artificial intelligence – lecture 8130.243.105.49/~mbl/ai/lectures.2010/ai-8.pdf · 2011. 12....

Artificial Intelligence – Lecture 8

2

● KnowledgeRepresentation

● Semantic Networks

● Prop. Logic

Lecture plan

• AI in general (ch. 1)

• Search based AI (ch. 4)

• search, games, planning, optimization

• Agents (ch. 8)

• applied AI techniques in robots, software agents, ...

• Knowledge representation (ch. 2)

• semantic networks, frames, logic, resolution

• Expert systems (ch. 3)

• forward/backward chaining, uncertainty, baysian networks

• Natural language processing (ch. 5)

• Machine learning (ch. 7)

• version spaces, decision trees, classification, neural networks

http://www.oru.se/start____2232.aspx?filter=0

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● Prop. Logic

Structured knowledge representation

• How can we represent knowledge as data in our programs?• Semantic networks

• Frames

• Propositional logic

• Predicate logic

• ...


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● Prop. Logic

Semantic networks

• Directed graph where: • nodes correspond to classes, instances or values.

• edges correspond to instance, subclass or attributes.

• Object oriented programming build on semantic networks!


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● Prop. Logic

Common semantic relations

• No “official” standard exists.

• Common relations:• instance vs. isa: Often these two are differentiated

• haspart


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● Prop. Logic

Reasoning on Semantic networks

• Assumption: general attributes (eg. livesin, has, ...) is closed under inheritance attributes (eg. isa, subclass, ...)

• Examples: • cat has vertebra since: cat isa mammal and mammal has vertebra

• mathias has vertebra since: ...

• There exists a path from X to Y consisting of zero or more isa steps and the has step as the last step.


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● Prop. Logic

Semantic networks in python

• List of nodes

Nodes=[“Mathias”, “Bobby”, “Human”, “Cat”, ...]

• List of edges aka. “relations”

Edges=[(“isa”,”cat”,”mammal”),(“has”,”mammal”,”vertebra”) ... ]

• Propagating relations

def reason():

Facts, NewFacts, i = Edges, list(Edges), 1

while True:

for E in Facts:

if E[0] in Inheritable:

R,A,B = E

for N in Nodes:

if ("isa",N,A) in Facts and (R,N,B) not in Facts:

NewFacts.append((R,N,B))

print("%d: %s %s %s"%(i,N,R,B))

if Facts == NewFacts:

break;

Facts=NewFacts ; i=i+1


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● Prop. Logic

Semantic networks in python

>>> reason()

1: cat has vertebra

1: human has vertebra

1: whale has vertebra

1: Bobby has fur

1: Bijou has fur

2: Bobby has vertebra

2: Bijou has vertebra

2: Mathias has vertebra


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● Prop. Logic

Nonbinary relations in semantic networks

• Nonbinary relations: convert the relation into an object• Called reification in logics

• eg. “Mathias gave cat food to Bobby”


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● Prop. Logic

Nonbinary relations in semantic networks

• Reasoning over reified relations, example rule:• If [object,X,R1] and [isa,X,Y] then:

• create new node R2

• add [object,Y,R2]

• for each [P,W,R1] where W != X and W != Y: add [P,W,R2]

• Same isa path(s), shows that Mathias gave food to Bobby, or to a cat, or that bobby was given food by a human.


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● Prop. Logic

Semantic networks

• Semantic networks allows us to represent relationships between entities

• Semantic networks allows us to propagate relationships between entities (reason)


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● Prop. Logic

Inference by association

• What is the common factor between sk1047 and rusty?• Both haspart wing, and both cando fly

• If two entities X,Y have the same relation R to an entity Z, ie. if X R Z and Y R Z then they are associated.


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● Prop. Logic

Default values

• Inheritance of properites through isa can be contradicted by direct links.

