Logics for Data and Knowledge Representation

1
Logics for Data and KnowledgeRepresentation

• Description Logics

2
Outline

• Overview
• Syntax
• Semantics
• Terminology
• Reasoning with Terminologies
• World Description
• Reasoning with World Descriptions
• Tableau Algorithm

3
Overview

• Description Logics (DLs) is a family of KR
  formalisms that represent the knowledge of an
  application domain (the world) by
• defining the relevant concepts of the domain
  (i.e., its terminology), and then
• using these concepts to specify the properties of
  objects in the domain (i.e., the worlds
  description).

4
Some History on DLs

• Descended from the structured inheritance
  networks (Brachman, 1977).
• Introduced to overcome the ambiguities of early
  semantic networks and frames.
• First realized in the system Kl-One by Brachman
  and Schmolze (1985).
• First DL presented in the B Ss paper.
• Some nicknames for DLs
• Terminological knowledge representation
  languages,
• Concept languages,
• Term subsumption languages,
• Kl-One-based knowledge representation languages.

5
Three Basic Features
Extended from ClassL

1. The basic syntactic building blocks are atomic
   concepts, atomic roles, individuals.
2. The expressive power of DLs is restricted to a
   rather small set of constructors for building
   complex concepts and relations.
3. Implicit knowledge about concepts and individuals
   can be inferred automatically with the help of
   specific reasoning services.

Syntax
Extended from ClassL to Complest Role
Logical
6
Architecture of a KR system on DL
TBox
Description Language
Reasoning
ABox
Application Programs
Rules
7
Language (Syntax)

• The first step in setting up a formal language
  (viz. a descriptive language) is to list the
  symbols, that is, the alphabet of symbols.
• We denote a generic alphabet of a descriptive (or
  description) language dS.
• Similarly to any logical language we can divide
  symbols in dS in descriptive (nonlogical) and
  non-descriptive (logical).

8
Outline

• Overview
• Syntax
• Semantics
• Terminology
• Reasoning with Terminologies
• World Description
• Reasoning with World Descriptions
• Tableau Algorithm

9
Syntax of DL

• Descriptive dS consists of concept names (set C),
  which denote sets of individuals, role names (set
  R), which denote binary relations between
  individuals, and individual names, (set I), which
  denote individuals.
• e.g.concept names Room, Person, Fruitrole
  names likeSkiing, hasChild, partOf,
  isA,..individual names I, you, apple, Fido, ...

10
DL Language and Previous Languages

• concept names are propositional variables
• (PL/ClassL)
• role names are binary predicate symbols
• (FOL)
• individual names are constants
• (FOL)

11
Language (Syntax)

• Non-descriptive dS provides concept constructors
  to build complex formulas, called concept
  descriptions and role descriptions, from atomic
  formulas.
• e.g. ?(negation),
• ?(conjunction),?(for all),
• ?(there exists)

12
AL-family Languages

• We shall now discuss various descriptive
  languages from the family of AL-languages.
• An AL-language ( Attributive Languages) is a
  minimal DL language of practical interest.
• More expressive descriptive languages are usually
  extensions of some AL-language.
• AL-languages do not deal with individuals.

13
AL Logical Symbols

• Universal concept symbol ?.
• Bottom concept symbol ?.
• Parentheses (auxiliary symbols) (, )
• Logical constants (concept constructors)?
  (atomic negation), ? (conjunction)
• ?R (for all atomic roles)?R (there exists an
  atomic role)

The SAME as in ClassL
New!
14
AL Non-logical Symbols
The SAME as in ClassL

• Atomic concept names A, B, ...
• Concept names C, D, ...
• Atomic role names R (generic)
• NOTE There is no logical symbol in AL for
  logical implication (as ? in PL and in FOL).
  For this, we will use the subsumption symbol ?
  instead (as in classL).

New!
15
Defined Symbols

• Similarly to ClassL, ? and ? can be defined
• For all concept names C,
• ? df C ? ? C
• ? df ? ? or also ? df U for U be a special
  coincept name denoting the Universal Concept.
• We prefer to consider ? and ? AL symbols.

