an object is instance of? (realization) • Which are the instances of a given concept ? (retrieval) 2 Building such a system we are confronted with the following questions: 1. How can the above properties found out at all ? And then if we know procedures that might do this: 2. How can we find out, if the procedures really do what they should do ? 3. How efficient are these procedures? Terminological logics based on concept description languages like KL-ONE [BS85] are such formalisms that make classification, description of relatiops among the classes and especially their hierarchical structure possible. However, concept description languages are not only one among a lot of possibilities, but meanwhile they offer compared to other KR-formalisms some fundamental advantages: • There is a well understood declarative semantics. This means that the meaning of the constructs is not given operationally, e .g. by the implementation ("John is a father", because my system answers to the question "What is John?" just "father"), but the meaning is given by its description and its models ("John is a father", because he is a father in all models- in all worlds- where the description suits to.) • There is a characterization of the tasks of the KR-systems by the declarative se-mantics. • There is a number of procedures and algorithms that realize these tasks and whose properties are well investigated now: 1. Correctness (If the system answers "John is a father", then John is a father within the meaning of the semantics- that is in all suitable worlds.) 2. Completeness (The system answers "John is a father", if John is a father within the meaning of the semantics.) 3. Complexity, Decidability (Are the services decidable and fast executable, respectively, at all ?) If we want to design a knowledge base, we first need a formal language that we can use. In the following we will present a proposal for a terminological language in both abstract form and machine readable form (LISP notation) . As a kernel, our language contains all the constructs provided by ACC [SS88] and some additional operators which (sometimes?) can be translated into ACCFNn [HN90]. 2 Symbols The terminological language IS based on the following primitives, the symbols of the alphabet: • Concept names: eN • Role names: RN 3 • Attribute names: AN • Individual names: IN • Object names: ON Examples with respect to our introductory example are: Person, Woman, Man, Parent are concept names, child is a role name, sex is an attribute name, Male and Female are individual names, and John and Mary are objects names. With this primitives we are allowed to form more complex expressions as specified in the next two sections: • Concept expressions: C • Role expressions: R • Attribute expressions: A The meaning of these is given by models or interpretations I. Those consist of a set ~I-the domain-and an interpretation function .I, that assigns a set to each concept name GN, a set-valued function (or equivalently a binary relation) to each role name RN, a single-valued partial function where domANI ~ ~I , to each attribute name AN, and an element to eac~ individual name IN and object name ON. We assume that different individuals and objects denote different elements in every interpretation. This property is called unique name assumption and is usually assumed in the database world. 3 Concept Forming Operators Besides the concept, role, and attribute names our alphabet includes a number of opera-tors, that permit to compose more complex concepts, roles, and attributes. We allow for the following concept forming operators: Concrete Form (and G1 ... Gn ) (or G1 ... Gn ) (not G) (all R G) (some R) (some R G) A bstract Form G1 n ... n Gn G1 U ... U Gn -,G VR.G 3R 3R.G Semantics Gf n ... nG~ Gt u ... UG~ ~I\GI {d E ~I I RI(d) ~ GI } {d E ~ I I RI (d) =f 0} {d E ~ I I RI (d) n GI =f 0} 4 (atleast n R) (atmost n R) (exact n R) (atleast n R G) (atmost n R G) (exact n R G) (eq RI R2 ) (neq RI R2 ) (subset RI R2 ) (supset RI R2 ) (in A G) (is A IN) (eq Al A2 ) (neq Al A2 ) (oneof INI ... INn) :l>nR :l<nR :l=nR :l>nR.G :l<nR.G :l=nR.G RI 1 R2 RI i R2 RI ---+ R2 RI +-- R2 A:G A: IN Al 1 A2 Al i A2 {INI, ... ,INn } {d E ~I IIRI(d) ~ nl} {d E ~I IIRI(d) ::; nl} {d E ~I IIRI(d) = nl} {d E ~I IIRI(d) n GI ~ nl} {d E ~I IIRI(d) n GI ::; nl} {d E ~I IIRI(d) n GI = nl} {d E ~I I Rf(d) = R{(d)} {d E ~I I Rf(d) of R{(d)} {d E ~I I Rf(d) ~ R{(d)} {d E ~I I Rf(d) 2 R{(d)} {d E domAI I AI(d) E GI } {d E domAI I AI(d) = INI} {d E domAf n domA{ I Af(d) = And)} {d E domAf n domA{ I Af(d) of And)} {IN{, ... , IN!