Logics for Data and Knowledge Representation

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Logics for Data and KnowledgeRepresentation

• Exercise 3 DLs

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Outline

• Modeling
• Previous Logics
• DL
• RelBAC
• OWL
• Comprehensive

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Modeling Procedure

• Abstraction of the world to a mental model
• Clarify the domain of interest
• Clarify the relations
• Choose/build a logic
• Build the theory of the mental model with the
  logic
• Reason about the theory

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What distincts DL from Previous Logics?

• PL
• Logical constructors
• Interpretations
• ClassL
• Logical constructors
• Interpretations
• Ground ClassL
• Individuals
• DL
• All?

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Summary of Previous Logics We Mentioned
PL ClassL Ground ClassL DL
Syn. Np Nc Nc, Ni Nc, Ni, Nr
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
?
Set Set
Fill Fill
??

Sem. ?true, false ?e1, ?e1, ?e1,
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Expressiveness of DL

• Binary Relations?
• YES!
• Subsumption?
• Of course!
• More than concept subsumption!
• Arbitrary!
• Else?
• Some
• Only
• Number

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Description Logics

• Propositional DL VS. ClassL
• DL VS. Ground ClassL
• Role Constructors
• ?
• ?
• ???

In addition to concept constructors
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DL Modeling

• Model the following NL sentences with DLs.
• Children with only a single parent and no
  siblings
• Child?1hasParent?1hasparent??hasSibling.?
• Friends that likes foreign movies but only
  Disney cartoons
• Friend??like.(Movie?Foreign)??like.(Cartoon?Disne
  y)
• A binary tree is a tree with at most two
  sub-trees that are themselves binary trees.
• BTreeTree?2hasSubTree.BTree
• The monkeys that can grasp the banana are those
  that have climbed onto the box at position of the
  banana
• Monkey??get.Banana??hasClimbedOnto.(Box??atPositi
  onOf.Banana)

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DL Reasoning TBox

• Prove the following tautology
• (C?D)C?D ?R.C?R.C
• Venn Diagrams
• Concepts
• Universal, Arbitrary non-empty set, Empty set
• Relations
• Intersection, union, disjoint
• Tableaux
• An algorithm to verify satisfiability.
• Rules
• and/or/some/only

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(C?D)C?D
C
D
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?R.C?R.C

• (?R.C)I
• ?-x ?y R(x,y)?C(y)
• x(?y R(x,y)?C(y) )
• x?y (R(x,y)?C(y) )
• x?y (R(x,y)?C(y) )
• x?y R(x,y)?C(y) )
• (?R.C)I

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DL Reasoning ABox (1)

• Given the interpretation I with the domain
  ?Id,e,f,g
• d,e,f?A B(f) R(d,e) R(e,g)
• S(g,d) S(g,g) S(e,f)
• In which A,B are concept and R,S are roles.
• Please find the instances of the ALC-concept C as
• A?B
• ?S.A
• ?S.A
• ?S.?S.?S.?S.A
• ?T.A??T.A

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DL Reasoning ABox (2)

• Let an ABox A consists of the following
  assertions
• Likes(Bob, Ann) Likes(Bob, Cate)
• Neighbor(Ann, Cate) Neighbor(Cate, David)
• Blond(Ann) Blond(David)
• where Neighbor is a symmetric and transitive
  role.
• Does A have a model?
• Is Bob an instance of the following concepts in
  all models of A?
• ?Likes.(Blond??Neighbor.Blond)
• ?Likes.(?Neighbor.(?Neighbor.Blond))

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Exercise on Tableaux

• The tableaux is an algorithm to check
  satisfiability.
• If all branches of your tableaux are open, then?
• You cannot say it is valid! Why?
• OWA!
• If all branches of your tableaux are closed,
  then?
• You can say it is unsatisfiable
• What can we do with tableaux?
• To prove the satisfiability of a concept.

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Tableaux cont.