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Formation Rules for AL

• Atomic Concepts
• 1. A, B,..., ?, ?.
• Concepts (concept descriptions) 2. All the
  atomic concepts3. ?A for A (atomic concept
  negation)4. C?D (intersection) 5. ?R.C (value
  restriction)6. ?R.?(limited existential
  quantification)
• Resulting language attributive language (AL).

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Examples

• Atomic concepts Person, Female, Room, ...Atomic
  roles hasChild, partOf, isIn, isA,...
• Concepts Person ? Female,Person ? ?hasChild.? (
  ?hasChild )Person ? ?hasChild.?
  (Not?hasChild.?) Person ? ?hasChild.Female
• Question What is the intended meaning?

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Solution

• Person ? Female
• persons that are female
• Person ? ?hasChild. ? (all those) persons that
  have a child
• Person ? ?hasChild. ? (all those) persons
  without a child
• Person ? ?hasChild.Female persons all of whose
  children are female

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ALs Extensions - ALU

• Extended Alphabet Logical constants (concept
  constructors) ? (disjunction).
• Extended concepts (descriptions) C ? D (union)
• The resulting new language (i.e. AL plus the new
  set of concepts) usually denoted ALU.
• E.g. Mother?Father describes the extension of
  parent.

20
ALs Extensions - ALE

• Extended Alphabet Logical constants (concept
  constructors)?R (there exists an arbitrary
  role)
• Extended concepts (descriptions) ?R.C (full
  existential quantification)
• The resulting new language (i.e. AL plus the new
  set of concepts) usually denoted ALE.
• E.g. ?hasChild.Female describes the extension of
  those parents who has at least a daughter.

21
ALs Extensions - ALN

• Extended Alphabet Logical constants (concept
  constructors)n, n for all n ? N
  (at-least/at-most n)
• Extended concepts (descriptions) nR (at-least
  number restriction)nR (at-most number
  restriction)
• The resulting new language (i.e. AL plus the new
  set of concepts) usually denoted ALN.
• E.g. 2 hasChild describes the extension those
  parents who has at least 2 children.

22
ALs Extensions - ALC

• Extended Alphabet Logical constants (concept
  constructors) ? (general negation)
• Extended concepts (descriptions) ? C (full
  concept negation)
• The resulting new language (i.e. AL plus the new
  set of concepts) usually denoted ALC. (C stands
  for Complement).
• E.g. ?(2 hasChild ) describes the extension
  those parents who has at most 1 children.

23
ALs Extensions (Summary)

• Extending AL by any subsets of the above
  constructors yields a particular DL language.
• Each language is denoted by a string of the form
  ALUENC, where a letter in the name stands
  for the presence of the corresponding
  constructor. Notation AL.
• ALC as the most important in many aspects.(Well
  see that ALU ? ALC and ALE ? ALC.)

24
ALs Sub-languages

• By eliminating some of the syntactical symbols,
  we get some sub-languages of AL.
• The most important sub-language obtained by
  elimination in the AL family is ClassL.
• Historically, another important sub-language is
  named as the Frame Language FL0.

25
From AL to ClassL

• Elimination (w.r.t. ALUEC!)
• Name Symbols atomic role names R
• Concept constructors ?R, ?R
• Descriptions ?R.C, ?R.C
• The new language is a description language
  without roles which is ClassL.
• NOTE So far, we are considering DL without
  individuals and ABox.

26
ALs Contractions FL-

• Elimination (w.r.t ALs alphabet)
• Symbols Universal and bottom symbols ?, ?
• Concept constructors (atomic negation)
• Descriptions ?, ? , ? A (atomic negation)
• The resulting new language (i.e. AL without
  negation) is denoted FL-.

27
ALs Contractions FL0

• Elimination(w.r.t FL-s alphabet)
• Concept constructors?R (there exists an atomic
  role)
• Descriptions ?R.?(limited existential
  quantification)
• The resulting new language (i.e. FL- without the
  limited existential quantification) is denoted
  FL0.
• FL Frame Language (for historical reasons)

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Summary

• DLs are a family of logic-based KR formalisms to
  describe a domain in terms of - concepts - roles
  - individuals (grounding).
• Strictly speaking we do not need DLs to represent
  concepts and roles, but the variable-free syntax
  of DLs is much more concise! (Thats good for
  automation!)
• ClassL is ALC without Role.