} Examples: The concept mother can be described as Person n sex: Female; MotheLwithJIlany _children can be described as Mother n :l~3child.Person; FatheLwith....sons_only can be described as Parent n sex: Male n child 1 son. 4 Role Forming and Attribute Forming Operators Similar as for concepts our terminological logic provides a couple of role forming and attribute forming operators: Concrete Form (and R1 ... Rn) (inverse R) (restrict R G) (domrange G1 G2 ) (trans R) (inverse A) (restrict A G) (compose A1 ... An) Abstract Form RI n ... n Rn R- 1 R Ie G1 x G2 R* A-I A Ie Al 0 ... 0 An Semantics Rf n ... n R~ {(d, d') I (d', d) E RI} {(d, d') E RI I d' E GI } G{ x Gi {(d, d') I :ldl , ... , dn(d, dd E RI , ... , (dn, d') E RI} ??? {(AI(d),d) IdE domAI} AI leI Af 0 ... 0 A~ Notice that the inverse of an attribute is a role, but in general not an attribute. Examples: The role daughter can be defined as femaleJelative n child; the role successor can be defined as (inverse predecessor). 5 5 Terminological Axioms The terminological axioms (definitions, specializations, and restrictions) are used to spec-ify the knowledge about the world or a part of the world. A set of terminological axioms specifies a terminology T. It selects from all possible interpretations of the language those models that satisfy the given axioms as described below. Concrete Form (defconcept CN C) (defrole RN R) (defattribute AN A) (defprimconcept CN C) (defprimrole RN R) (defprimattribute AN A) (defdisjoint CNI ... CNn) (definvpair ANI AN2 ) Abstract Form CN=C RN=R AN=A CN~C RN~R AN~A GNI 11··.11 CNn ANI = ANi l Semantics CNI = CI RNI = RI ANI = AI CNI ~ CI RNI ~ RI ANI ~ AI CNI n CNI = 0 i -I- J" , J ,r AN! = (ANn- 1 Example (our introductory example in formal notation): Person ~ sex: {Male, Female} Woman = Person n sex: Female Man = Person n sex: Male Woman II Man Parent = Person n :3child.Person n Vchild.Person Mother = Parent n sex: Female Father = Parent n sex: Male Mother_wi thJllany _children = Mother n :3~3child.Person FatheLwi th_sons_only = Father n child 1 son. 6 Assertional Axioms In order to fill our world with objects we allow for assertional axioms which have the following forms. Concrete Form (C ON) (R ON ON') (A ON ON') Examples: (John Father) (John child Tom) (Mary Woman). 7 Services A bstract Form (ON: C) (ON RON') (ON A ON') Semantics aNI E GI (aNI, ON,I) E RI aNI E dom AI, (aNI, ON,I) E AI Now we are able to give a formal specification ofthe services mentioned in the introduction. 6 1. Satisfiability of a concept C in a terminology T: Does there exist a model I of T with CI =f 0 ? (Man n Woman is not satisfiable.) 2. Subsumption within a terminology T: C ~T D iff in all models I of T: CI ~ DI (e.g. Mother ~T Woman). 3. Equivalence of concepts within a terminology T: C ';::jT D iff in all models I of T: CI = DI 4. Classification in T: Find all minimal concepts D w.r.t. the subsumption relation with D ~T C. 5. Find the smallest relation on the concepts in T such that their transitive closure is the subsumption relation (modulo ';::jT). 6. Consistency of the represented knowledge. Does there exist a model I for the terminological and assertional axioms? 