First Order Logic (3)

8/13/2019 First Order Logic (3)

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ITCS 3153

Artificial Intelligence

Lecture 13

First-Order Log ic

Chapter 8


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First-order logic

We saw how propo si t ional logic can create

intel l igent behavior

Bu t propos i t ional log ic is a poor representat ionfor complex envi ronm ents

Why?

First -order log ic is a more exp ressive andpowerfu l representation


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What do we like about propositional

logic?

It is :

Declarative

Relationships between variables are described

A method for propagating relationships Expressive

Can represent partial information using disjunction

Compositional

IfA means foo and B means bar,A ^ Bmeans foo and ba


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What dont we like about

propositional logic?

Lacks expressive power to descr ibe the

envi ronm ent conc isely

Separate rules for every square/square relationship in

Wumpus world


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Natural Language

Engl ish appears to be express ive

Squares adjacent to pits are breezy

Bu t natural language is a medium o f communicat ion, not

a know ledge representat ion

Much of the information and logic conveyed by language is dependent

on context

Information exchange is not well defined

Not compositional (combining sentences may mean something different

It is ambiguous


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But we borrow representational

ideas from natural language

Natural language syn tax

Nouns and noun phrases refer to objects

People, houses, cars

Verbs and verb phrases refer to relationshipsbtw objects

Red, round, nearby, eaten

Some relationships are clearly defined functions where there is only one

output for a given input

Best friend, first thing, plus

We bui ld f i rst order logic around objects and relat ions


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Ontology

a specification of a conceptualization

A description of the objects and relationships that can exist

Propositional logic had only true/false relationships

First-order logic has many more relationships

The ontological commitmentof languages is different

How much can you infer from what you know?

Temporal logicdefines additional ontologicalcommitments because of timing constraints


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Higher-order logic

First-order logic is first because you relate objects (thef i rst-order enti t ies that actual ly exist in the world )

There are 10 chickens chickens.number=10

There are 10 ducks ducks.number=10

You canno t bu i ld relationships between relat ions or

funct ions

There are as many chickens as ducks chickens.number =

ducks.number

the number of objects belonging to a group must be a property of the

group, and not the objects themselves

Cannot represent Leibnizs law: Ifxand yshare all properties,xisy


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Another characterization of a logic

Epistemolog ica l comm itments

The possible states of knowledge permitted with respect to

each fact

In first-order logic, each sentence is a statement that is True, false, or unknown


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Formal structure of first-order logic

Models of f i rst-order logic contain: A set of objects (its domain)

Alice, Alices left arm, Bob, Bobs hat

Relationshipsbetween objects

Represented as tuples

Sibling (Alice, Bob), Sibling (Bob, Alice)

On head (Bob, hat)

Person (Bob), Person (Alice)

Some relationships are functionsif a given object is related to exactl

one object in a certain way

Alice -> Alices left arm


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First-order logic syntax

Constant Symbols

A, B, Bob, Alice, Hat

Predicate Symbols

Is, onHead, hasColor, person

Funct ion Symbols

Mother, leftLeg

Each predicate and func t ion s ymbol has an ar i ty

A constant the fixes the number of arguments


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First-order logic syntax

Names of thin gs are abitrary

Knowledge base adds meaning

Number o f poss ib le domain elements is

unbounded

Number of models is unbounded

Checking enumeration by entailment is impossible


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Syntax

Term

A logical expression that refers to an object

Constants

We could assign names to all objects, like providing aname for every shoe in your closet

Function symbols

Used in place of a constant symbol OnLeftFoot(John))


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Atomic Sentences

Formed by a predicatesym bo l fo l lowed by parenthesizedl ist of terms

Sibling (Alice, Bob)

Married (Father(Alice), Mother(Bob))

An atom ic sentence is true in a given m odel , under a

given interpretat ion , i f the relat ion referred to by the

predicate sym bol ho lds among the objects referred to by

the arguments


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Complex sentences

We can use logical connect ives

~Sibling(LeftLeg(Alice), Bob)

Sibling(Alice, Bob) ^ Sibling (Bob, Alice)


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Quantifiers

A way to express propert ies o f ent i re co l lect ions

of objects

Universal quantification (forall, )

The power of first-order logic

ForallxKing(x) => Person(x)

xis a variable

)()(: xPersonxKingx


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Universal Quantification

Forall x, P

P is true for every object x

Forall x, King(x) => Person(x)

Richard the Lionheart King John

Richards left leg

Johns left leg

The crown


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Universal Quantification

Note that al l of th ese are true Implication is true if premise is false

Using AND instead of implication is overly strong

By assert ing a un iversal ly quant i f ied sentence,you assert a who le l is t of indiv idual impl ications


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Existential Quantification

There exists ,

There exists an x such that Crown(x) ^ OnHead(x, John)

It is true for at leastone object

AND, ^, is the appropriate connective


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Existential Quantification

What if we used impl icat ion as the connect ive?

Implication is true if both premise and conclusion are true or

if premise is false

Richard the Lionheart is not a crown, first assertion is true

and existential is satisfied


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Nested Quantifiers

Bui ld ing m ore complex sentences

Everybody loves somebody

There is someone who is loved by everyone

Use un ique variable names and parenth eses whenappropriate


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Combining

Everyone who dislikes parsnips ==there does not exist someone who likes parsnips

Everyone likes ice cream ==

there is no one who does not like ice cream


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Combining

De Morgans rules apply


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Equality

Two terms refer to the same object

Father (John) = Henry

Richard has at least two brothers

Notice this sentence is not the same


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An Example

A Tel l/Ask interface for a first-order know ledge base

Sentences are added with Tell

These are called assertions

Tell (KB, King(John))

Tell (KB, Forall x: King(x) => Person(x))

Queries are made with Ask

Ask (KB, King(John))

Ask (KB, Person(John))


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An Example

Quantified queries

Ask (KB, exist x: Person(x))

KB should return a list of variable/term pairs that satisfy the

query


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The Wumpus World

More precise axiom s than with pro pos i t ional logic Percept has five values

Time is important

A typical sentence

Percept ([Stench, Breeze, Glitter, None, None], 5)

The actions are terms

Turn(right), Turn(left), Forward, Shoot, Grab, Release

Computing best action with a query

Exist a: BestAction(a, 5)


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The Wumpus World

After executing query, KB responds with variable/term list:{a/Grab}

Then tell the KB the action taken

Raw percept data is easily encoded

Reflexes are easily encoded


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Wumpus World

Defining the environment with[x,y] reference instead ofalternative atomic name

Adjacency between two squares

Location of Wumpus is constant: Home (Wumpus)

Location of agent changes: At (Agent, [ ], t)


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Diagnostic Rules

Rules leading from observed ef fects to hiddencauses

Breezy implies pits

Not breezy implies no pits

Combining


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Causal Rules

Some hidd en property causes percepts to begenerated

A pit causes adjacent squares to be breezy

If all squares adjacent to a square a pitless, it will not be

breezy


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Causal Rules

The causal ru les fo rmulate a model

Knowledge of how the environment operates

Model can be very useful and important and replace

straightforward diagnostic approaches


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Conclusion

If the axiom s correct ly and completely descr ibe the waythe wor ld works and the way percepts are produced,

then any com plete logical inference procedure w i l l infer

the strongest possib le descr ipt ion of the wo r ld state

given the avai lable percepts

The agent designer can focus on gett ing the knowledge

r ight wi thout worry ing about the processes of deduct ion

First Order Logic (3)

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