This will be discussed later.
29
Outline

• Overview
• Syntax
• Semantics
• Terminology
• Reasoning with Terminologies
• World Description
• Reasoning with World Descriptions
• Tableau Algorithm

30
Semantics

• The elements of the description languages in
  AL-family (AL) are plain strings of symbols
• The meaning which is intended to be attached to
  concept, role, and individual names form an
  informal interpretation of the given AL
  languages expressions.

No (formal) meaning.
31
Recall Interpretation (?, I)

• Def. An interpretation of a language L is a pair
  I (?,I), where- ? (domain) is a non-empty set
  of objects - I(interpretation function) is a
  mapping from L to ?.
• NOTE sometimes, the mapping I is written as
  I but of the same function.
• E.g. I(Person) pp is a person is written as
• PersonI pp is a person.

32
AL Interpretation (?,I)

• I(?) Ø and I(?) ? (full domain, Universe)
• For every concept name A of L, I(A) ? ?.
• I(C) ? \ I(C).
• I(C?D) I(C) n I(D) I(C ? D) I(C) ? I(D).
• For every role name R of L, I(R) ? ? ?.
• I(?R.C) a ? ? for all b, if (a,b)?I(R) then
  b?I(C).
• I(?R. ?) a ? ? exists b s.t. (a,b) ? I(R).
• I(?R.C) a ? ? exists b s.t. (a,b) ? I(R), b
  ? I(C).
• I(nR) a ? ? b (a, b) ? I(R) n.
• I(nR) a ? ? b (a, b) ? I(R) n.

The SAME as in ClassL
33
Interpretation of Existential Quantifier

• I(?R.C) a ? ? exists b s.t. (a,b) ? I(R), b
  ? I(C)
• Those that have some value in C with role R.
• Remark a ? ?

34
Example

• Those who have a daughter.
• I(?R.C) I(?hasChild.Female) a ? ? exists b
  s.t. (a,b) ? I(hasChild), b ? I(Female)

hasChild
Anna has a child betty and betty is a female,
then Anna is an individual of the concept
?hasChild.Female.
35
Interpretation of Value Restriction

• I(?R.C) a ? ? for all b, if (a,b)?I(R) then
  b?I(C)
• Those that have only value in C with role R.
• Remark a ? ?

36
Example

• Those who have only daughter(s)
• I(?R.C) I(?hasChild.Female) a ? ? for all
  b, if (a,b)?I(R) then b?I(C)

hasChild
37
Interpretation of Number Restriction

• I(nR) a?? b (a, b) ? I(R) n
  
• Those that have relation R to at least n
  individuals.

b (a, b) ? I(R) n
38
Example

• Those who has at least 2 children
• I(nR) I(2hasChild)a ? ? b (a, b) ?
  I(R)n

?
hasChild
Annas children betty, bianca, babara,... are
2. Anna is one individual of those who has at
least 2 children.
39
Interpretation of Number Restriction Cont.

• I(nR) a ? ? b (a, b) ? I(R) n
  
• Those that have relation R to at most n
  individuals.

?
b
b'

a
b (a,b) ? I(R) n
40
Example

• Those who as at most 2 children
• I(nR) I(2hasChild)a?? b (a, b) ?
  I(R)n

?
hasChild
Anna has only one child Betty, so Anna belongs to
the concept of those who has at most 2 children.
41
Simple Exercises

• Verify the following equivalences hold for all
  interpretations (?,I)
• I((C ? D)) I(C ? D)
• I((C ? D)) I(C ? D)
• I(?R.C) I(?R.C)
• I(?R.C) I(?R.C)
• E.g.
• I(?R.C) a ? ? not exists b s.t. (a,b) ?
  I(R), b ? I(C) a ? ? ?b. R(a,b)?C(b)
• a ? ? ?b. (R(a,b)?C(b))
• a ? ? ?b. R(a,b)?C(b)
• I(?R.C)

42
Outline

• Overview
• Syntax
• Semantics
• Terminology
• Reasoning with Terminologies
• World Description
• Reasoning with World Descriptions
• Tableau Algorithm

43
Terminological Axioms

• Inclusion Axiom
• C?D
• e.g.
• Master ? 18Pass.Exam?1Finish.Project,
• Professor??Give.Lecture??Organize.(Exam?Experiment
  )
• Equivalence (Equality) Axiom
• CD
• e.g.
• PhD Postgraduate?3Publish.Paper,
• Parent Man??hasChild.?
• Is the TBox definition the SAME here as in ClassL?

