Knowledge Representation and Reasoning using Description Logic Presenter Shamima Mithun

Overview Introduction to Description Logics (DL) Introduction to Description Logics (DL) DL Syntax and Semantic DL Syntax and Semantic DL Systems Architecture DL Systems Architecture Reasoning Techniques Reasoning Techniques KB Construction and Access using PowerLoom KB Construction and Access using PowerLoom

Introduction Description Logics are a family of logic based knowledge representation formalisms based on concepts and roles: Description Logics are a family of logic based knowledge representation formalisms based on concepts and roles: Concepts (classes) are interpreted as sets of objects e.g. person. Concepts (classes) are interpreted as sets of objects e.g. person. Roles are interpreted as binary relations on objects e.g. has-child. Roles are interpreted as binary relations on objects e.g. has-child. Key features of DLs are Key features of DLs are a well defined semantics a well defined semantics Inference services Inference services

Background Description Logics: Introduced to resolve the ambiguities of Semantic Networks Introduced to resolve the ambiguities of Semantic Networks Descendants of Semantic Networks, frame based systems and KL-ONE Descendants of Semantic Networks, frame based systems and KL-ONE Decidable fragment of FOL [2] Decidable fragment of FOL [2] Useful for Semantic Web Language (DAML +OIL) [4] Useful for Semantic Web Language (DAML +OIL) [4]

DL Applications & Systems Application Areas Terminological KR and Ontologies Terminological KR and Ontologies Semantic Web Semantic Web Software Information Systems Software Information Systems Database Applications Database ApplicationsSystems LOOM/PowerLoom, RACER, CLASSIC, and others LOOM/PowerLoom, RACER, CLASSIC, and others

DL Constructors DLs are characterized by a set of constructors. These are used to construct complex concepts and roles from simple ones. DLs are characterized by a set of constructors. These are used to construct complex concepts and roles from simple ones. Common constructors: Common constructors: Conjunction (П), disjunction (  ), negation (  ) Restricted forms of quantification ( ,  ) Restricted forms of quantification ( ,  ) Example in ALC: the concept Happy-Father Example in ALC: the concept Happy-Father Happy-Father  Man П  has- child. Male Happy-Father  Man П  has- child. Male

DL Syntax and Semantic Ian Horrocks [2] Figure is taken from Ian Horrocks [2]

Other DL Constructors

DL and Other Logical Formalism: FOPL Syntactic feature of DL: variable free notation. Syntactic feature of DL: variable free notation. Most DLs are fragments of FOL, e.g. ACL. Most DLs are fragments of FOL, e.g. ACL. ACL expressions can be translated into FOL: ACL expressions can be translated into FOL: A unary predicate Φ A is introduced for each concept C, and a binary relation ρ R for each role R. A unary predicate Φ A is introduced for each concept C, and a binary relation ρ R for each role R. Translation ACL →FOL: Translation ACL →FOL: artist П (  CREATES. song ) →  x  y: artist ( x ) Λ (CREATES ( x, y ) Λ song ( y )) Why not use FOL? Why not use FOL? The expressive power is too high for having good computational properties and efficient inference procedures.

DL and Other Logical Formalism: Modal Logics DLs are notational variants of Modal Logics DLs are notational variants of Modal Logics ALC  multi-modal K: ALC  multi-modal K: C П D  C Λ D, C  D  C ν D  C   C,  R.C  C,  R.C  [R]C  R.C  C,  R.C  [R]C transitive roles  transitive frames (e.g., K4) transitive roles  transitive frames (e.g., K4) inverse roles  converse programs (e.g., C-PDL) inverse roles  converse programs (e.g., C-PDL) number restrictions  deterministic programs (e.g., D-PDL) number restrictions  deterministic programs (e.g., D-PDL) No TBoxes available in modal logics: No TBoxes available in modal logics: internalise" axioms using a universal role u: C  D  [u] (C  D) No ABoxes available in modal logics: Use nominals No ABoxes available in modal logics: Use nominals

DL Systems Architecture Knowledge Base TBox (schema) П Man ≡ Human П Male Happy-Father ≡ П Man П has-child. Female ABox (data) John : Happy-Father John, Mary> : has-child Inference System Interface Figure after Horrocks [3]

DL TBox

DL ABox

Knowledge to Reasoning Knowledge Base Knowledge Base TBox TBox ABox Reasoning Reasoning about the knowledge Add new knowledge to the KB that follows logically. Add new knowledge to the KB that follows logically. Ask KB if a statement is valid. Ask KB if a statement is valid. Ian Horrocks [3] Figure is taken from Ian Horrocks [3]

Reasoning / Inference Basic Inference Problems, Basic Inference Problems, for TBox T: Consistency: Consistency: “A concept C is consistent with respect to T, if there exists a model I of T with C I  . [ I is a model of C]”. Inconsistent: songwriter  artist П (  CREATES. song ) song    IS _ CREATED _ BY. songwriter Subsumption: Subsumption: “A concept C is subsumed by a concept D with respect to T if C I  D I for every model I of T”. male  person Definitions taken from [6], p. 66.