• Only treat inheritance as default values


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● Prop. Logic

Wordnet

• Lexical database over English language• Nouns, verbs, adjectives and adverbs

• Grouped into cognitive synonyms (synsets)

• Relations (eg. hyponym's and hypernym's) between words

Synsets

HyponymsHypernyms


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● Prop. Logic

Wordnet

• Wordnet is a semantic network

• Freely available, download online

• 150.000 words organised in 115.000 synsets (meanings)

• Relations

• hypernyms:

• Nouns: Y is a hypernym of X if every X is a (kind of) Y

• Verbs: the verb Y is a hypernym of the verb X if the activity X is a kind of Y (cf. troponym)

• hyponyms: Y is a hyponym of X if every Y is a (kind of) X

• coordinate terms: Y is a coordinate term of X if X and Y share a hypernym

• holonym: Y is a holonym of X if X is a part of Y

• meronym: Y is a meronym of X if Y is a part of X

• entailment: the verb Y is entailed by X if by doing X you must be doing Y

• adjectives: related nouns, similar to, participle of verb

• adverbs: root adjectives

http://en.wikipedia.org/wiki/Hypernym

http://en.wikipedia.org/wiki/Hyponym

http://en.wikipedia.org/wiki/Holonymy

http://en.wikipedia.org/wiki/Meronymy


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● Prop. Logic

Frames

• More advanced than semantic networks

• A frame is a collection of attributes (slots) and values that describe some entity

• Constraints on attributes

• Procedural attachments: Associated function to slots

• Generic frames (eg. classes) vs. instances (eg. objects)

Team;

* hasmember: Human

Hockeyteam;

isa: Team

Largeteam;

isa: Team

(> 10) hasmember: Human

LHC;

instanceof: {Largeteam, Hockeyteam}

hasmember: { .... }


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● Prop. Logic

Frames : inheritance

• Proper inheritance• Slots and values inherited by subclass and instances

• Inheritance with exceptions• Overriding values given from superclass

• Multiple inheritance• Inheriting from multiple parents

• Problem with order of inheritance


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● Prop. Logic

Frames : example

Animal:

Flies: False

Alive: True

Mammal:

isa: Animal

Legs: 4

Cats:

isa: Mammal

Bird:

isa: Animal

Flies: True

Wings: True

Penguin:

isa: Birds

Flies: False

Bill:

instanceof: {Cat}

Friends: {Opus}

Opus:

instanceof: {Penguin}

Friends: {Bill}


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● Prop. Logic

Frames : metaslots

• Frames that describe other slots

• Example

Spotty:

instanceof: dog

owner: mathias

Owner:

value: person

inverse: owns

Conclusions:

person(mathias)

owns(mathias,spotty)


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● Prop. Logic

Frames : Procedural attachments

• Functions attached to metaslots• trigger when adding/removing/replacing slots

• trigger when needing (but not having) a slot

• ...


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● Prop. Logic

Cyc

• Cyc

• Large AI project to create “complete” database of common sense knowledge

• Goal: enable human like reasoning

• Technologies

• Representation language

• Knowledge base

• Inference mechanism

• Natural language processing

• Developer tools


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● Prop. Logic

OpenCyc

• OpenCyc• Free subset of knowledge base

• Free subset of tools, inference mechanism ...

• Uses same representational (CycL) language

• Links with WordNet• Which Cyc concept correspond to which word?

• Links with Wikipedia• Link concepts with articles, browse Wikipedia with OpenCyc

tools (find common concepts etc)

• ...


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● Prop. Logic

Lecture plan

• AI in general (ch. 1)

• Search based AI (ch. 4)

• search, games, planning, optimization

• Agents (ch. 8)

• applied AI techniques in robots, software agents, ...

• Knowledge representation (ch. 2)

• semantic networks, frames, logic, resolution

• Expert systems (ch. 3)

• forward/backward chaining, uncertainty, baysian networks

• Natural language processing (ch. 5)

• Machine learning (ch. 7)

• version spaces, decision trees, classification, neural networks


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● Prop. Logic

Logic

• What is logic?• The study of inference – how new assertions can be

produced from already established ones

• Requirements• Representation (syntax) and interpretation (semantics)

• Basic axioms (something to start reasoning with)• cf. René Descartes (cogito ergo sum)

• An inference mechanism• Soundness (truthpreserving)

• Complete


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● Prop. Logic

Logic – the AI dream of the 60's

• Logic allows us to express everything “formally”

• Logic also allows us to prove theorems based on given information• Can we exploit this to build the ultimate automated

reasoning systems?

• Different logics allow different representation and inference mechanisms• Reasoning about propositions

• Reasonings about predicates, quantifiers, ...

• Representing time, and changes over time

• ...


