Logics: one, no one and one hundred thousand

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Oct 22, 2013 (4 years and 19 days ago)

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7. Juni 2006

Juri De Coi

1

Logics:

one, no one and one hundred thousand


Juri De Coi

L3S Research Seminar

Hannover, 09
-
06
-
2006

7. Juni 2006

Juri De Coi

2

Please, forgive me!

7. Juni 2006

Juri De Coi

3

Please, forgive me!


Please, help me!

7. Juni 2006

Juri De Coi

4

Why did I do what I did?

7. Juni 2006

Juri De Coi

5

Why did I do what I did?

WP0: State of the art investigation.


Description Logic
-
based Policy Specification Languages (KAoS, REI)


Logic Programming
-
based Policy Specification Languages
(PeerTrust, Protune)

7. Juni 2006

Juri De Coi

6

Why did I do what I did?

WP0: State of the art investigation.


Description Logic
-
based Policy Specification Languages (KAoS, REI)


Logic Programming
-
based Policy Specification Languages
(PeerTrust, Protune)


WP1: Study of DL and LP.

7. Juni 2006

Juri De Coi

7

Why did I do what I did?

WP0: State of the art investigation.


Description Logic
-
based Policy Specification Languages (KAoS, REI)


Logic Programming
-
based Policy Specification Languages
(PeerTrust, Protune)


WP1: Study of DL and LP.


WP2: Mapping DL and LP to a common formalism.

7. Juni 2006

Juri De Coi

8

Grosof et al.,
Description Logic Programs: Combining Logic Programs with Description Logic

7. Juni 2006

Juri De Coi

9

Description of Work

WP0: State of the art investigation.


Description Logic
-
based Policy Specification Languages (KAoS, REI)


Logic Programming
-
based Policy Specification Languages
(PeerTrust, Protune)


WP1: Study of DL, LP
and FOL
.


WP2: Mapping DL and LP
to FOL (as far as possible)
.

7. Juni 2006

Juri De Coi

10

Description of Work

WP0: State of the art investigation.


Description Logic
-
based Policy Specification Languages (KAoS, REI)


Logic Programming
-
based Policy Specification Languages
(PeerTrust, Protune)


WP1: Study of DL, LP and FOL.


WP2: Mapping DL and LP to FOL (as far as possible).


WP3: Identify the set of features we are interested in.


Identify the
sustainable overhead.

7. Juni 2006

Juri De Coi

11

Description of Work

WP0: State of the art investigation.


Description Logic
-
based Policy Specification Languages (KAoS, REI)


Logic Programming
-
based Policy Specification Languages
(PeerTrust, Protune)


WP1: Study of DL, LP and FOL.


WP2: Mapping DL and LP to FOL (as far as possible).


WP3: Identify the set of features we are interested in


Identify the
sustainable overhead.

7. Juni 2006

Juri De Coi

12

WP1: Study of DL, LP and FOL.


First
-
order Logic




Description Logic


Logic Programming


7. Juni 2006

Juri De Coi

13

WP1: Study of DL, LP and FOL.

Propositional logic

First
-
order Logic

(Horn
-
clause Logic, Definite Horn
-
clause Logic, Equality
-
free
Horn
-
clause Logic, Datalog Horn
-
clause Logic, def
-
Horn Logic, Description
Horn Logic)

Description Logic
(ℓ, ℓ
h
, ℓ
b
)

Logic Programming
(Definite Logic Programming, Equality
-
free Logic
Programming, Datalog, def
-
Logic Programming, Description Logic
Programming, Rules with Contextually Scoped Negation)

Resource Description Framework
-

RDF

Web Ontology Language
-

OWL
(Lite, Description Logic
-

DL, Full)

Semantic Web Rule Language
-

SWRL

Aℓ
-
log

7. Juni 2006

Juri De Coi

14

Propositional Logic

Propositional variables:
x
i

(a denumerable set)

Negation:


Conjunction:


Disjunction:


Implication:



EX:
(x
1



x
2
)


(

x
1



x
2
)

7. Juni 2006

Juri De Coi

15

First
-
order Logic

n
-
ary predicates:
P
ni
(_, _, ... _)

(a denumerable set)

Individual constants:
a
i

(a denumerable set)

Individual variables:
x
i

(a denumerable set)

Universal quantifier:


Existential quantifier:



EX:

x
1
(P
2,1
(a
1
, x
1
)


P
2,2
(x
2
, a
2
))





x
2
(

P
2,1
(x
1
, a
2
)


P
2,2
(a
1
, x
2
))

7. Juni 2006

Juri De Coi

16

Open issues


What are parameters?

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Juri De Coi

17

Open issues


What are parameters?


