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DCP 1172Introduction to Artificial Intelligence

Lecture notes for Ch.8 [AIAM-2nd Ed.]

First-order Logic (FOL)Chang-Sheng Chen

DCP1172, Ch.8 2

Midterm format

• Date: 11/19/2004 from 10:15 am – 11:55 am • Location: EC016• Credits: 25% of overall grade• Approx. 5 problems, several questions in each.• Material: everything so far (or, up to Sec. 8.3).• No books (or other material) are allowed.• Duration will be 100 minutes.

DCP1172, Ch.8 3

Question 1 – problems with short answer

• Problem 1-1: Forward search vs. backward search. Please explain with your example.

• Problem 1-2: Translate the following English sentences to FOL. …

• Problem 1-3: Please explain the phrase " Truth depends on interpretation.” by showing with an example.…

DCP1172, Ch.8 4

Question Type 2: multiple-choices

• Please choose the topics that are candidates for the midterm exam ?

(a) Informed search(b) Logical programming(c) Constraint Searching Program(d) Decision making under uncertainty(e) Probability Reasoning System

Answer: (a), (c)

DCP1172, Ch.8 5

Question Type 3- Matching

• Here are a few things that are related to NCTU and NTHU, please match the correct ones to each of them.(a) http://www.nctu.edu.tw(b) http://www.ust.edu.tw(c) IPv4 address range, 140.114.*.*(d) The current administrator of TANet Hsinchu-Miaoli

regional center (e) The university with the largest number of students in

Taiwan

Answer: <NCTU, {(a), (b), (d)} >, <NTHU, {(b),(c)} >

DCP1172, Ch.8 6

Question type 4 – calculation or inference(with much more detailed involved)

• Problem 4-1: Please find an optimal solution to go from NCTU to the Hsinchu Railway Station. Here is the related information …• the most economic way (e.g., less money)• the most efficient way (e.g., less time)...

DCP1172, Ch.8 7

Last time: Logic and Reasoning

• Knowledge Base (KB): contains a set of sentences expressed using a knowledge representation language• TELL: operator to add a sentence to the KB

• ASK: operator to query the KB • Logics are KRLs where conclusions can be drawn

• Syntax• Semantics

• Entailment: KB |= α iff a is true in all worlds where KB is true

• Inference: KB |–i α, sentence α can be derived from KB using procedure i

• Sound: whenever KB |–i α then KB |= α is true

• Complete: whenever KB |= a then KB |–i a

DCP1172, Ch.8 8

Last Time: Syntax of propositional logic

DCP1172, Ch.8 9

Last Time: Semantics of Propositional logic

DCP1172, Ch.8 10

Last Time: Inference rules for propositional logic

DCP1172, Ch.8 11

This time

• First-order logic• Syntax• Semantics• Wumpus world example

• Ontology (ont = ‘to be’; logica = ‘word’):

• kinds of things one can talk about in the language

DCP1172, Ch.8 12

Why first-order logic?

• We saw that propositional logic is limited because it only makes the ontological commitment that the world consists of facts.

• Difficult to represent even simple worlds like the Wumpus world; e.g., “don’t go forward if the Wumpus is in front of you” takes 64 rules

DCP1172, Ch.8 13

First-order logic (FOL)

• Ontological commitments:• Objects: wheel, door, body, engine, seat, car,

passenger, driver• Relations: Inside(car, passenger),

Beside(driver, passenger)• Functions: ColorOf(car)• Properties: Color(car), IsOpen(door),

IsOn(engine)

• Functions are relations with single value for each object

DCP1172, Ch.8 14

Semantics

there is a correspondence between • functions, which return values•predicates, which are true or false

Function: father_of(Mary) = BillPredicate: father_of(Mary, Bill)

DCP1172, Ch.8 15

Examples:

• “ One plus two equals three”Objects: one, two, three, one plus twoRelations: equalsProperties: --Functions: plus

(“one plus two” is the name of the object obtained by applying function plus to one and two; three is another name for this object)

• “Squares neighboring the Wumpus are smelly”Objects: Wumpus, squareRelations: neighboringProperties: smellyFunctions: --

DCP1172, Ch.8 16

FOL: Syntax of basic elements

• Constant symbols: 1, 5, A, B, Alex, Manos, …

• Predicate symbols: >, Friend, Student, Colleague, …

• Function symbols: +, sqrt, SchoolOf, TeacherOf, ClassOf, …

• Variables: x, y, z, next, first, last, …

• Connectives: , , ,

• Quantifiers: ,

• Equality: =

DCP1172, Ch.8 17

Syntax of Predicate Logic

• Symbol set• constants• Boolean connectives• variables• functions• predicates (relations)• quantifiers

