Alternative representations: Semantic networks

1
Alternative representations Semantic networks

• A semantic net is a labeled directed graph, where
  each node represents an
• object (a proposition), and each link represents
  a relationship between two
• objects.
• Example
• 
  sister-of
  husband-of
• wife-of
• 
  
  wife-of
• husband-of
• mother-of
  father-of
  mother-of
• 
  wife-of
  father-of
• 
  
• 
  husband-of
• 
  mother-of
  father-of
• Note that only binary relationships can be
  represented in this model.

Ann
Carol
David
Bill
Tom
Susan
John
2

• Semantic nets represent propositional
  information. Relations between propositions are
• of primary interest because they provide the
  basic structure for organizing knowledge.
• Some important relations are
• IS-A (is an instance of). Refers to a member of
  a class, where a class is a group of objects
  with one or more common attributes (properties).
  For example, Tom IS-A bird.
• A-KIND-OF. Relates one class to another, for
  example Birds are A-KIND-OF animals.
• HAS-A. Relates attributes to objects, for
  example Mary HAS-A cat.
• CAUSE. Expresses a causal relationship, for
  example Fire CAUSES smoke.
• Note that semantic nets can be easily converted
  into a set of FOL formulas, and
• vice versa. Semantic nets, however, have two
  important advantages
• A very simple execution model.
• Very readable representation, which makes it easy
  to visualize inference steps.

3
Inference in semantic networks.

• The inference procedure in semantic nets is
  called inheritance, and it allows
• one nodes characteristics to be duplicated by a
  descendent node.
• Example Consider a class aircraft, and assume
  that balloons, propeller-driven objects and
  jets are subclasses of it, i.e.
• Balloons are A-KIND-OF aircrafts
• Propeller-driven objects are A-KIND-OF
  aircrafts, etc.
• Assume that the following attributes are
  assigned to aircrafts Aircraft IS-A flying
  object, Aircraft HAS-A wings, Aircraft HAS-A
  engines
• All properties assigned to the superclass,
  aircraft, will be inherited
• by its subclasses, unless there is an
  exception link capturing a
• non-monotonic inference relation.

4
Multiple inheritance may result in a conflicting
inference

• In some semantic networks, one class can inherit
  properties of more than one
• superclass.
• The Nixon diamond example It is widely
  accepted that Quakers tend to be pacifists, and
  Republicans tend not to be. Nixon is known to be
  both - a Quaker, and a Republican.
• Not IS-A
  IS-A
• 
  IS-A IS-A
• The resulting conflict can be resolved only if
  additional information stating a preference to
• one of the conflicting inferences is provided.

Pacifists
Republicans
Quakers
Nixon
5
Object-attribute-value triplets

• One problem with semantic nets is that there is
  no standard definition of link
• names. To avoid this ambiguity, we can restrict
  this formalism to a very simple
• kind of a semantic network, which has only two
  types of links, HAS-A and
• IS-A. Such a formalism is called
  Object-Attribute-Value (OAV) triplets, and it
• is a widely used mode of knowledge representation
  (especially for representing
• declarative knowledge). It is also the base of
  the Semantic Web KR model, called RDF.
• Example Consider object airplane. Some of its
  attributes are
• number of engines
• type of engines
• type of wing design.
• Possible values of these attributes are
• number of engines 2, 3, 4.
• type of engines jet, propeller-driven.
• type of wing design conventional, swept-back.