FOL does not deal with equality


FOL does not deal with function symbols

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Juri De Coi

18

Grosof et al.,
Description Logic Programs: Combining Logic Programs with Description Logic

7. Juni 2006

Juri De Coi

19

Description Logics (I)

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Juri De Coi

20

Description Logics (I)

Concept

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Juri De Coi

21

Description Logics (I)

Concept


Role

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Juri De Coi

22

Description Logics (I)

Concept


Role


Instance

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Juri De Coi

23

Description Logics (II)

Atomic concept:
A

(default concepts
Τ

and

)

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Juri De Coi

24

Description Logics (II)

Atomic concept:
A

(default concepts
Τ

and

)


Negation:

C

Intersection:
C


D

Union:
C


D

(Full) existential quantification:

R.C

7. Juni 2006

Juri De Coi

25

Description Logics (II)

Atomic concept:
A

(default concepts
Τ

and

)


Negation:

C

Intersection:
C


D

Union:
C


D

(Full) existential quantification:

R.C

Number restriction (cardinality constraint):


≤n R.C


≥n R.C

7. Juni 2006

Juri De Coi

26

Description Logics (II)

Atomic concept:
A

(default concepts
Τ

and

)


Negation:

C

Intersection:
C


D

Union:
C


D

(Full) existential quantification:

R.C

Number restriction (cardinality constraint):


≤n R.C


≥n R.C

Value restriction:

R.C

7. Juni 2006

Juri De Coi

27

Open issue


Do people without children belong to the Concept of "people whose
children are only female" (

hasChild.Female
)?

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Juri De Coi

28

Open issue


Do people without children belong to the Concept of "people whose
children are only female" (

hasChild.Female
)?


Grosof et al.: yes

7. Juni 2006

Juri De Coi

29

Open issue


Do people without children belong to the Concept of "people whose
children are only female" (

hasChild.Female
)?


Grosof et al.: yes


Baader et al.: no (?)

7. Juni 2006

Juri De Coi

30

Description Logics (III)


Atomic role:
R

Intersection:
R


S

Inverse role:
R
-

Transitive closure:
R
+


Concept assertion:
a:C

Role assertion:
<a, b>:R

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Juri De Coi

31

Description Logics (IV)

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Juri De Coi

32

Description Logics (IV)

Inclusion axiom:


C


D


R


S

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Juri De Coi

33

Description Logics (IV)

Inclusion axiom:


C


D


R


S


Symmetrical property:

Transitive property:

Functional property:

Inverse functional property:


C is range of R:

C is domain of R:

7. Juni 2006

Juri De Coi

34

Description Logics (IV)

Inclusion axiom:


C


D


R


S


Symmetrical property:
R
-



R

Transitive property:

Functional property:

Inverse functional property:


C is range of R:

C is domain of R:

7. Juni 2006

Juri De Coi

35

Description Logics (IV)

Inclusion axiom:


C


D


R


S


Symmetrical property:
R
-



R

Transitive property:
R
+



R

Functional property:

Inverse functional property:


C is range of R:

C is domain of R:

7. Juni 2006

Juri De Coi

36

Description Logics (IV)

Inclusion axiom:


C


D


R


S


Symmetrical property:
R
-



R

Transitive property:
R
+



R

Functional property:
Τ



≤1 R.
Τ

Inverse functional property:


C is range of R:

C is domain of R:

7. Juni 2006

Juri De Coi

37

Description Logics (IV)

Inclusion axiom:


C


D


R


S


Symmetrical property:
R
-



R

Transitive property:
R
+



R

Functional property:
Τ



≤1 R.
Τ

Inverse functional property:
Τ



≤1 R
-
.
Τ


C is range of R:

C is domain of R:

7. Juni 2006

Juri De Coi

38

Description Logics (IV)

Inclusion axiom:


C


D


R


S


Symmetrical property:
R
-



R

Transitive property:
R
+



R

Functional property:
Τ



≤1 R.
Τ

Inverse functional property:
Τ



≤1 R
-
.
Τ


C is range of R:

R.
Τ




C

(according to Grosof et al.
Τ




R.C
)

C is domain of R:

7. Juni 2006

Juri De Coi

39

Description Logics (IV)

Inclusion axiom:


C


D


R


S


Symmetrical property:
R
-



R

Transitive property:
R
+



R

Functional property:
Τ



≤1 R.
Τ

Inverse functional property:
Τ



≤1 R
-
.
Τ


C is range of R:

R.
Τ




C

(according to Grosof et al.
Τ




R.C
)

C is domain of R:

R
-
.
Τ



C

(according to Grosof et al.
Τ




R
-
.C
)

7. Juni 2006

Juri De Coi

40

Resource Description Framework (RDF)
and RDF Schema (RDFS)



7. Juni 2006

Juri De Coi

41

Resource Description Framework (RDF)
and RDF Schema (RDFS)


Support for



definition of atomic Concepts/Roles (
Τ

is called
rdfs:Resource
)


Concept/Role assertions


Concept/Role inclusion axioms


domain/range specification


Open issue



Unique
-
ID assumption?