7. What facts are deducible from the knowledge? (e.g. a fact ex is deducible from the knowledge iff all models for the terminological and assertional axioms satisfy ex.) 8. Realization. Given an object ON occurring in an assertional axiom. Which are most specific concepts of T w.r.t. the subsumption relation ON is instance of? 9. Retrieval. Given an concept C. Which objects occurring in the assertional axioms are instances of C ? Thus with this formalization of our services we can develop procedures or algorithms for the services and prove their correctness, completeness, complexity, decidability. References [BS85] [HN90] [SS88] R. J. Brachman, J. G. Schmolze. "An Overview of the KL-ONE knowledge representat ion system." Cognitive Science, 9(2):171-216, April 1985. B. Hollunder, W. Nutt. Subsumption Algorithms for Concept Languages. DFKI Research Report RR-90-04, DFKI, Postfach 2080, D-6750 Kaiser-slautern, West Germany. M. Schmidt-SchauB, G. Smolka. Attributive Concept Descriptions with Unions and Complements. SEKI Report SR-88-21, FB Informatik, Univer-sitat Kaiserslautern, D-6750, Kaiserslautern, West Germany,1988. To appear in Artificial Intelligence, 47, 1991. 7 Deutsches Forschungszentrum fOr KOnstilche Intelllgenz GmbH DFKI Publikationen Die folgenden DFKI Veroffentlichungen oder die aktuelle Liste von erhaltlichen Publikationen konnen bezogen werden von der oben angegebenen Adresse. DFKI Research Reports RR-90-0l Franz Baader Tenninological Cycles in KL-ONE-based Knowledge Representation Languages 33 pages RR-90-02 Hans-Jurgen Burckert A Resolution Principle for Clauses with Constraints 25 pages RR-90-03 Andreas Dengel & Nelson M. Mattos Integration of Document Representation, Processing and Management 18 pages RR-9Q-04 Bernhard Hol/under & Werner Nutt Subsumption Algorithms for Concept Languages 34 pages RR-90-05 Franz Baader A Fonnal Definition for the Expressive Power of Knowledge Representation Languages 22 pages RR-90-06 Bernhard Hol/under Hybrid Inferences in KL-ONE-based Knowledge Representation Systems 21 pages DFKI -Bibliothek-Stuhlsatzenhausweg 3 6600 Saarbrticken 11 FRG DFKI Publications The following DFKI publications or the list of currently available publications can be ordered from the above address. RR-90-07 Elisabeth Andre, Thomas Rist Wissensbasierte Infonnationsprasentation: Zwei Beitrage zum Fachgesprach Graphik und KI: 1. Ein planbasierter Ansatz zur Synthese illustrierter Dokumente 2. Wissensbasierte Perspektivenwahl fUr die automatische Erzeugung von 3D-Objektdarstellungen 24 pages RR-90-08 Andreas Dengel A Step Towards Understanding Paper Documents 25 pages RR-90-09 Susanne Biundo Plan Generation Using a Method of Deductive Program Synthesis 17 pages RR-90-I0 Franz Baader, Hans-Jurgen Burckert, Bernhard Hol/under, Werner Nutt, Jorg H. Siekmann Concept Logics 26 pages RR-90-11 Elisabeth Andre, Thomas Rist Towards a Plan-Based Synthesis of Illustrated Documents 14 pages RR-90-12 Harold Boley Declarative Operations on Nets 43 pages RR-90-13 Franz Baader Augmenting Concept Languages by Transitive Closure of Roles: An Alternative to Tenninological Cycles 40 pages RR-90-14 Franz Schmalhofer, Otto Kuhn , Gabriele Schmidt Integrated Knowledge Acquisition from Text, Previously Solved Cases, and Expert Memories 20 pages RR-90-15 Harald Trost The Application of Two-level Morphology to Non-concatenative GenTIan Morphology 13 pages DFKI Technical Memos TM-89-01 Susan Halbach-Weber Connectionist Models and Figurative Speech 27 pages TM-90-01 Som Bandyopadhyay Towards an Understanding of Coherence in Multimodal Discourse 18 pages TM-90-02 Jay C. Weber The Myth of Domain-Independent Persistence 18 pages TM-90-03 Franz Baader, Bernhard Hal/under KRIS: Knowledge Representation and Inference System -System Description-15 pages TM-90-04 Franz Baader, Hans-Jurgen Burckert, Jochen Heinsohn, Bernhard Hal/under, Jurgen Muller, Bernhard Nebel, Werner Nutt, Hans-Jilrgen Profitlich Tenninological Knowledge Representation: A Proposal for a Tenninological Logic 7 pages DFKI Documents D-89-01 Michael H. Malburg & Rainer Bleisinger HYPERBIS: ein betriebliches Hypennedia-Informationssystem 43 Seiten D-90-01 DFKI Wissenschaftlich-Technischer Jahresbericht 1989 45 pages D-90-02 Georg Seul Logisches Programmieren mit Feature-Typen 107 Seite n D-90-03 Ansgar Bernardi, Christoph Klauck, Ralf Legleitner AbschluBbericht des Arbeitspaketes PROD 36 Seiten D-90-04 Ansgar Bernardi, Christoph Klauck, Ralf Legleitn.,!r STEP: Uberblick tiber eine zukiinftige Schnittstelle zum Produktdatenaustausch 69 Seiten D-90-05 Ansgar Bernardi, Christoph Klauck, Ralf Legleitner FonTIalismus zur Reprasentation von Geometrie- und Technologieinfonnationen als Teil eines Wissensbasierten Prod uktmodells 66 Seiten Terminological Knowledge Representation: A Proposal for a Terminological Logic F. Baader, H.-J. Biirckert, J. Heinsohn, B. Hollunder, J. Miiller, B. Nebel, W. Nutt, H.-J. Profitlich TM-90-04 Technical Memo 