6
Object-attribute-value triples (example contd)

• Object Attribute
  Value
• Airplane NumberOfEngines
  2
• Airplane NumberOfEngines
  3
• Airplane NumberOfEngines
  4
• Airplane TypeOfEngines
  Jet
• Airplane TypeOfEngines
  Propeller
• Airplane TypeOfWings
  Conventional
• Airplane TypeOfWings
  SweptBack
• Or, as predicates
• NumberOfEngines (Airplane, 2),
• TypeOfEngines (Airline, Jet),
• TypeOfWings (Airlines, Propeller),

7
Problems with semantic nets and OAV triplets

• There is no standard definition of link and node
  names. This makes it difficult to understand the
  network, and whether or not it is designed in a
  consistent manner.
• Inheritance is a combinatorially explosive
  search, especially if the response to a query is
  negative. Plus, it is the only inference
  mechanism built in semantic nets, which may be
  insufficient for some applications.
• Initially, semantic nets were proposed as a model
  of human associative memory (Quinllian, 1968).
  But, are they an adequate model? It is believed
  that human brain contains about 1010 neurons,
  and 1015 links. Consider how long it takes for a
  human to answer NO to a query Are there trees
  on the moon? Obviously, humans process
  information in a very different way, not as
  suggested by the proponents of semantic networks.
• Semantic nets are logically and heuristically
  very weak. Statements such as Some books are
  more interesting than others, No book is
  available on this subject, If a fiction book is
  requested, do not consider books on history,
  health and mathematics cannot be represented in
  a semantic network.

8
Frames (Minsky, 1975)

• Semantic nets represent shallow knowledge,
  because all of the information must be
• represented in terms of nodes and links which are
  propositions. What if the objects in
• the domain are, in turn, complex structures? For
  example, consider object animal.
• We may want to incorporate as part of the
  objects description all of the important
• properties of this object.
• Example AIMA (first edition), page 318, Figure
  10.7
• Note that frames comprising the nodes of
  this frame-based network represent typical
  examples or stereotypes of described objects.
  Such typical example are called concepts, and
  they are like data structures where data, in
  turn, contain data.
• The underlying assumption of the frame theory is
  that when one encounters a
• new situation (or a substantial change in ones
  view of the situation occurs),
• one selects from his/her memory the frame
  representing a given concept and
• changes it to reflect the new reality.
• Note Case-based reasoning systems exclusively
  utilize frames as a knowledge
• representation formalism.

9
Problems with Frames

• Frames cannot represent exceptions. By
  definition, frames represent typical objects.
  But, consider Clyde, an elephant with 3 legs. Is
  he NOT an elephant?
• Frames lack a well-defined semantics. Consider
  Sam, a bat. Is he a typical mammal?
• It is difficult to incorporate heuristic
  information into the frame. For example, consider
  a medical ES which uses frames to represent a
  typical patient or a typical disease. It is
  difficult to represent how specific symptoms
  relate to each other and how they must be used in
  the diagnostic process.
• In general, frames are good for representing
  class hierarchies to model domains where objects
  are well defined and the classification is clear
  cut.

10
Description logics

• Description logics are decidable fragments of FOL
  intended to represent categories
• (classes/concepts), the relations between classes
  (roles) and individuals.
• Example Consider category Student. One
  possible definition of this category is
• Student ? And(takes-classes,
  does-homeworks, is-responsible)
• Note that this definition can be translated
  into the following FOL sentence
• ? x (Student(x) ltgt Takes-classes(x)
  Does-homeworks(x)
• 
  Is-responsible(x))
• Example Consider category of men with at least
  three sons who are all unemployed and married to
  doctors, and at most two daughters who are all
  professors in physics or chemistry departments.
• Man3SS2SD ? And(Man, At-Least(3, Son),
  At-Most(2, Daughter),
• All(Son, And(Unemployed, Married,
  All(Spouse, Doctor))),
• All(Daughter, And(Professor,
  Fills(Department, Physics, Chemistry)))).
• DLs is a family of logics which defer by their
  expressivity depending on the
• constructors employed to build complex
  descriptions from simple ones.