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Juri De Coi

42

Additional features

Facilities to deal with



common data
-
types (the predefined Concept
rdfs:Literal
)


collections


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Juri De Coi

43

RDF(S): Example


<
a
,
b
>:
R


<rdf:Statement>


<rdf:subject rdf:resource="
a
" />


<rdf:predicate rdf:resource="
R
" />


<rdf:object rdf:resource="
b
" />

</rdf:Statement>

7. Juni 2006

Juri De Coi

44

RDF(S): Example




<rdf:Statement>


<rdf:subject rdf:resource="
R
" />


<rdf:predicate rdf:resource="
R
" />


<rdf:object rdf:resource="
R
" />

</rdf:Statement>

7. Juni 2006

Juri De Coi

45

RDF(S): Example




<rdf:Statement
rdf:ID="S"
>


<rdf:subject rdf:resource="
S
" />


<rdf:predicate rdf:resource="
R
" />


<rdf:object rdf:resource="
R
" />

</rdf:Statement>

7. Juni 2006

Juri De Coi

46

Web Ontology Language (OWL)


Extension of RDF(S)


Available in three flavours (Lite, Description Logic
-

DL, Full)


OWL expressiveness


varies according to the chosen flavour


can reach (and pass) the one of the above
-
described DL languages

7. Juni 2006

Juri De Coi

47

Web Ontology Language (OWL)


Extension of RDF(S)


Available in three flavours (Lite, Description Logic
-

DL, Full)


OWL expressiveness


varies according to the chosen flavour


can reach (and pass) the one of the above
-
described DL languages


Additional features



No Role intersection


No unique
-
ID assumption


Two kinds of Roles
(
owl:DatatypeProperty

and
owl:ObjectProperty
)

7. Juni 2006

Juri De Coi

48

Logic Programming

7. Juni 2006

Juri De Coi

49

Logic Programming

n
-
ary predicates:
P
ni
(_, _, ... _)

n
-
ary functions:
F
ni
(_, _, ... _)

Constants:
a
i

Variables:
x
i

7. Juni 2006

Juri De Coi

50

Logic Programming

n
-
ary predicates:
P
ni
(_, _, ... _)

n
-
ary functions:
F
ni
(_, _, ... _)

Constants:
a
i

Variables:
x
i


Negation
-
as
-
failure:
~

7. Juni 2006

Juri De Coi

51

Logic Programming

n
-
ary predicates:
P
ni
(_, _, ... _)

n
-
ary functions:
F
ni
(_, _, ... _)

Constants:
a
i

Variables:
x
i


Negation
-
as
-
failure:
~


P
0



P
a1
, ... P
an
, ~P
b1
, ... ~P
bm

with
m,n≥0

and
m+n>0

P
0



P
a1
, ... P
an
, ~P
b1
, ... ~P
bm

with
m,n≥0

and
m+n>0

7. Juni 2006

Juri De Coi

52

Open issues (among others)


What is an atom?

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Juri De Coi

53

Open issues (among others)


What is an atom?


What are procedural attachments?

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Juri De Coi

54

WP2: Mapping DL to FOL (?)


cf. File

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Juri De Coi

55

WP2: Mapping LP to FOL (??)

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Juri De Coi

56

WP2: Mapping LP to FOL (??)

(i)
P
0

(ii)


P
1
, ... P
n

with
n>0

(iii)
P
0



P
1
, ... P
n

with
n>0

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Juri De Coi

57

WP2: Mapping LP to FOL (??)

(i)
P
0

(ii)


P
1
, ... P
n

with
n>0

(iii)
P
0



P
1
, ... P
n

with
n>0


Let
x
1
, ... x
n

be the variables appearing in (iii) (resp. (ii) or (i)):


(iii)

x
1
, ... x
n

(P
0



P
1



... P
n
)

with
n>0

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Juri De Coi

58

WP2: Mapping LP to FOL (??)

(i)
P
0

(ii)


P
1
, ... P
n

with
n>0

(iii)
P
0



P
1
, ... P
n

with
n>0


Let
x
1
, ... x
n

be the variables appearing in (iii) (resp. (ii) or (i)):


(iii)

x
1
, ... x
n

(P
0



P
1



... P
n
)

with
n>0


(iii')

x
1
, ... x
n

(P
0




P
1



...

P
n
)

7. Juni 2006

Juri De Coi

59

WP2: Mapping LP to FOL (??)

(i)
P
0

(ii)


P
1
, ... P
n

with
n>0

(iii)
P
0



P
1
, ... P
n

with
n>0


Let
x
1
, ... x
n

be the variables appearing in (iii) (resp. (ii) or (i)):


(iii)

x
1
, ... x
n

(P
0



P
1



... P
n
)

with
n>0


(iii')

x
1
, ... x
n

(P
0




P
1



...

P
n
)

(ii)

x
1
, ... x
n

(

P
0



...

P
n
)

(i)

x
1
, ... x
n

P
0

7. Juni 2006

Juri De Coi

60

Further work


Understand what I did not understand


Map what I did not map


inverse role


transitive closure


RTF collections


RTF data
-
type facilities


owl:DatatypeProperty

and
owl:ObjectProperty


lack of unique
-
ID assumption


Extend the set of considered logic languages


WP3: Identify the set of features we are interested in


Identify the
sustainable overhead

7. Juni 2006

Juri De Coi

61