Formal languages

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On the foundations of semantic technologies.

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Formal languages

  1. 1. Vestlandsforsking’sSemantic Technologies Seminars 2010-11 Sponsored by
  2. 2. The Next Seminars01.12. kl 13.00The Semantic Web by Robert Engels, Vestlandsforsking/ESIS RDF:triples Linked Open Data15.12. kl 13.00Topic Maps by Lars Marius Garshol, Bouvet Human-oriented semantics? Topics, Associations, Occurences – The TAO of Topic Maps
  3. 3. What are semantic technologies?• Declarative languages for representing data that can be “understood” by software systems – i.e. common terminologies (ontologies) that interpret data from disparate sources and turn them into information• Rules that allow software to retrieve and reason about information on the basis of the ontologies
  4. 4. Why semantic technologies?• Semantic technologies – better knowledge representation and management – enhance human-computer communication – improve information retrieval – make possible system interoperability and automated data exchange
  5. 5. Application Areas• The (Semantic) Web – Linked Open Data – more efficient information retrieval• Control and monitoring systems – situation awareness – alert rules• Robotics – context awareness – improved communication• ….
  6. 6. Formal Languages Terje Aaberge taa@vestforsk.no Vestlandsforsking
  7. 7. Objective• Construct languages in which to – express unambiguous sentences – make valid inferences
  8. 8. Cost/Benefit• Cost – loss of expressivitivity, readability and flexibility• Benefit – precision – tailoring
  9. 9. Kinds of Formal Languages • Imperative – to express commands • Declarative – to formulate descriptions
  10. 10. Subject-Object Partition The logical paradox: “this statement is false”  The strict separation between language and domain of discourse
  11. 11. Characteristics of Languages • Syntax • Logic • Semantic • Pragmatic
  12. 12. Content• Elements of a formal declarative language• Propositional calculus• First order languages• Description logics
  13. 13. Elements of a Formal Language • Vocabulary – Names, Variables, Predicates – Logical constants • Rules of syntax • Formulas - sentences • Logical axioms • Rules of deduction • Ontology – Axioms – Terminological definitions • Interpretation
  14. 14. Roles of Rules• Rules of syntax ascertain meaning: the meaning of a sentence is determined by the meaning of the words composing it provided the sentence is well-formed• Rules of deduction preserves truth: if the premises are true then the conclusion is true
  15. 15. Propositional Calculus• ”Vocabulary” – atomic propositions – logical connectives – Complex propositions composed from atomic propositions and logical connectives Symbol Symbol names Example Read  conjunction A B and  disjunction A B or  implication A B If .. then  negation A not
  16. 16. Deduction• Modus ponens A B A B
  17. 17. Semantics• Semantics consists in assigning truth values to the atomic propositions• Truth tables = decision procedure A B A∧B T T T T F F F T F F F F
  18. 18. SyllogismsAll humans are MortalSocrates is a HumanSocrates is Mortal
  19. 19. First Order Languages• Notation , syntax and deduction• Formal semantics – extensional interpretations – intensional interpretations• Expressiveness• Decidability
  20. 20. Notation, Syntax and Deduction• Let H be a 1-place predicate, K a 2-place predicate, n and m names, and u, v variables• ’Hn’ and ’Knm’ are atomic sentences reading ”n is H” and ”n is K-related to m”• atomic sentences = propositions• formulas: Hv, Kuv,uHu , uHu etc.• example:   Hu  Mu   u Hs Ms
  21. 21. Extensional Semantics• Let N denote the set of names, P and R the sets of 1-place and 2-place predicates• Let D   D   D  D be the conceptual model of the domain for  D being the set of subsets• The semantics is defined by an injective map  : N  P  R  D   D    D  D  such that n  n  D p   p    D  r   r    D  D 
  22. 22. Extensional Truth Conditions• An individual named n belongs to the extension of a one-place predicate p if and only if the sentence ‘pn’ is true, according to the truth condition: ’pn’ is true if and only if pn, e.g. ’snow is white’ is true if and only if snow is white
  23. 23. Conceptual Model for Intensional Semantics: Directed Graph Individual relation
  24. 24. Intensional Semantics• Object language for D: LD(NV, PR)• Interpretation :D  N; d   d  n  isomorphism  :D  P; d   d  p  observable • For each observable there exists a unique map defined by the condition of commutativity of the diagrams  N  P        d    d  , d  D D• Extension of a predicate is its inverse image by the observable
  25. 25. Intensional Truth Conditions• pn expresses an atomic fact if  n  p for p    d and n    d• An atomic sentence is true iff it states the result of a measurement.
  26. 26. Decidability• A language is said to be decidable if there exist a procedure that determine in a finite number of steps that a sentence follows from the axioms• Whether a first order language is decidable depends on the axiom system
  27. 27. Description Logics• A-Box• T-Box• Expressiveness versus decidability• Notation• Naming convention of DLs
  28. 28. A-Box and T-box• A-Box – assertions about individuals of the domain• T-Box – axioms and terminological definitions
  29. 29. Expressiveness versus Decidability• A descriptions logic has a weaker syntax than first order predicate logic• Therefore only axiom systems that are decidable can be formulated
  30. 30. NotationSymbol Symbol names Example Read all concept names top empty concept bottom intersection or C and D conjunction of concepts union or disjunction of C or D concepts negation or complement not C of concepts universal restriction all R-successors are in C existential restriction an R-successor exists in C
  31. 31. Naming ConventionFunctional propertiesFull existential qualificationConcept unionComplex concept negationAn abbreviation for with transitive rolesRole hierarchy (subproperties - rdfs:subPropertyOf)Limited complex role inclusion axiomsNominals
  32. 32. Example• Attributive language. This is the base language which allows: – Atomic negation (negation of concepts that do not appear on the left hand side of axioms) – Concept intersection – Universal restrictions – Limited existential quantification
  33. 33. Synonyms OWL DL FOL Domainclass concept 1-place predicate propertyproperty role 2-place predicate relationobject individual name/singular term individual

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