11
General Architecture of Description Logics

• DL knowledge base consists of 3 parts
• The TBox, which contains terminological knowledge
  (i.e. knowledge about concepts in a domain).
• Examples Student ? Person ? ?
  takesCourse . Course
• Student ? Person
• The ABox, which contains assertion knowledge
  (i.e. knowledge about individuals).
• Example Student (Bob), takesCourse (Bob,
  ComputerScience)
• The RBox, R, which contains role-centric
  knowledge (i.e. knowledge about interdependences
  between roles/properties)
• Example teachesGradCourse ? teachesAtUniv
• RBox is not required only more
  expressive DLs have constructors to handle
  RBoxes.
• The smallest deductively complete DL is called
  ALC. It contains ?, ?, and ? constructors, and
  quantifiers restricting the domain and the range
  of roles, i.e. ? R . C and ? R . C
• It also contains class inclusion axiom (ex.
  Student ? Person) and class equivalence axiom
  (ex. PhDThesis ? Dissertation)

12
ALC (Attributive Language with Complements)
Syntax

• ALC Atomic types
• Concept names A, B, C,
• Top and bottom concepts, ? and ?
• Role names R, S,
• ALC Constructors ?, ?, ?, ? R . C and ? R . C
• Class inclusion and equivalence axioms ?, ?
• Complex class relations are built from atomic
  types and ?, ?, ?
• Example Instructor ? (Staff ? ?Professor) ?
  Lecturer
• In FOL ?x Instructor(x) ? (Staff(x) ?
  ?Professor(x)) ? Lecturer(x)
• Quantifiers on Roles
• Strict binding of the range of a role to a class.
  Example A thesis must be authored by a student,
  i.e. Thesis ? ? author . Student
• In FOL ?x (Thesis(x) ? ?y(author(x, y) ?
  Student(y)))
• Open binding of the range of a role to a class.
  Example Every student has at least one advisor,
  i.e. Student ? ? advisor . Professor
• In FOL ?x (Student(x) ? ?y (advisor(x, y)
  ? Professor(y)))

13
ALC Formal Syntax

• Production rules for creating classes in ALC
• C, D A T ? ?C C ? D C ? D ? R .
  C ? R . C
• where A is an atomic class, C and D are
  complex classes, and R
• is a role.
• An ALC TBox contains assertions of the forms C ?
  D and C ? D
• An ALC ABox contains assertions of the form C(a)
  and R(a, b), where a and b are individuals.
• An ALC Knowledge Base TBox, ABox.

14
ALC Semantics

• An interpretation, I (?I, .I), is defined by
• A domain of individuals, ?I
• An interpretation function .I which maps
• Individual names, a, to domain elements aI ? ?I
• Class names, C, to a set of domain elements CI ?
  ?I
• Role names, R, to a set of pairs of domain
  elements
• RI ? ?I ? ?I
• An interpretation, I, for axioms
• C(a) holds iff aI ? CI
• R(a, b) holds iff (aI, bI) ? RI
• C ? D holds iff CI ? DI
• C ? D holds iff CI DI
• An interpretation, I, for complex classes is
  defined as follows
• TI ?I and ?I ?
• (C ? D)I CI ? DI , (C ? D)I CI ? DI ,
  (? C)I ?I \ CI
• ? R . C a ? ?I ( ? b ? ?I ) ((a, b) ? RI ?
  b ? CI )
• ? R . C a ? ?I (?b ? ?I ) ((a, b) ? RI ? b
  ? CI )

15
More DL constructors that are beyond ALC

• Number restrictions for roles. Example ? 20
  hasStudent
• Qualified number restrictions for roles. Example
  
• ?6 hasStudent . Graduate
• Nominals (definition by enumeration) CS151,
  CS152, CS253
• Concrete domains (datatypes) hasStudent . (? 20)
• Inverse roles hasChild- ? hasParent
• Transitive roles has Ancestor ? has Ancestor
• Role composition hasParent . hasBrother (uncle)
• Constructors allowed define the specific DL.
  Examples
• S ALC Transitive roles
• R Role constructors
• O Nominals
• I Inverse roles
• Q Qualified number restrictions
• (D) datatypes.
• SROIQ(D) is the DL behind the latest version of
  OWL 2.