DCP1172, Ch.8 18

Syntax of Predicate Logic

• Terms: a reference to an object• variables, • constants, • functional expressions (can be arguments to

predicates)

• Examples: • first([a,b,c]), sq_root(9), sq_root(n), tail([a,b,c])

DCP1172, Ch.8 19

Syntax of Predicate Logic

• Sentences: make claims about objects• Well-formed formulas, (wffs)

• Atomic Sentences (predicate expressions):• Loves(John,Mary), Brother(John,Ted)

• Complex Sentences (Atomic Sentences connected by booleans):• ¬ Loves(John,Mary)• Brother(John,Ted) ^ Brother(Ted,John)• Mother(Alice, John) ⇒ Mother(Alice,Ted)

DCP1172, Ch.8 20

Examples of Terms: Constants, Variables and Functions

• Constants: object constants refer to individuals• Alan, Sam, R225, R216

• Variables• PersonX, PersonY, RoomS, RoomT

• Functions• father_of(PersonX)• product_of(Number1,Number2)

DCP1172, Ch.8 21

Examples of Predicates and Quantifiers

• Predicates• In(Alan,R225)• PartOf(R225,BuildingEC3)• FatherOf(PersonX,PersonY)

• Quantifiers• All dogs are mammals.• Some birds can’t fly.• 3 birds can’t fly.

DCP1172, Ch.8 22

Semantics

• Referring to individuals• Jackie • son-of(Jackie), Sam

• Referring to states of the world• person(Jackie), female(Jackie)• mother(Sam, Jackie)

DCP1172, Ch.8 23

FOL: Atomic sentences

AtomicSentence Predicate(Term, …) | Term = Term

Term Function(Term, …) | Constant | Variable

• Examples: • SchoolOf(Manos)• Colleague(TeacherOf(Alex),

TeacherOf(Manos))• >((+ x y), x)

DCP1172, Ch.8 24

FOL: Complex sentences

Sentence AtomicSentence | Sentence Connective Sentence| Quantifier Variable, … Sentence| Sentence| (Sentence)

• Examples:

• S1 S2, S1 S2, (S1 S2) S3, S1 S2, S1 S3

• Colleague(Paolo, Maja) Colleague(Maja, Paolo) Student(Alex, Paolo) Teacher(Paolo, Alex)

DCP1172, Ch.8 25

Semantics of atomic sentences

• Sentences in FOL are interpreted with respect to a model

• Model contains objects and relations among them

• Terms: refer to objects (e.g., Door, Alex, StudentOf(Paolo))

• Constant symbols: refer to objects

• Predicate symbols: refer to relations

• Function symbols: refer to functional Relations

• An atomic sentence predicate(term1, …, termn) is true iff the relation referred to by predicate holds between the objects referred to by term1, …, termn

DCP1172, Ch.8 26

Example model

• Objects: John, James, Marry, Alex, Dan, Joe, Anne, Rich

• Relation: sets of tuples of objects{<John, James>, <Marry, Alex>, <Marry, James>, …}{<Dan, Joe>, <Anne, Marry>, <Marry, Joe>, …}

• E.g.: Parent relation -- {<John, James>, <Marry, Alex>, <Marry, James>}

then Parent(John, James) is trueParent(John, Marry) is false

DCP1172, Ch.8 27

Quantifiers

• Expressing sentences about collections of objects without enumeration (naming individuals)

• E.g., All Trojans are clever

Someone in the class is sleeping

• Universal quantification (for all):

• Existential quantification (three exists):

DCP1172, Ch.8 28

Universal quantification (for all):

<variables> <sentence>

• “Every one in the DCP1172 class is smart”: x In(DCP1172, x) Smart(x)

P corresponds to the conjunction of instantiations of PIn(DCP1172, Manos) Smart(Manos) In(DCP1172, Dan) Smart(Dan) …In(DCP1172, Clinton) Smart(Clinton)

DCP1172, Ch.8 29

Universal quantification (for all):

is a natural connective to use with

• Common mistake: to use in conjunction with e.g: x In(DCP1172, x) Smart(x)means “every one is in DCP1172 and everyone is smart”

DCP1172, Ch.8 30

Existential quantification (there exists):

<variables> <sentence>

• “Someone in the dcp1172 class is smart”: x In(dcp1172, x) Smart(x)

P corresponds to the disjunction of instantiations of PIn(dcp1172, Manos) Smart(Manos) In(dcp1172, Dan) Smart(Dan) …In(dcp1172, Clinton) Smart(Clinton)

DCP1172, Ch.8 31

Existential quantification (there exists):

is a natural connective to use with

• Common mistake: to use in conjunction with e.g: x In(dcp1172, x) Smart(x)is true if there is anyone that is not in dcp1172!(remember, false true is valid).