Classification “ Classification is the task of inserting new concepts or instances in a taxonomy” [ 3]. Classification of concepts Classification of concepts allows to structure the terminology in the form of a subsumption hierarchy allows to structure the terminology in the form of a subsumption hierarchy Classification of instances Classification of instances determines whether an individual is an instance of a certain concept. determines whether an individual is an instance of a certain concept. Fixed- Furniture Movable- Furniture DoorWindowChairSofa Is-a

Reasoning Algorithms Structural subsumption algorithms Subsumption of concepts can be computed. Subsumption of concepts can be computed. They are complete for simple languages with little expressivity. They are complete for simple languages with little expressivity. Used by KL-ONE, LOOM and other systems. Used by KL-ONE, LOOM and other systems. Tableau–based algorithms Turned out to be very efficient reasoning algorithms. Turned out to be very efficient reasoning algorithms. Sound, complete and decidable. Sound, complete and decidable. Used e.g. in RACER. Used e.g. in RACER.

Structural Subsumption Algorithm Proceed in two phases 1)The descriptions to be tested for subsumption are normalized. 2)Then the syntactic structure of the normal forms is compared with each other. An FL o - concept description is in normal form iff it is of the form A 1 П… ПA m П  R 1.C 1 П… П  R n.C n Where A 1,.., A m are distinct concept names, R 1,..., R n are distinct roles names, and C 1,…, C n are concept descriptions in normal from.

Structural Subsumption Algorithm (contd.) Given is the normal form of the concept description C A 1 П… ПA m П  R 1.C 1 П… П  R n. C n and the normal form of the concept description D B 1 П… ПB k П  S 1.D 1 П… П  S l. D l D subsumes C: C  D iff  i, 1≤ i ≤k,  j, 1 ≤j ≤m: B i = A j  i, 1≤ i ≤l,  j, 1 ≤j ≤n:S i = R j and C j  D i

Tableau-based Algorithms Construct a model for the input concept description C 0. Construct a model for the input concept description C 0. Model is represented by tree form. Model is represented by tree form. The formula has been translated into Negation Normal Form (NNM). The formula has been translated into Negation Normal Form (NNM). To decide satisfiability of the concept C 0, start with the initial tree (root node). To decide satisfiability of the concept C 0, start with the initial tree (root node). Repeatedly apply expansion rules until find contradiction or expansion completed. Repeatedly apply expansion rules until find contradiction or expansion completed. Satisfiable iff a complete and contradiction-free tree is found. Satisfiable iff a complete and contradiction-free tree is found.

Tableau-based Algorithms - Example Determine the satisfiability of the concept-definition: ( (  CHILD. Male ) П (  CHILD.  Male ) ) ( (  CHILD. Male ) П (  CHILD.  Male ) ) ( (  CHILD. Male ) П (  CHILD.  Male ) ) (  CHILD. Male ) П -rule (  CHILD.  Male ) П –rule CHILD  -rule  Male  -rule Male  -rule

Reasoning (contd.) Reasoning services like subsumption and consistency Speed-up the inference procedures for query. Speed-up the inference procedures for query. Help to infer implicitly represented knowledge from the explicitly contained knowledge of KB. Help to infer implicitly represented knowledge from the explicitly contained knowledge of KB. T-Box A-Box Female  Male  Human Mary: Mother Mother  Female John: Father Father  Male Mary:  parent.Child Child   has.Mother П  has.Father John:  parent.Child Able to deduce implicit knowledge, like Mary is a Human. Able to deduce implicit knowledge, like Mary is a Human.

