The document discusses various methods for representing knowledge, including: - Using categories, objects, and properties to represent knowledge about the world in first-order logic - Representing actions, events, and how objects and properties change over time using the situation calculus - Addressing challenges like the frame and qualification problems by developing successor state axioms that characterize how fluents change with actions

1. 1
Chapter 10
Knowledge Representation

2. 2
KR
• previous chapters: syntax, semantics,
and proof theory of propositional and
first-order logic
• Chapter 10: what content to put into
an agent’s KB
• How to represent knowledge of the
world

3. 3
Natural Kinds
• Some categories have strict definitions
(triangles, squares, etc)
• Natural kinds don’t
• Define a cup (distinguishing it from bowls,
mugs, glasses, etc)
• Bachelor: is the Pope a bachelor?
• But logical treatment can be useful (can
extend with typicality, uncertainty,
fuzziness)

4. 4
Upper Ontologies
• An ontology is similar to a dictionary
but with greater detail and structure
• Ontology: concepts, relations, axioms
that formalize a field of interest
• Upper ontology: only concepts that
are meta, generic, abstract; cover a
broad range of domain areas

5. 5
Anything
AbstractObjects GeneralizedEvents
Sets Numbers RepresentationalObjects Interval Places PhysicalObjects Processes
Categories Sentences Measurements Moments things stuff
times weights animals agents solid liquid gas
Lower concepts are specializations of their parents

6. 6
Categories and Objects
• I want to marry a Swedish woman
– Category of Swedish woman?
– A particular woman who is Swedish?
• Choices for representing categories:
predicates or reified objects
• basketball(b) vs
member(b,basketballs)
• Let’s go with the reified version…

7. 7
Facts about categories and
objects in FOL
• An object is a member of a category
• A category is a subclass of another
category
• All members of a category have some
properties
• Members of a category can be recognized
by some properties
• A category as a whole has some properties
Necessary versus sufficient properties?
Note: simplification of real categories

8. 8
Other Relationships
• disjoint (no members in common)
• exhaustive decomposition of a
category (all members are in at least
one of the sets)
• Partition: disjoint, exhaustive
decomposition

9. 9
Composite Objects
• partof(england,europe)
• All X,Y,Z partof(X,Y) ^ partof(Y,Z) 
partof(X,Z)
• Heavy(bunchOf({apple1,apple2,apple3}))
• Before continuing: inspiration for
creative reification!
• From Through the Looking Glass

10. 10
Measures
• Diameter(basketball12) = inches(9.5)
• All XY member(X,dimestore) ^ sells(X,Y) 
cost(Y) = $(1)
• member(db1,dollarbills)
• member(db2,dollarbills)
• denomination(db1) = $(1)
• denomination(db2) = $(1)
There are multiple dollar bills, but a single
$(1)

11. 11
Ordinal Comparisons
• But often scales are not so precisely
defined
• Often, ordinal comparisons among members
of categories are useful
• member(p1,poems) ^ member(p2,poems) ^
beauty(p1) < beauty(p2)
We don’t have to say p1 has beauty 54.321
Qualitative physics: reasoning about physical
systems without detailed equations and
numerical simulations.

12. 12
Stuff versus Things
• Suppose some ice cream and a cat in
front of you. There is one cat, but no
obvious number of ice-cream things in
front of you.
• A piece of an ice-cream thing is an
ice-cream thing (until you get down to
very low level)
• A piece of a cat is not a cat

13. 13
Stuff versus Things
• Linguistically distinguished, in English
through mass versus count noun phrases
• “a cat”
• “an ice-cream” (you have to coerce this to a
thing, such as an ice-cream bar, or a
variety of ice cream)
• “a sand”, “an energy”
• “some cat” (you have to coerce this to a
substance; eeewww)

14. 14
Actions, Situations, and Events
The Situation Calculus
• The robot is in the kitchen.
– in(robot,kitchen)
• It walks into the living room.
– in(robot,livingRoom)
• Oops…
• in(robot,kitchen,2:02pm)
• in(robot,livingRoom,2:17pm)
• But what if you are not sure when it was?
• We can do something simpler than rely on
time stamps…

15. 15
Situation Calculus Ontology
• Actions: terms, such as “forward”
and “turn(right))”
• Situations: terms; initial situation s0
and all situations that are generating
by applying an action to a situation.
result(a,s) names the situation
resulting when action a is done in
situation s.

16. 16
Situation Calculus Ontology
continued
• Fluents: functions and predicates that
vary from one situation to the next. By
convention, the situation is the last
argument of the fluent.
~holding(robot,gold,s0)
• Atemporal or eternal predicates and
functions do not change from situation to
situation. gold(g1).
lastName(wumpus,smith).
adjacent(livingRoom,kitchen).

17. 17
Sequences of Actions
• Also useful to reason about action
sequences
• All S resultSeq([],S) = S
• All A,Se,S resultSeq([A|Se],S) =
resultSeq(Se,result(A,S))
resultSeq([a,b,a2,a3],so) is
result(a3,result(a2,result(b,result(a,s0)

18. 18
Modified Wumpus World
• Fluent predicates: at(O,X,S) and
holding(O,S)
• Initial situation: at(agent,[1,1],s0) ^
at(g1,[1,2],s0)
• But we want to exclude possibilities
from the initial situation too…

19. 19
Initial KB
• All O,X at(O,X,s0)  [O=agent ^
X = [1,1]) v (O=g1 ^ X = [1,2])]
• All O ~holding(O,s0)
• Eternals:
– gold(g1) ^ adjacent([1,1],[1,2]) ^
adjacent([1,2],[1,1]).