Terminological knowledge representation : a proposal for a terminological logic

Universaar

Scientific documents from the Saarland University,

Deutsches 
Forschungszentrum 
fOr KOnstliche 
Inte li lgenz GmbH 
Technical 
Memo 
TM-90-04 
Terminological Knowledge Representation: 
A Proposal for a Terminological Logic 
F. Baader, H.-J. Burckert, 
J. Heinsohn, B. Hollunder, 
J. Muller, B. Nebel, 
W. Nutt, H.-J. Profitlich 
December 1990 
Deutsches Forschungszentrum fur Kunstliche Intelligenz 
GmbH 
Postfach 20 80 
D-6750 Kaiserslautern 
Tel.: (+49631) 205-3211/13 
Fax: (+49631) 205-3210 
Stuhlsatzenhausweg 3 
D-6600 Saarbrilcken 11 
Tel.: (+49681) 302-5252 
Fax: (+49681) 302-5341 
Deutsches Forschungszentrum 
fur 
KOnstliche Intelligenz 
The German Research Center for Artificial Intelligence (Deutsches Forschungszentrum fOr 
KOnstliche Intelligenz, DFKI) with sites in Kaiserslautern und SaarbrOcken is a non-profit 
organization which was founded in 1988 by the shareholder companies ADV/Orga, AEG, IBM, 
Insiders, Fraunhofer Gesellschaft, GMD, Krupp-Atlas, Mannesmann-Kienzle, Philips, Siemens 
and Siemens-Nixdorf. Research projects conducted at the DFKI are funded by the German 
Ministry for Research and Technology, by the shareholder companies, or by other industrial 
contracts. 
The DFKI conducts application-oriented basic research in the field of artificial intelligence and 
other related subfields of computer science. The overall goal is to construct systems with 
technical knowledge and common sense which - by using AI methods - implement a problem 
solution for a selected application area. Currently, there are the following research areas at the 
DFKI : 
Cl Intelligent Engineering Systems 
Cl Intelligent User Interfaces 
Cl Intelligent Communication Networks 
Cl Intelligent Cooperative Systems. 
The DFKI strives at making its research results available to the scientific community. There exist 
many contacts to domestic and foreign research institutions, both in academy and industry. The 
DFKI hosts technology transfer workshops for shareholders and other interested groups in 
order to inform about the current state of research. 
From its beginning, the DFKI has provided an attractive working environment for AI researchers 
from Germany and from all over the world. The goal is to have a staff of about 100 researchers at 
the end of the building-up phase. 
Prof. Dr. Gerhard Barth 
Director 
Terminological Knowledge Representation: 
A Proposal for a Terminological Logic 
Franz Baader, Hans-Jiirgen Biirckert, Jochen Heinsohn, 
Bernhard Hollunder, Jiirgen Miiller, Bernhard Nebel, 
Werner Nutt, Hans-Jiirgen Profitlich 
DFKI-TM-90-04 
© Deutsches Forschungszentrum fUr Kunstliche Intelligenz 1990 
This work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to 
copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes 
provided that all such whole or partial copies include the follow ing: a notice that such copying is by 
permission of Deutsches Forschungszentrum fur Kunstliche Intelligenz, Kaiserslautern, Federal Republic 
of Germany; an acknowledgement of the authors and individual contributors to the work; all applicable 
portions of this copyright notice. Copying, reproducing, or republish ing for any other purpose shall require 
a licence with payment of fee to Deutsches Forschungszentrum fUr Kunstliche Intelligenz. 
Terminological Knowledge Representation: 
A Proposal for a Terminological Logic 
Franz Baader, Hans-Jurgen Burckert, Jochen Heinsohn, 
Bernhard Hollunder, Jurgen Muller, Bernhard Nebel, 
Werner Nutt, Hans-Jurgen Profitlich 
Deutsches Forschungszentrum fur Kunstliche Intelligenz (DFKI) 
Postfach 2080, D-6750 Kaiserslautern 
Stuhlsatzenhausweg 3, D-6600 Saarbrucken 11 
West Germany 
Abstract 
This paper contains a proposal for a terminological logic. The formalisms for 
representing knowledge as well as the needed inferences are described. 