16
Questions DLs inference must address

• Does the knowledge base make sense?
• Are there empty classes?
• Are two classes equivalent?
• Does one class subsume another class?
• Are two classes disjoined?
• Is a given individual a member of a specified
  class?
• Find all individuals of a given class.
• The so-called tableaux algorithm (initially
  developed for FOL) is adapted to DLs to guarantee
  that these questions are answered in finite time.
  Tableaux algorithm proves unsatisfiability of a
  KB using the refutation method and transformation
  rules similar to Wangs algorithm (called here
  tableaux extension rules)

17
Tableaux algorithm PL example

• Given an expression in disjunctive normal form,
  construct the tableaux / decision tree where each
  node is a logical formula, each path from the
  root note to a leaf note is a conjunction of
  formulas, and each branch of a path is a
  disjunction of formulas. To build the tree, we
  apply the following rules
• 
  -- ?-Rules (A
  B) gt A, B
• Example
  
  ?(A ? B) gt ?A, ?B
• 
  
  ?(A ? B) gt A, ?B
• (A B C) ? (D ?A) ? (B ?D)
  -- ?-Rules A
• ?-rule
  
  A ? B
• 
  
  B
• (A B C) (D ?A) ? (B ?D)
  ?A
• ?-rule ?-rule
  ?(A B)
  
• 
  
  ?B
• A (D ?A) (B
  ?D)
  ?A
• 
  
  A ? B
• B D
  B
  B
• 
  -- Double
  negation rule ??A gt A
• C ?A
  ?D -- ?True gt False,
  ?False gt True

18
Tableaux Algorithm for ALC

• Step 1 Translate the DL KB to a negation normal
  form (NNF) using the following transformations
• Substitute C ? D by C ? D and D ? C
• Substitute C ? D by ?C ? D
• Apply all possible NNF transformations to complex
  classes
• NNF (? ?C) ? NNF (C)
• NNF (C ? D) ? NNF (C) ? NNF (D)
• NNF (C ? D) ? NNF (C) ? NNF (D)
• NNF (?(C ? D)) ? NNF (?C) ? NNF (?D)
• NNF (?(C ? D)) ? NNF (?C) ? NNF (?D)
• NNF (? R . C) ? (? R) . NNF (C)
• NNF (? R . C) ? (? R) . NNF (C)
• NNF (?? R . C) ? (? R) . NNF (?C)
• NNF (?? R . C) ? (? R) . NNF (?C)

19
Tableaux Algorithm for ALC (contd.)

• Step 2 Apply the following tableaux expansion
  rules to derive new ABox facts until no more
  rules can be applied (in which case we say that
  the tableaux is fully expanded).
• ? -Rule if (C ? D) (a) ? ABox and C(a), D(a)
  ? ABox,
• then ABox Abox ? C(a),
  D(a)
• ? -Rule if (C ? D) (a) ? ABox and C(a), D(a)
  ? ABox ?,
• then ABox ABox ? C(a)
  and ABox ABox ? D(a)
• ?-Rule if ? R . C(a) ? ABox and there is no
  b such that C(b) ? ABox and
• R(a, b) ? ABox
• then Abox Abox ? C(z),
  R(a, z) for a new individual z ? ABox
• ?-Rule if ?R . C(a) ? ABox and R(a, b) ?
  Abox, but C(b) ?ABox
• then Abox Abox ?
  C(b)
• The algorithm returns TRUE if there is a
  clash-free tableaux and FALSE if the tableaux is
  closed. The tableaux is closed if all its paths
  are closed, where a path is closed if a formula
  and its negation occur along that path, or if
  false occurs. To prove X, we must obtain a closed
  tableaux for ?X.

