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### Logics for Data and KnowledgeRepresentation

Exercise 3: DLs

Outline

- Modeling
- Previous Logics
- DL
- RelBAC
- OWL
- Comprehensive

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

<?xml version=“1.0”?>

<!DOCTYPE rdf:RDF[…

]>

<rdf:RDF…

>

<Owl:Ontology…>

…

</rdf:RDF>

What distincts DL from Previous Logics?

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

Expressiveness of DL

- Binary Relations?

YES!

- Subsumption?

Of course!

More than concept subsumption!

Arbitrary!

- Else?
- Some
- Only
- Number

Description Logics

- Propositional DL VS. ClassL
- DL VS. Ground ClassL
- Role Constructors
- ∃
- ∀
- ≤
- ≥
- ⊓⊔¬≡⊑

In addition to concept constructors

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⊓Disney)

- “A binary tree is a tree with at most two sub-trees that are themselves binary trees.”

BTree≡Tree⊓≤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⊓∃atPositionOf.Banana)

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

¬∀R.C≡∃R.¬C

(¬∀R.C)I

=Δ-{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

DL Reasoning: ABox (1)

- Given the interpretation I with the domain ΔI={d,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

A

A,B

S

e

f

R

R

A

d

S

S

g

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))

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.

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 T={C≡ (∃R.B)⊓¬A, D≡¬(∃R.A)⊓∃R.(∃R.B)}

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

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).

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

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

⊔1≤i<j≤n ∃Pi.Ai⊓∃Pj.Aj⊑⊥

SoD

- Separation of Duties…
- Intuition
- Definition
- Semantic Details
- MEP
- MEO
- FA
- IFA

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 mostone 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.

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.’

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.’

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 OPon the same object.

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 MEO’s.

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

Exercise of FA and IFA

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

…

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.

Exercise of OWL

- Refer to document specification…

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