R?S
RS
44
Terminology (TBox)

• A terminology (or TBox) is a set of a
  (terminological) axioms.
• e.g. T is
• PhD Postgraduate?3Publish.Paper,
• Parent Person??hasChild.Person,
• hasGrandChild?hasChild
• NOTE the punctuations mean logical conjunctions.
• .

45
Outline

• Overview
• Syntax
• Semantics
• Terminology
• Reasoning with Terminologies
• World Description
• Reasoning with World Descriptions
• Tableau Algorithm

46
TBox Reasoning

• Let T be a TBox
• Satisfiability(with respect to T)
• T satisfies P?
• Subsumption (with respect to T)
• T P ? Q?
• Equivalence (with respect to T)
• (T P Q) T P ? Q and T P ? Q?
• Disjointness (with respect to T)
• T P ? Q ? ??
• NOTE they are the SAME as in ClassL with more
  concepts built by the role constructors.

47
Outline

• Overview
• Syntax
• Semantics
• Terminology
• Reasoning with Terminologies
• World Description
• Reasoning with World Descriptions
• Tableau Algorithm

48
ABox

• In a ABox, one introduces individuals, by giving
• them names, and one asserts properties about
• these individuals.
• We denote individual names as a, b, c,
• An assertion with Role R is called Role assertion
  (in contrast to Concept assertion C(a) in ClassL)
  as
• R(a, b), R(b, c),
• e.g.
• Study(Tin, ldkr)
• NOTE the other part of the ABox is the Concept
  assertions the SAME as in Ground ClassL.

The SAME as in ClassL
49
Semantics of the ABox

• We give a semantics to ABox by extending
  interpretations to individual names.
• An interpretation I (?, I) not only maps atomic
  concepts to sets, but in addition maps each
  individual name a to an element aI ??., namely
• I (C(a)) aI ?CI,
• I(R(a, b))(aI, bI)?RI
• We still assume unique name assumption as in
  ClassL.
• set constructor is the SAME as in ClassL, which
  allows individual description in TBox.

The SAME as in ClassL
50
Outline

• Overview
• Syntax
• Semantics
• Terminology
• Reasoning with Terminologies
• World Description
• Reasoning with World Descriptions
• Tableau Algorithm

51
Consistency

• Consistency An ABox A is consistent with respect
  to a TBox T if there is an interpretation I which
  is a model of both A and T.
• e.g. TParent?1hasChild.Person
• AhasChild(mary, bob), hasChild(mary, cate)
• is consistent but Not consistent with respect
  the above TBox.
• NOTE it is the SAME as in ClassL, with respect
  to more complex TBox (Role axioms).

52
Other ABox Reasoning

• Instance Checking Checking whether an assertion
  C(a) or R(a, b) is entailed by an ABox A (and
  TBox via expansion)
• A ? C(a) or A ? R(a, b)
• Instance retrieval Given an ABox A and a concept
  C retrieve all instance ai which satisfies C.
• A ? C(ai)
• Concept Realization Given an ABox A, a set of
  concepts and an individual a find the most
  specific concepts C in the set such that A ? C(a)
• NOTE the last two services are the SAME as in
  ClassL, with respect to more complex TBox (Role
  axioms).

53
Outline

• Overview
• Syntax
• Semantics
• Terminology
• Reasoning with Terminologies
• World Description
• Reasoning with World Descriptions
• Tableau Algorithm

54
Tableaux Calculus

• The Tableaux calculus is a decision procedure
  solving the problem of satisfiability.
• If a formula is satisfiable, the procedure will
  constructively exhibit a model of the formula.
• The basic idea is to incrementally build the
  model by looking at the formula, by decomposing
  it in a top-down fashion. The procedure
  exhaustively tries all the possibilities so that
  it can prove eventually that no model could be
  found for an unsatisfiable formula.