1 
1 Introduction 
An important aspect of intelligence is the use of existing knowledge. In order to realize 
this in AI-Systems we need both adequate methods to represent knowledge and effective 
procedures to retrieve and reuse the needed knowledge. One of the basic mechanisms of 
human knowledge representation and processing is the division of the world into classes 
or concepts ("find the right pigeonhole") which usually are given with a hierarchical 
structure. 
Let us consider some knowledge base about families and relationships. We have to 
deal with persons which are of sex male or female. We have parents, mothers, fathers etc. 
A verbal description of this knowledge might be as follows: 
• Persons are of sex Male or Female. 
• Woman is defined as Person with sex Female. 
• Man is defined as Person with sex Male. 
• Nobody can be both Man and Woman. 
• Parents are defined as Persons which have some child (which is also a P erson) . 
• Mothers are defined to be Parents with sex Female. 
• Fathers are defined to be Parents with sex Male. 
• Mother_with_many_children is defined as Mother with at least three children. 
We also have individuals (or objects) which are instances of concepts. For example, 
• John is a Father. 
• Tom is a child of John. 
• Mary is a Woman. 
Now every knowledge representation system should offer a couple of services that allow 
to arrange, manage, modify or retrieve information of the above kind . It should be able 
to answer the following questions: 
• Is an introduced concept defined in a meaningful way at all (or does it denote the 
empty concept in all worlds) ? (satisfiability) 
• Is a concept more general than another one? (subsumption) 
• Where exactly is the concept situated in a concept hierarchy? (classification) 
• Is the represented knowledge consistent? (consistency) 
• What facts are deducible from the knowledge? (instantiation) 
• Which are the concepts an object is instance of? (realization) 
• Which are the instances of a given concept ? (retrieval) 
2 
Building such a system we are confronted with the following questions: 
1. How can the above properties found out at all ? 
And then if we know procedures that might do this: 
2. How can we find out, if the procedures really do what they should do ? 
3. How efficient are these procedures? 
Terminological logics based on concept description languages like KL-ONE [BS85] are 
such formalisms that make classification, description of relatiops among the classes and 
especially their hierarchical structure possible. However, concept description languages 
are not only one among a lot of possibilities, but meanwhile they offer compared to other 
KR-formalisms some fundamental advantages: 
• There is a well understood declarative semantics. 
This means that the meaning of the constructs is not given operationally, e .g. by 
the implementation ("John is a father", because my system answers to the question 
"What is John?" just "father"), but the meaning is given by its description and its 
models ("John is a father", because he is a father in all models- in all worlds- where 
the description suits to.) 
• There is a characterization of the tasks of the KR-systems by the declarative se-
mantics. 
• There is a number of procedures and algorithms that realize these tasks and whose 
properties are well investigated now: 
1. Correctness 
(If the system answers "John is a father", then John is a father within the 
meaning of the semantics- that is in all suitable worlds.) 
2. Completeness 
(The system answers "John is a father", if John is a father within the meaning 
of the semantics.) 
3. Complexity, Decidability 
(Are the services decidable and fast executable, respectively, at all ?) 
If we want to design a knowledge base, we first need a formal language that we can 
use. In the following we will present a proposal for a terminological language in both 
abstract form and machine readable form (LISP notation) . As a kernel, our language 
contains all the constructs provided by ACC [SS88] and some additional operators which 
(sometimes?) can be translated into ACCFNn [HN90]. 
2 Symbols 
The terminological language IS based on the following primitives, the symbols of the 
alphabet: 
• Concept names: eN 
• Role names: RN 
3 
• Attribute names: AN 
• Individual names: IN 
• Object names: ON 
Examples with respect to our introductory example are: Person, Woman, Man, Parent are 
concept names, child is a role name, sex is an attribute name, Male and Female are 
individual names, and John and Mary are objects names. 
With this primitives we are allowed to form more complex expressions as specified in 
the next two sections: 
• Concept expressions: C 
• Role expressions: R 
• Attribute expressions: A 
The meaning of these is given by models or interpretations I. Those consist of a set 
~I-the domain-and an interpretation function .I, that assigns a set 
to each concept name GN, a set-valued function (or equivalently a binary relation) 
to each role name RN, a single-valued partial function 
where domANI ~ ~I , to each attribute name AN, and an element 
to eac~ individual name IN and object name ON. We assume that different individuals 
and objects denote different elements in every interpretation. This property is called 
unique name assumption and is usually assumed in the database world. 
