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Chapter 10Knowledge Representation

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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

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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)

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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

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AnythingAbstractObjects 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

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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…

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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

propertiesNecessary versus sufficient properties?Note: simplification of real categories

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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

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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

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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)

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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 54.321Qualitative physics: reasoning about

physical systems without detailed equations and numerical simulations.

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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

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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)

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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…

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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.

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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).

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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)

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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…

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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]).

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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?...

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Possibility and Effect Axioms

• Possibility axioms: – Preconditions poss(A,S)

• Effect axioms:– poss(A,S) changes that result from

that action

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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.

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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

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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

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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

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Frame

• A data structure• Knowledge about an object or

concept• Natural way for concise

representation of knowledge• Organizes knowledge into a set of

slots

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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).

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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

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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)

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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”

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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]).

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Qualification Problem

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

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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

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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

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Mammals

Persons

Female Male

Mary John

SubsetOf

SubsetOf SubsetOf

MemberOf MemberOf

SisterOf

Legs

2

Legs

1

HasMother

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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)

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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)

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Semantic Networks

• cat – label value• All X X in cat label(X,value)

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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

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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