55
The Tableau Algorithm

• Construct a model for the input concept
  description C0.
• Model is represented by tree form.
• The formula has been translated into Negation
  Normal Form (NNM).
• To decide satisfiability of the concept C0 ,
  start with the initial tree (root node).
• Repeatedly apply expansion rules until find
  contradiction or expansion completed.
• Satisfiable iff a complete and contradiction-free
  tree is found

56
Rules for Tableau Algorithm
57
?

• When there are not both C1(x) and C2(x) in the
  ABox.
• Add to the ABox C1(x) and C2(x) .
• ABoxABox?C1(x), C2(x)

58
? cont.

• TBoxMother Female ??hasChild.Person
• ABoxMother(Anna)
• Is (?hasChild.Person?? hasParent.Person)
  satisfiable?
• Mother(Anna)
• (Female ??hasChild.Person)(Anna)
• Female(Anna)
• (?hasChild.Person)(Anna)
• (?hasChild.Person?? hasParent.Person)(Anna)
• (?hasChild.Person)(Anna)
• ( ? hasParent.Person)(Anna)

59
?

• When there are neither C1(x) nor C2(x) in the
  ABox.
• Split into multiple branches
• Add in branch 1 to the ABox C1(x)
• and add in branch 2 to the Abox C2(x) .
• ABoxABox?C1(x)
• ABoxABox?C2(x)

60
? cont.

• TBoxParent?hasChild.Female??hasChild.Male, Pers
  onMale?Female,
• MotherParent ?Female
• ABoxMother(Anna)
• Is (?hasChild.Person) satisfiable?
• Mother(Anna)
• Parent(Anna)
• (?hasChild.Female??hasChild.Male)(Anna)
• (?hasChild.Female)(Anna) (?hasChild.Male)(Anna
  )

61
?

• When there are no individual name z such that
  both C(z) and R(x,z) in the ABox.
• Add to the ABox C(z) and R(x,z) .
• ABoxABox?C(z), R(x,z)

62
? cont.

• TBoxParent?hasChild.Female??hasChild.Male,
  PersonMale?Female,
• MotherParent ?Female
• ABoxMother(Anna), hasChild(Anna,Bob),
  Female(Bob)
• Is (?hasChild.Person) satisfiable?
• Mother(Anna)
• Parent(Anna)
• (?hasChild.Female??hasChild.Male)(Anna)
• (?hasChild.Male)(Anna)
• hasChild(Anna,Bob)
• Male(Bob)

63
?

• When there is R(x,z) but not C(z) in the ABox.
• Add to the ABox C(z) .
• ABoxABox?C(z)

64
? cont.

• TBoxDaughterParent?hasChild.Female,
  Male?Female??
• ABoxhasSibling(Anna,Bob), Female(Bob)
• Is DaughterMather satisfiable?
• DaughterMather(x)
• ?hasChild.Female(x)
• hasSibling(Anna,Bob)
• Female(Bob)
• Female(Bob)

?
65
Example of Tableau Reasoning

• Is ?hasChild.Male??hasChild.Male satisfiable?
• (?hasChild.Male??hasChild.Male)(x)
• (?hasChild.Male)(x)
• (?hasChild.Male)(x)
• hasChild(x,y)
• Male(y)
• Male(y)

?
?
?
66
Additional Rules
67
Exercise

• Suppose we have the following knowledge base
• TBox
• Female?Male?Human
• Mother?Female
• Father?Male
• Child?has.Mother? ?has.Father
• Male?Female??
• ABox
• Mother(Anna)
• Father(Bob)
• has (Cate,Anna)
• has (Cate,Bob)

Prove Human(Anna) Female(Bob) Child(Cate) Father
?Mother??
68
Exercise

• Prove with Tableau algorithm the satisfiability
  of the following formula.

69
Complexity of Tableau Algorithms

• The satisifiability algorithm of ALC may need
  exponential time and space. It is
  PSPACE-complete.
• A modified algorithm needs only polynomial space
  as it assumes a depth-first search and store only
  the correct path.
• The consistency and instance checking problem for
  ALC are also PSPACE-complete.
• The complexity results for other Description
  Logics varies according to corresponding
  constructors.