3 Concept Forming Operators 
Besides the concept, role, and attribute names our alphabet includes a number of opera-
tors, that permit to compose more complex concepts, roles, and attributes. We allow for 
the following concept forming operators: 
Concrete Form 
(and G1 ... Gn ) 
(or G1 ... Gn ) 
(not G) 
(all R G) 
(some R) 
(some R G) 
A bstract Form 
G1 n ... n Gn 
G1 U ... U Gn 
-,G 
VR.G 
3R 
3R.G 
Semantics 
Gf n ... nG~ 
Gt u ... UG~ 
~I\GI 
{d E ~I I RI(d) ~ GI } 
{d E ~ I I RI (d) =f 0} 
{d E ~ I I RI (d) n GI =f 0} 
4 
(atleast n R) 
(atmost n R) 
(exact n R) 
(atleast n R G) 
(atmost n R G) 
(exact n R G) 
(eq RI R2 ) 
(neq RI R2 ) 
(subset RI R2 ) 
(supset RI R2 ) 
(in A G) 
(is A IN) 
(eq Al A2 ) 
(neq Al A2 ) 
(oneof INI ... INn) 
:l>nR 
:l<nR 
:l=nR 
:l>nR.G 
:l<nR.G 
:l=nR.G 
RI 1 R2 
RI i R2 
RI ---+ R2 
RI +-- R2 
A:G 
A: IN 
Al 1 A2 
Al i A2 
{INI, ... ,INn } 
{d E ~I IIRI(d) ~ nl} 
{d E ~I IIRI(d) ::; nl} 
{d E ~I IIRI(d) = nl} 
{d E ~I IIRI(d) n GI ~ nl} 
{d E ~I IIRI(d) n GI ::; nl} 
{d E ~I IIRI(d) n GI = nl} 
{d E ~I I Rf(d) = R{(d)} 
{d E ~I I Rf(d) of R{(d)} 
{d E ~I I Rf(d) ~ R{(d)} 
{d E ~I I Rf(d) 2 R{(d)} 
{d E domAI I AI(d) E GI } 
{d E domAI I AI(d) = INI} 
{d E domAf n domA{ I Af(d) = And)} 
{d E domAf n domA{ I Af(d) of And)} 
{IN{, ... , IN!} 
Examples: The concept mother can be described as 
Person n sex: Female; 
MotheLwithJIlany _children can be described as 
Mother n :l~3child.Person; 
FatheLwith....sons_only can be described as 
Parent n sex: Male n child 1 son. 
4 Role Forming and Attribute Forming Operators 
Similar as for concepts our terminological logic provides a couple of role forming and 
attribute forming operators: 
Concrete Form 
(and R1 ... Rn) 
(inverse R) 
(restrict R G) 
(domrange G1 G2 ) 
(trans R) 
(inverse A) 
(restrict A G) 
(compose A1 ... An) 
Abstract Form 
RI n ... n Rn 
R- 1 
R Ie 
G1 x G2 
R* 
A-I 
A Ie 
Al 0 ... 0 An 
Semantics 
Rf n ... n R~ 
{(d, d') I (d', d) E RI} 
{(d, d') E RI I d' E GI } 
G{ x Gi 
{(d, d') I :ldl , ... , dn(d, dd E RI , ... , (dn, d') E RI} ??? 
{(AI(d),d) IdE domAI} 
AI leI 
Af 0 ... 0 A~ 
Notice that the inverse of an attribute is a role, but in general not an attribute. 
Examples: The role daughter can be defined as 
femaleJelative n child; 
the role successor can be defined as 
(inverse predecessor). 
5 
5 Terminological Axioms 
The terminological axioms (definitions, specializations, and restrictions) are used to spec-
ify the knowledge about the world or a part of the world. A set of terminological axioms 
specifies a terminology T. It selects from all possible interpretations of the language those 
models that satisfy the given axioms as described below. 