20. 20
Goal: g1 is in [1,1]
At(g1,[1,1],resultSeq(
[go([1,1],[1,2]),grab(g1),go([1,2],[1,1])],
s0)
Or, planning by answering the query:
Exists S at(g1,[1,1],resultSeq(S,s0))
So, what has to go in the KB for such queries to
be answered?...

21. 21
Possibility and Effect Axioms
• Possibility axioms:
– Preconditions  poss(A,S)
• Effect axioms:
– poss(A,S)  changes that result from
that action

22. 22
Axioms for our Wumpus World
• For brevity: we will omit universal
quantifies that range over entire
sentence. S ranges over situations, A
ranges over actions, O over objects
(including agents), G over gold, and
X,Y,Z over locations.

23. 23
Possibility Axioms
• The possibility axioms that an agent
can
– go between adjacent locations,
– grab a piece of gold in the current
location, and
– release gold it is holding

24. 24
Effect Axioms
• If an action is possible, then certain
fluents will hold in the situation that
results from executing the action
– Going from X to Y results in being at Y
– Grabbing the gold results in holding the
gold
– Releasing the gold results in not holding
it

25. 25
Frame Problem
• We run into the frame problem
• Effect axioms say what changes, but
don’t say what stays the same
• A real problem, because (in a non-toy
domain), each action affects only a
tiny fraction of all fluents

26. 26
Frame
• A data structure
• Knowledge about an object or concept
• Natural way for concise
representation of knowledge
• Organizes knowledge into a set of
slots

27. 27
Frame Problem (continued)
• One solution approach is writing explicit
frame axioms, such as:
At(O,X,S) ^ ~(O=agent) ^ ~holding(O,S) 
at(O,X,result(Go(Y,Z),S))
• With F fluent predicates and A actions,
need O(AF) frame axioms
• But if an action has at most E effects,
then need only O(AE).

28. 28
Representational Frame
Problem
• What stays the same?
• A actions, F fluents, and E
effects/action (worst case).
Typically, E << F
• Want O(AE) versus O(AF) solution

29. 29
Solving the Representational
Frame Problem
• Instead of writing the effects of each
action, consider how each fluent predicate
evolves over time
• Successor-state axioms:
• Action is possible 
(fluent is true in result state 
action’s effect made it true v
it was true before and action left it alone)

30. 30
Ramification Problem
• Implicit effects, such as: if an agent
moves from X to Y, then any gold it is
carrying will move too
• For our specific domain, we can solve
this by writing a more general
successor-state axiom for “at”

31. 31
Initial KB (reminder)
• All O,X at(O,S,s0)  [O=agent ^
X = [1,1]) v (O=g1 ^ X = [1,2])]
• All O ~holding(O,s0)
• Eternals:
– gold(g1) ^ adjacent([1,1],[1,2]) ^
adjacent([1,2],[1,1]).

32. 32
Qualification Problem
• Ensuring that all necessary conditions
for an action’s success have been
specified. No complete solution.

33. 33
Inheritance
• If a property is true of a class, it is true
of all subclasses of that class
• If a property is true of a class, it is true
of all objects that are members of that
class
• (If a property is true of a class, it is true
of all objects that are members of
subclasses of that class)
• There are exceptions

34. 34
Semantic Networks
• Graphical aids for visualizing the knowledge
base
• Efficient algorithms for inferring
properties based on category membership
• Often, correspond to a subset of first-
order logic
• Many variants
• All distinguish among individual objects,
categories of objects and relations among
objects

35. 35
Mammals
Persons
Female Male
Mary John
SubsetOf
SubsetOf SubsetOf
MemberOf MemberOf
SisterOf
Legs
2
Legs
1
HasMother

36. 36
Example
• See previous slide
• Specify what edges and nodes mean
• In previous slide, indivs and categories look
the same
• memberOf(indiv,category)
• sisterOf(indiv,indiv)
• subsetOf(category,category)
• hasMother(indiv,indiv)

37. 37
Semantic Networks
• How about
hasMother(persons,femalePersons)?
• Nope: hasMother is a relation between
individuals
• cat1-- label  cat2 means:
• all X X in cat1  [all Y label(X,Y)  Y in
cat2]
(So, this does not say that each person has a
mother)

38. 38
Semantic Networks
• cat – label  value
• All X X in cat  label(X,value)

39. 39
Inheritance
• Inheritance is efficient and
convenient
• Trace paths from individuals to
categories, inheriting properties as
you go
• In Example slide, how many legs does
John have? Most specific (nearest)
information wins

40. 40
Semantic Networks
• In a semantic network, only binary
relations are possible
• A richer representation is possible by
reifying propositions and events
• This forces creation of a rich
ontology of reified concepts; many
current ideas originated in semantic
network systems