Reasoning: Decidability vs. Expressivity KR system should KR system should answer queries in a reasonable time. answer queries in a reasonable time. The reasoning procedures should terminate. The reasoning procedures should terminate. Trade-off between the expressivity of DLs and the complexity of their reasoning. Trade-off between the expressivity of DLs and the complexity of their reasoning. Very expressive DLs can have inference problems of high complexity, they may even be undecidable. Very expressive DLs can have inference problems of high complexity, they may even be undecidable. Very Weak DLs my not be sufficiently expressive to represent the important concepts of an application. Very Weak DLs my not be sufficiently expressive to represent the important concepts of an application. Quest for a highly expressive DL with decidable/ practical inference problems. Quest for a highly expressive DL with decidable/ practical inference problems.

Conclusion DL are logic based knowledge representation formalisms. DL are logic based knowledge representation formalisms. DL systems provide efficient inference services like consistency, subsumption. DL systems provide efficient inference services like consistency, subsumption. DLs are effective in a range of applications. DLs are effective in a range of applications.

PowerLoom

PowerLoom Commands Defines concept, relation, function and rules using Defines concept, relation, function and rules using defconcept, defrelation, deffunction, defrule. Add/Remove facts from KB with Add/Remove facts from KB with assert and retract Query KB Query KB ask, retrieve Note: Relations have to defined, before they are used in assertions or queries.

Model Domain Color Fixed- Furniture Movable- Furniture Age Size Big Small GreenRedNewOld IS-A DoorWindowChairSofa

Define Concept |=(defconcept furniture) |c| FURNITURE |= (defconcept movable-furniture (?f furniture)) |c| MOVABLE-FURNITURE |= (defconcept fixed-furniture (?f furniture)) |c| FIXED-FURNITURE |=(defconcept chair (?f movable-furniture)) |c| CHAIR |= (defconcept sofa (?f movable-furniture)) |c| SOFA

Define Concept (contd.) |= (defconcept window (?ff fixed-furniture)) |c| WINDOW |= (defconcept door (?ff fixed-furniture)) |c| DOOR |= (defconcept age (?a) : (member-of ?a (setof new old))) |c| AGE |= (defconcept color (?c) : (member-of ?c (setof green red blue))) |c| COLOR |= (defconcept size (?s) : (member-of ?s (setof small big))) |c| SIZE

Model Domain Color Fixed- Furniture Movable- Furniture Age Size Big Small GreenRedNewOld IS-A DoorWindowChairSofa has-color has-age has-size

Define Relation and Function |= (defrelation has-age ((?f furniture) (?a age))) |r| HAS-AGE |= (defrelation has-color ((?f furniture) (?c color))) ) |r| HAS-COLOR |= (defrelation has-size ((?f furniture) (?s size))) |r| HAS-SIZE |= (deffunction has-price ((?f furniture)):-> (?n INTEGER)) |r| HAS-PRICE

Model Domain Color Fixed- Furniture Movable- Furniture Age Size has-size has-age has-color Big Small GreenRedNewOld IS-A DoorWindowChairSofa

Value Assertion |= (assert (has-size chair big)) |P|(HAS_SIZE CHAIR BIG) |= (assert (has-color sofa red)) |P|(HAS_COLOR SOFA RED) |= (assert (has-age chair new)) |P|(HAS_AGE CHAIR NEW) |= (assert (has-age door old)) |P|(HAS_AGE DOOR OLD) |= (assert (forall (?x ?y) (=> (has-color ?x ?y) (and (furniture ?x) (color ?y)))))

Query the KB: Retrieve, Retract, and Ask |=(retrieve (has-color sofa ?x)) There is 1 solution so far: #1: ?X=RED #1: ?X=RED |= (retract (has-color sofa red)) |P?|(HAS-COLOR SOFA RED) |= (retrieve (has-color sofa ?x)) No solutions. No solutions. |= (ask (has-size chair big)) TRUE |= (ask (has-color door red)) UNKNOWN

References 1. PowerLoom Homepage 2. Ian Horrocks and Ulrike Sattler: Description Logics - Basics, Applications, and More. Aavilable at 3.Ian Horrocks: Reasoning with Expressive Description Logics: Theory and Practice. Available at: 4.Christel Kemke. Lecture Notes on Artificial Intelligence. Available at 5.Daniele Nardi and Ronald J. Brachman. An introduction to Description Logics. In Baader, Calvanese, McGuinnes Nardi and Patel-Schneider, (eds). The Description Logics Handbook, chapter 1. Cambridge University Press, Franz Baader and Werner Nutt. Basic Description Logics. In Baader, Calvanese, McGuinnes Nardi and Patel-Schneider, (eds). The Description Logics Handbook, chapter 2. Cambridge University Press, 2003.

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