Concrete Form 
(defconcept CN C) 
(defrole RN R) 
(defattribute AN A) 
(defprimconcept CN C) 
(defprimrole RN R) 
(defprimattribute AN A) 
(defdisjoint CNI ... CNn) 
(definvpair ANI AN2 ) 
Abstract Form 
CN=C 
RN=R 
AN=A 
CN~C 
RN~R 
AN~A 
GNI 11··.11 CNn 
ANI = ANi l 
Semantics 
CNI = CI 
RNI = RI 
ANI = AI 
CNI ~ CI 
RNI ~ RI 
ANI ~ AI 
CNI n CNI = 0 i -I- J" 
, J ,r 
AN! = (ANn- 1 
Example (our introductory example in formal notation): 
Person ~ sex: {Male, Female} 
Woman = Person n sex: Female 
Man = Person n sex: Male 
Woman II Man 
Parent = Person n :3child.Person n Vchild.Person 
Mother = Parent n sex: Female 
Father = Parent n sex: Male 
Mother_wi thJllany _children = Mother n :3~3child.Person 
FatheLwi th_sons_only = Father n child 1 son. 
6 Assertional Axioms 
In order to fill our world with objects we allow for assertional axioms which have the 
following forms. 
Concrete Form 
(C ON) 
(R ON ON') 
(A ON ON') 
Examples: 
(John Father) 
(John child Tom) 
(Mary Woman). 
7 Services 
A bstract Form 
(ON: C) 
(ON RON') 
(ON A ON') 
Semantics 
aNI E GI 
(aNI, ON,I) E RI 
aNI E dom AI, (aNI, ON,I) E AI 
Now we are able to give a formal specification ofthe services mentioned in the introduction. 
6 
1. Satisfiability of a concept C in a terminology T: 
Does there exist a model I of T with CI =f 0 ? 
(Man n Woman is not satisfiable.) 
2. Subsumption within a terminology T: 
C ~T D iff in all models I of T: CI ~ DI 
(e.g. Mother ~T Woman). 
3. Equivalence of concepts within a terminology T: 
C ';::jT D iff in all models I of T: CI = DI 
4. Classification in T: 
Find all minimal concepts D w.r.t. the subsumption relation with D ~T C. 
5. Find the smallest relation on the concepts in T such that their transitive closure is 
the subsumption relation (modulo ';::jT). 
6. Consistency of the represented knowledge. 
Does there exist a model I for the terminological and assertional axioms? 
7. What facts are deducible from the knowledge? 
(e.g. a fact ex is deducible from the knowledge iff all models for the terminological 
and assertional axioms satisfy ex.) 
8. Realization. 
Given an object ON occurring in an assertional axiom. Which are most specific 
concepts of T w.r.t. the subsumption relation ON is instance of? 
9. Retrieval. 
Given an concept C. Which objects occurring in the assertional axioms are instances 
of C ? 
Thus with this formalization of our services we can develop procedures or algorithms 
for the services and prove their correctness, completeness, complexity, decidability. 
References 
[BS85] 
[HN90] 
[SS88] 
R. J. Brachman, J. G. Schmolze. "An Overview of the KL-ONE knowledge 
representat ion system." Cognitive Science, 9(2):171-216, April 1985. 
B. Hollunder, W. Nutt. Subsumption Algorithms for Concept Languages. 
DFKI Research Report RR-90-04, DFKI, Postfach 2080, D-6750 Kaiser-
slautern, West Germany. 
M. Schmidt-SchauB, G. Smolka. Attributive Concept Descriptions with 
Unions and Complements. SEKI Report SR-88-21, FB Informatik, Univer-
sitat Kaiserslautern, D-6750, Kaiserslautern, West Germany,1988. To appear 
in Artificial Intelligence, 47, 1991. 
7 
Deutsches 
Forschungszentrum 
fOr KOnstilche 
Intelllgenz GmbH 
DFKI Publikationen 
Die folgenden DFKI Veroffentlichungen 
oder die aktuelle Liste von erhaltlichen 
Publikationen konnen bezogen werden von 
der oben angegebenen Adresse. 
DFKI Research Reports 
RR-90-0l 
Franz Baader 
Tenninological Cycles in KL-ONE-based 
Knowledge Representation Languages 
33 pages 
RR-90-02 
Hans-Jurgen Burckert 
A Resolution Principle for Clauses with 
Constraints 
25 pages 
RR-90-03 
Andreas Dengel & Nelson M. Mattos 
Integration of Document Representation, 
Processing and Management 
18 pages 
RR-9Q-04 
Bernhard Hol/under & Werner Nutt 
Subsumption Algorithms for Concept 
Languages 
34 pages 
RR-90-05 
Franz Baader 
A Fonnal Definition for the Expressive 
Power of Knowledge Representation 
Languages 
22 pages 
RR-90-06 
Bernhard Hol/under 
Hybrid Inferences in KL-ONE-based 
Knowledge Representation Systems 
21 pages 
DFKI 
-Bibliothek-
Stuhlsatzenhausweg 3 
6600 Saarbrticken 11 
FRG 
DFKI Publications 
The following DFKI publications or the list 
of currently available publications can be 
ordered from the above address. 
