On the foundations of semantic technologies.

1. Vestlandsforsking’s
Semantic Technologies Seminars 2010-11
Sponsored by

2. The Next Seminars
01.12. kl 13.00
The Semantic Web by Robert Engels, Vestlandsforsking/ESIS
RDF:triples
Linked Open Data
15.12. kl 13.00
Topic Maps by Lars Marius Garshol, Bouvet
Human-oriented semantics?
Topics, Associations, Occurences – The TAO of Topic Maps

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. 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. 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
• ….

7. Formal Languages
Terje Aaberge
taa@vestforsk.no
Vestlandsforsking

8. Objective
• Construct languages in which to
– express unambiguous sentences
– make valid inferences

9. Cost/Benefit
• Cost
– loss of expressivitivity,
readability and flexibility
• Benefit
– precision
– tailoring

10. Kinds of Formal Languages
• Imperative
– to express commands
• Declarative
– to formulate descriptions

11. Subject-Object Partition
The logical paradox:
“this statement is false”

The strict separation between
language and domain of
discourse

12. Characteristics of Languages
• Syntax
• Logic
• Semantic
• Pragmatic

13. Content
• Elements of a formal
declarative language
• Propositional calculus
• First order languages
• Description logics

14. 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

15. 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

16. 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

17. Deduction
• Modus ponens
A B
A
B

18. 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

19. Syllogisms
All humans are Mortal
Socrates is a Human
Socrates is Mortal

20. First Order Languages
• Notation , syntax and deduction
• Formal semantics
– extensional interpretations
– intensional interpretations
• Expressiveness
• Decidability

21. 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

22. 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 

23. 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

24. Conceptual Model for Intensional
Semantics: Directed Graph
Individual
relation

25. 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

26. 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.

27. 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

28. Description Logics
• A-Box
• T-Box
• Expressiveness versus
decidability
• Notation
• Naming convention of DLs

29. A-Box and T-box
• A-Box
– assertions about individuals of the
domain
• T-Box
– axioms and terminological
definitions

30. 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

31. Notation
Symbol 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

32. Naming Convention
Functional properties
Full existential qualification
Concept union
Complex concept negation
An abbreviation for with transitive roles
Role hierarchy (subproperties - rdfs:subPropertyOf)
Limited complex role inclusion axioms
Nominals

33. 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

34. Synonyms
OWL DL FOL Domain
class concept 1-place predicate property
property role 2-place predicate relation
object individual name/singular term individual