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● Prop. Logic

Example:

The following knowledge is given :

1. Marcus was a man.

2. Marcus was a Pompeian.

3. All Pompeians were Romans.

4. Caesar was a ruler.

5. All Romans were either loyal to Caesar or hated him.

6. Everyone is loyal to someone.

7. People only try to assassinate rulers to whom they are not loyal.

8. Marcus tried to assassinate Caesar.

Can we automatically answer the following questions? Was Marcus loyal to Caesar?

Did Marcus hate Caesar?


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● Prop. Logic

Conversion to the First Order Logic:

Representation of facts:

1. Marcus was a man.man(Marcus)

2. Marcus was a Pompeian.Pompeian(Marcus)

4. Caesar was a ruler.ruler(Caesar)

8. Marcus tried to assassinate Caesar.try_assassinate(Marcus, Caesar)


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● Prop. Logic

5. All Romans were either loyal to Caesar or hated him.

Conversion to the First Order Logic (2):

General representation (representation of rules):

3. All Pompeians were Romans.∀∀x Pompeian(x)x Pompeian(x) →→ Roman(x) Roman(x)

6. Everyone is loyal to someone.∀∀xx ∃∃yy loyal_to(x,y)loyal_to(x,y)

7. People only try to assassinate rulers to whom they are not loyal.

∀∀xx∀∀y person(x)y person(x) ∧∧ ruler(y)ruler(y) ∧∧ try_assassinate(x,y)try_assassinate(x,y) →→ ~loyal_to(x,y)~loyal_to(x,y)

∀∀x Roman(x) x Roman(x) →→ loyal_to(x,Caesar) loyal_to(x,Caesar) ∨∨ hates(x,Caesar)hates(x,Caesar)


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● Prop. Logic

Prove that he did:Prove that he did:

The “theorem” ?

Was Marcus loyal to Caesar?Was Marcus loyal to Caesar?

Did Marcus hate Caesar?Did Marcus hate Caesar?

hates(Marcus,Caesar)hates(Marcus,Caesar)

Try, for example, to prove that he was not Try, for example, to prove that he was not :: ~loyal_to(Marcus,Caesar)~loyal_to(Marcus,Caesar)


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● Prop. Logic

Logic

• Interpretation

• Truth assignments to propositions “a possible world”

• Not neccessarily the world

• Valid sentence

• True in every interpretation

• Satisfiable sentence

• True in some interpretation

• Unsatisfiable sentence

• True in no interpretation


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● Prop. Logic

Propositional logic (sv. satslogik)

• Syntax


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● Prop. Logic

Propositional logic: Well formed formulas


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● Prop. Logic

Propositional logic: interpretation

• Interpretation• An assignment of true / false to each propositional

variable

• Indirectly gives true / false to all other formulas


34



● Prop. Logic

Interpretation:

pp qq ~p~p p p ∧∧ q q p p ∨∨ q q p p →→ q q p p ↔↔ q qTT TTTT FFFF TTFF FF

FF TT TT TT TTFF FF TT FF FFTT FF TT TT FFTT FF FF TT TT

Truth tableTruth table

• A function that assigns truth value to each “atomic” formula• Also provides (indirectly) truth values for each wff


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● Prop. Logic

Model

p p →→ ~ q ~ q

ppq q ∨∨ r r

SSExample:Example:

p p →→ ~q ~q q q ∨∨ r rTT TT TT

pppp qq rrTT FF TT

Model:Model:

• Given a set of formulas S• A model of S, is an interpretation that makes all

formulas of S true


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● Prop. Logic

Logical entailment:

p p →→ ~ q ~ q

ppq q ∨∨ r r

SSExample:Example:

F:F:

r r →→ p ppp qq rrTT FF TT

(the only) Model:(the only) Model:

r r →→ p pTT

• Given a set of formulas S and a formula F:• F is logically entailed by S ( S |= F ), if all models of S

also make F true.


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● Prop. Logic

Propositional logic: Example 1

• Use a truth table to figure out in which interpretations the following formula is true

• Thus, that formula is a valid statement (since it is always true)


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● Prop. Logic

Inference rules for prop. logic


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● Prop. Logic

Propositional logic: Example 2


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● Prop. Logic

Examples

• 1.Given X (Y (¬X ⇒ ⇒ ∧ Z) and Y, prove ¬X

• 2. Given A (B⇒ ∨C) and C ¬A, prove ¬A⇒ ∨B


artificial intelligence – lecture 8130.243.105.49/~mbl/ai/lectures.2010/ai-8.pdf · 2011. 12....

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