RR-90-07 
Elisabeth Andre, Thomas Rist 
Wissensbasierte Infonnationsprasentation: 
Zwei Beitrage zum Fachgesprach Graphik 
und KI: 
1. Ein planbasierter Ansatz zur Synthese 
illustrierter Dokumente 
2. Wissensbasierte Perspektivenwahl fUr 
die automatische Erzeugung von 3D-
Objektdarstellungen 
24 pages 
RR-90-08 
Andreas Dengel 
A Step Towards Understanding Paper 
Documents 
25 pages 
RR-90-09 
Susanne Biundo 
Plan Generation Using a Method of 
Deductive Program Synthesis 
17 pages 
RR-90-I0 
Franz Baader, Hans-Jurgen Burckert, 
Bernhard Hol/under, Werner Nutt, Jorg H. 
Siekmann 
Concept Logics 
26 pages 
RR-90-11 
Elisabeth Andre, Thomas Rist 
Towards a Plan-Based Synthesis of 
Illustrated Documents 
14 pages 
RR-90-12 
Harold Boley 
Declarative Operations on Nets 
43 pages 
RR-90-13 
Franz Baader 
Augmenting Concept Languages by 
Transitive Closure of Roles: An Alternative 
to Tenninological Cycles 
40 pages 
RR-90-14 
Franz Schmalhofer, Otto Kuhn , Gabriele 
Schmidt 
Integrated Knowledge Acquisition from 
Text, Previously Solved Cases, and Expert 
Memories 
20 pages 
RR-90-15 
Harald Trost 
The Application of Two-level Morphology 
to Non-concatenative GenTIan Morphology 
13 pages 
DFKI Technical Memos 
TM-89-01 
Susan Halbach-Weber 
Connectionist Models and Figurative 
Speech 
27 pages 
TM-90-01 
Som Bandyopadhyay 
Towards an Understanding of Coherence in 
Multimodal Discourse 
18 pages 
TM-90-02 
Jay C. Weber 
The Myth of Domain-Independent 
Persistence 
18 pages 
TM-90-03 
Franz Baader, Bernhard Hal/under 
KRIS: Knowledge Representation and 
Inference System 
-System Description-
15 pages 
TM-90-04 
Franz Baader, Hans-Jurgen Burckert, 
Jochen Heinsohn, Bernhard Hal/under, 
Jurgen Muller, Bernhard Nebel, Werner 
Nutt, Hans-Jilrgen Profitlich 
Tenninological Knowledge Representation: 
A Proposal for a Tenninological Logic 
7 pages 
DFKI Documents 
D-89-01 
Michael H. Malburg & Rainer Bleisinger 
HYPERBIS: ein betriebliches Hypennedia-
Informationssystem 
43 Seiten 
D-90-01 
DFKI Wissenschaftlich-Technischer 
Jahresbericht 1989 
45 pages 
D-90-02 
Georg Seul 
Logisches Programmieren mit Feature-
Typen 
107 Seite n 
D-90-03 
Ansgar Bernardi, Christoph Klauck, Ralf 
Legleitner 
AbschluBbericht des Arbeitspaketes PROD 
36 Seiten 
D-90-04 
Ansgar Bernardi, Christoph Klauck, Ralf 
Legleitn.,!r 
STEP: Uberblick tiber eine zukiinftige 
Schnittstelle zum Produktdatenaustausch 
69 Seiten 
D-90-05 
Ansgar Bernardi, Christoph Klauck, Ralf 
Legleitner 
FonTIalismus zur Reprasentation von 
Geometrie- und Technologieinfonnationen 
als Teil eines Wissensbasierten 
Prod uktmodells 
66 Seiten 


Terminological Knowledge Representation: 
A Proposal for a Terminological Logic 
F. Baader, H.-J. Biirckert, J. Heinsohn, B. Hollunder, 
J. Miiller, B. Nebel, W. Nutt, H.-J. Profitlich 
TM-90-04 
Technical Memo 


http://scidok.sulb.uni-saarland.de/volltexte/2011/3857/pdf/TM_90_04.pdf

Terminological knowledge representation : a proposal for a terminological logic

Abstract

Similar works

Full text

Available Versions

Acronym

Universaar

Scientific documents from the Saarland University,