• Rules
• ?
• ?
• ?
• ?
• Exercises
• Are these subsumptions valid?
• ?R.A??R.B??R.(A?B) ?R.A??R.B??R.(A?B)
• Decide whether the following subsumption holds
• ?R.A??R.C?T?R.D
• with TC (?R.B)?A, D(?R.A)??R.(?R.B)

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RelBAC Domain Specific DLs

• Syntax
• Nc subject groups, object types
• Ni individual subjects, individual objects
• Nr permissions
• DL constructors and formation rules
• Semantics
• Hierarchy
• Permission assignment
• Ground assignment
• Chinese Wall
• SoD
• High Level SoD
• Queries

Policies
Properties
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RelBAC Modeling

• The LDKR course consists of
• For persons Prof. Giunchiglia and TA Zhang as
  lecturers, Student Tin, Hoa, Parorali, Sartori,
  Chen, Gao, Lu, Zhang
• For online materials syllabus, slides for
  lectures, references, exercises and keys, exam
  questions, results and marks.
• We know that,
• Slides can be updated only by professors or TA
• Students can download all materials but only
  update keys to exercises.
• Each student should upload exam result to the
  site that TA can read and check for propose marks
  which will be finally decided by professor(s).

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LDKR Modeling Answer
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Upload
Update
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Chinese Wall Property

• Chinese Wall (CW)
• Originally no one has any access to anything
  then some requests are accepted and someone is
  allowed to perform some operation on something
  from then on, those has been allowed to access
  should not be allowed to access on those things
  arousing conflict of interests.
• Conflict of Interests (COI)
• Resources in COI should be avoided access for
  disclosure of information about competing
  parties.

COI
COI
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Modeling of the Chinese Wall Property

• Given a COI A1, , An, if one can access Ai,
  then s/he should not be allowed to access the
  rest.
• Suppose for Ai, the permission is Pi, then
• ?1iltjn ?Pi.Ai??Pj.Aj??

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SoD

• Separation of Duties
• Intuition
• Definition
• Semantic Details
• MEP
• MEO
• FA
• IFA

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Mutually Exclusive Positions

• A position is an organizational role denoting a
  group of subjects such as employees, managers,
  CEOs, etc. Given a set of positions P P1, ,
  Pn, where each Pi is a concept name
• To enforce that a subject can be assigned to at
  most one position among P.
• To enforce that no subject can be assigned to all
  the positions in P.
• To enforce that a subject can be assigned to at
  most m positions among P.

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Exercise of MEP

• In a bank scenario, customers sign checks bank
  clerks cash out the checks and managers monitor
  the checks.
• MEP one can play at most one of the positions
  as customer, clerk and manager.

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Exercise of MEP cont.

• MEP no one can play more all of the positions
  as customer, clerk and manager.
• MEP one can play at most 1 of the positions as
  customer, clerk and manager.

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Mutually Exclusive Operations

• An operation is a kind of permission that
  subjects may be allowed to perform some act on
  objects, such as Read, Download, etc. Given a set
  of operations giving rise to a MEO, OP Op1, ,
  Opn (where each Opi is a DL role name), then, we
  distinguish two different kinds of MEO
• To enforce that a subject cannot perform any two
  operations in OP.
• To enforce that a subject cannot perform any two
  operations in OP on the same object.

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Exercise of MEO

• Suppose a common file repository scenario files
  are objects, users are clients that visit the
  repository and permissions are read or write.
• MEO one cannot read and write at the same
  time.
• MEO one cannot read and write at the same time
  on the same file.
• Notice the difference between the two MEOs.

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Functional Access and Inverse

• FA A permission is functional iff it connects at
  most one object in the range.
• If each user in U, has an FA, P, to an object in
  O, then
• IFA A permission is inverse functional iff it
  connects at most one subject in the domain.
• If each object in O, has and IFA, P-, from a user
  in U, then

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Exercise of FA and IFA

• Give a scenario where FA and IFA are necessary.
• Desktop usage in lab.
• Bank private manager/clerk service

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OWL

• OWL Lite originally intended to support those
  users primarily needing a classification
  hierarchy and simple constraints.
• OWL DL to provide the maximum expressiveness
  possible while retaining computational
  completeness, decidability and the availability
  of practical reasoning algorithms.
• OWL Full designed to preserve some compatibility
  with RDF Schema.

Syntax Semantics Expressiveness Computability
OWL Lite simple DL subset Hierarchy, simple constraints Efficient
OWL DL SHIO(D) DL Maximum possible expressiveness Exist
OWL RDF RDF Compatible with RDF Non
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Exercise of OWL

• Refer to document specification