20
Example

• Let TBox Professor ? (Person ? Staff) ?
  (Person ? ?Student)
• Prove Professor ? Person
• Step 1 Negate the conclusion you want to prove.
• ? (Professor ? Person)
• Step 2 Convert all formulas in the NNF
• TBox ?Professor ? (Person ? Staff) ?
  (Person ? ?Student)
• NNF (? (Professor ? Person)) NNF (?
  (?Professor ? Person))
• NNF (Professor) ? NNF (? Person)
  Professor ? ? Person
• Step 3 Apply the tableaux extension rules to
  derive new ABox facts from the TBox extended with
  the negated theorem.

21
Example step 3 (contd.)

• Professor(a) ? ? Person(a) (Consider a
  new individual a for whom you are
• 
  disproving the theorem)
• (2) Apply ?-rule to (1) Professor(a)
  closed path
• Apply ?-rule to (1) ? Person(a)
  closed
  path
• ?Professor(a) ? (Person(a) ? Staff(a)) ?
  (Person(a) ? ?Student(a))
• Apply ?-rule to (4) ?Professor(a)
• (6) Apply ?-rule
  to (4) (Person(a) ? Staff(a)) ? (Person(a) ?
  ?Student(a))
• (7) Apply
  ?-rule to (6) (Person(a) ? Staff(a))
• 
  
• 
  (8) Apply ?-rule to (6)
  (Person(a) ? ?Student(a))
• closed path (9) Apply ?-rule to (7) Person(a)
• (10) Apply ?-rule to (7)
  Staff(a)
• 
  
  (11) Apply ?-rule to (8) Person(a)
• 
  
  (12) Apply ?-rule to (8) ?Student(a)
• No more rules can be applied the tableaux is
  fully expanded and closed. Therefore, the
  original theorem is proved.

22
Advantages and disadvantages of description logics

• Advantages
• The two inference tasks in DLs, subsumption (i.e.
  checking if one category is a subset of another
  based on their descriptions) and classification
  (i.e. checking if an object belongs to a
  category) have polynomial complexity. Some DLs
  may include also consistency checking, i.e.
  whether a category definition is satisfiable.
• DLs have clear semantics, which makes them a good
  KR formalism for applications with large
  declarative component. The Semantic Web language
  OWL is build upon DLs.
• Disadvantages
• Polynomial inference is assured by the
  definitions of categories. If they are small and
  correctly defined, then the inference tasks are
  easy to carry out. But, if categories are hard to
  define in a concise manner, then their
  descriptions may become (exponentially) large.
• Weak inference capabilities (to preserve
  decidability), which cannot handle imprecisely
  specified concepts and not fully enforced
  relations between them.

23
Subsumption example

• Given the following KB
• Woman ? Person ? Female
• Man ? Person ? ?Woman
• Mother ? Woman ? ?hasChild . Person
• Father ? Man ? ?hasChild . Person
• Parent ? Mother ? Father
• Grandmother ? Woman ? ?hasChild .
  Parent
• Prove Grandmother ? Person.
• 1. Negate the goal and add it to the KB,
  i.e. add (Grandmother ? ?Person).
• 2. Translate the KB into the negation
  normal form
• i.) substitute A ? B with A ? B and B ?
  A
• Woman ? Person ? Female
  Person ? Female ? Woman
• Man ? Person ? ?Woman
  Person ? ?Woman ? Man
• Mother ? Woman ? ?hasChild . Person
  Woman ? ?hasChild . Person ? Mother
• Father ? Man ? ?hasChild . Person
  Man ? ?hasChild . Person ? Father
• Parent ? Mother ? Father
  Mother ? Father ? Parent
• Grandmother ? Woman ? ?hasChild.Parent Woman ?
  ?hasChild.Parent ? Grandmother