DCP1172, Ch.8 32

Properties of quantifiers

Proof?

Not all by one person but each one at least by one

DCP1172, Ch.8 33

Proof

• In general we want to prove:

x P(x) <=> ¬ x ¬ P(x)

x P(x) = ¬(¬( x P(x))) = ¬(¬(P(x1) ^ P(x2) ^ … ^ P(xn)) ) = ¬(¬P(x1) v ¬P(x2) v … v ¬P(xn)) )

x ¬P(x) = ¬P(x1) v ¬P(x2) v … v ¬P(xn)

¬ x ¬P(x) = ¬(¬P(x1) v ¬P(x2) v … v ¬P(xn))

DCP1172, Ch.8 34

Example sentences

• Brothers are siblings x, y Brother(x, y) Sibling(x, y)

• Sibling is transitive x, y, z Sibling(x, y) Sibling(y, z) Sibling(x, z)

• One’s mother is one’s sibling’s mother m, c,d Mother(m, c) Sibling(c, d) Mother(m, d)

• A first cousin is a child of a parent’s sibling c, d FirstCousin(c, d)

p, ps Parent(p, d) Sibling(p, ps) Parent(ps, c)

DCP1172, Ch.8 35

Example sentences

• One’s mother is one’s sibling’s mother m, c,d Mother(m, c) Sibling(c, d)

Mother(m, d) c,d m Mother(m, c) Sibling(c, d)

Mother(m, d)

c d

m

Mother of

sibling

DCP1172, Ch.8 36

Translating English to FOL

• Every gardener likes the sun. x gardener(x) => likes(x,Sun)

• You can fool some of the people all of the time. x t (person(x) ^ time(t)) => can-fool(x,t)

DCP1172, Ch.8 37

Translating English to FOL

• You can fool all of the people some of the time. x t (person(x) ^ time(t) => can-fool(x,t)

• All purple mushrooms are poisonous. x (mushroom(x) ^ purple(x)) => poisonous(x)

DCP1172, Ch.8 38

Translating English to FOL…

• No purple mushroom is poisonous.

¬( x) purple(x) ^ mushroom(x) ^ poisonous(x)

or, equivalently,

( x) (mushroom(x) ^ purple(x)) => ¬poisonous(x)

DCP1172, Ch.8 39

Translating English to FOL…

• There are exactly two purple mushrooms.

( x)( y) mushroom(x) ^ purple(x) ^ mushroom(y) ^ purple(y) ^ ¬(x=y) ^ ( z) (mushroom(z) ^ purple(z)) => ((x=z) v (y=z))

• Deb is not tall.

¬tall(Deb)

DCP1172, Ch.8 40

Translating English to FOL…

• X is above Y if X is on directly on top of Y or else there is a pile of one or more other objects directly on top of one another starting with X and ending with Y.

( x)( y) above(x,y) <=> (on(x,y) v ( z) (on(x,z) ^ above(z,y)))

DCP1172, Ch.8 41

Equality

DCP1172, Ch.8 42

Higher-order logic?

• First-order logic allows us to quantify over objects (= the first-order entities that exist in the world).

• Higher-order logic also allows quantification over relations and functions.e.g., “two objects are equal iff all properties applied to them are equivalent”:

x,y (x=y) ( p, p(x) p(y))

• Higher-order logics are more expressive than first-order; however, so far we have little understanding on how to effectively reason with sentences in higher-order logic.

DCP1172, Ch.8 43

Logical agents for the Wumpus world

1. TELL KB what was perceivedUses a KRL to insert new sentences, representations of facts, into KB

2. ASK KB what to do.Uses logical reasoning to examine actions and select best.

Remember: generic knowledge-based agent:

DCP1172, Ch.8 44

Using the FOL Knowledge Base

Set of solutions

DCP1172, Ch.8 45

Wumpus world, FOL Knowledge Base

DCP1172, Ch.8 46

Deducing hidden properties

DCP1172, Ch.8 47

Situation calculus

DCP1172, Ch.8 48

Describing actions

May result in too many frame axioms

DCP1172, Ch.8 49

Describing actions (cont’d)

DCP1172, Ch.8 50

Planning

DCP1172, Ch.8 51

Generating action sequences

[ ] = empty plan

Recursively continue until it gets to empty plan [ ]

DCP1172, Ch.8 52

Summary