24
Subsumption example (contd.)

• ii.) substitute A ? B with ?A ? B and then
  apply all possible transformations to
• complex classes to bring negation down
  to atomic classes (see Lecture 7)
• ?Woman ? (Person ? Female)
• ?(Person ? Female) ? Woman ? ?Person ? ?Female ?
  Woman
• ?Man ? (Person ? ?Woman)
• ? (Person ? ?Woman) ? Man ? ?Person ? Female ?
  Man
• ?Mother ? (Woman ? ?hasChild . Person)
• ? (Woman ? ?hasChild . Person) ? Mother ? ?Woman
  ? ?hasChild.Parent ? Mother
• ?Father ? (Man ? ?hasChild . Person)
  
• ? (Man ? ?hasChild . Person) ? Father ? ?Man ?
  ?hasChild.Parent ? Father
• ?Parent ? (Mother ? Father) ? ?Parent ? Mother ?
  Father
• ? (Mother ? Father) ? Parent ? (?Mother ?
  ?Father) ? Parent
• ?Grandmother ? (Woman ? ?hasChild.Parent)
• ?(Woman ? ?hasChild.Parent) ? Grandmother ?
• 
  ?Woman ? ?hasChild.Parent ?
  Grandmother

25
The (beginning of the) proof

• Grandmother(a) ? ?Person(a) An individual a
  for whom we are disproving the
• 
  theorem.
• (2) Apply ?-rule to (1) Grandmother(a)
• (3) Apply ?-rule to (1) ?Person(a)
  closed path
• (4) ?Grandmother ? (Woman ? ?hasChild.Parent)
• (5) Apply ?-rule to (4) ?Grandmother(a)
• (6) Apply ?-rule to (4) Woman
  ? ?hasChild.Parent
• (7) Apply ?-rule to (6)
  Woman(a)
• (8) Apply ?-rule to (6)
  ?hasChild.Parent
• (9) ?Woman ? (Person ? Female)
  closed path
• (10) Apply ?-rule
  to (9) ?Woman(a)
• (11) Apply ?-rule
  to (9) Person ? Female closed path
• (12) Apply ?-rule
  to (11) Person(a)
• (12) Apply ?-rule
  to (11) Female(a)
• (13) ?Person ?
  ?Female ? Woman
  all three paths
• (14)
  Apply ?-rule to (13) ?Person(a)
  are closed
• (14)
  Apply ?-rule to (13) ?Female(a)
• (14)
  Apply ?-rule to (13) Woman(a)

26
Other reasoning tasks

• Note the tableau above is not fully expanded.
  There are more formulas that can be transformed
  by means of ?-rule, ?-rule, ?-rule and ?-rule. To
  prove the original goal, we must show that all
  paths of the fully expanded tableau are closed.
  Although fully mechanical, this process is very
  tedious for a human automated theorem provers
  like Pellet and Fact (both employed in Protégé)
  do a much better job.
• The classification task can be reduced to
  inconsistency checking as well. To prove that
  individual x is a member of class A, we can
  prove that ?A(x) is inconsistent instead. To find
  all members of a class (task called instance
  generation/retrieval), we must show ?A(x) for all
  x.
• To prove that a class A is inconsistent (i.e. has
  no members), we can add A(x) to the KB and show
  that the resulting set is inconsistent (i.e.
  assume that there is a member x ? A and proof a
  contradiction).
• To prove A ? B (class disjointness), we can prove
  that adding A ? B to the KB makes it
  inconsistent.

27
Inconsistency example

• Consider the following KB
• Bear ? Animal ? Omnivore
• Omnivore ? ?eat . Animal
• Panda ? Bear ? Vegetarian
• Vegetarian ? ?eat . ?Animal
• It is easy to see that this KB is inconsistent.
  Assume that there exists an individual x which is
  a member of the class Panda. Then
• Panda(x) ? Bear(x) AND
  Vegetarian(x)
• Vegetarian(x) ? x does not
  eat animals
• Bear(x) ? Animal(x) AND
  Omnivore(x) Contradiction,
• Omnivore(x) ? x eats
  animals therefore
  class
• 
  
  Panda must be empty.
• Note reasoning over defined classes ONLY!