[PDF] - Prolog Reading Sethi Chapter Overview Predicate Calculus PDF Document

SLIDE 1 Prolog Reading Sethi Chapter

Overview
Predicate

Calculus

Substitution

and Unication

Intro

duction to Prolog

Prolog

Inference Rules

Programming

in Prolog

Recursion
List

Pro cessing

Arithmetic
Highero

rder p rogramming

Miscellaneous

functions

Conclusion

SLIDE 2 Prolog Programming in Log ic

Idea

emerged in ea rly s most w

rk

done at Univ

f

Edinburgh

Based
n

a subset

f

rsto rder logic

F

eed it theo rems and p

se

queries system do es the rest

main

uses

Originally
mainly

fo r natural language p ro cessing

No

w nding uses in database systems and even rapid p rotot yping systems

f

industrial soft w a re

P
pula

r languages Prolog XSB LDL Co ral Datalog SQL

SLIDE 3 Logic Programming F ramew

rk

Answ er Query Programming En vironmen t Is qX

XN

true Y esNo

variable

bindings Kno wledge Base Pro

f

Pro cedure F acts

R

ules

SLIDE 4 Decla rative Languages In its purest fo rm Logic p rogramming is an example

f

decla rative p rogramming P

pula

r in database systems and a rticial in telligence Decla rative sp ecications Sp ecify what y

u

w ant but not ho w to compute it Example Find X and Y such that

X
Y
X
Y
A

metho d p rogram fo r solving these is ho w to get values fo r X and Y But all w e gave w as a sp ecication

r

decla ration

f

what w e w ant Hence the name

SLIDE 5 Examples

Retrieve

the telephone numb er

f

the p er son whose name is T

m

Smith easy

Retrieve

the telephone numb er

f

the p er son whose address is

Black

St ha rd

Retrieve

the name

f

the p erson whose telephone numb er is

ha

rd Each command sp ecies what w e w ant but not ho w to get the answ er A database sys tem w

uld

use a dierent algo rithm fo r each

f

these cases Can also return multiple answ ers

Retrieve

the names

f

all p eople who live

n

Oak St

SLIDE 6 Algo rithm

Logic
Control
Users

sp ecify logic

what

the algo rithm do es

using

logical rules and facts

Control

ho w the algo rithm is to b e im plemented

is

built into Prolog ie Sea rch p ro cedures a re built into Prolog They apply logical rules in a pa rticula r

rder

to answ er user questions Example P if Q

and

Q

and
and

Q k can b e read as to deduce P deduce Q

deduce

Q

deduce

Q k Users sp ecify what they w ant using classical rsto rder logic p redicate calculus

SLIDE 7 Classical FirstOrder Logic

The

simplest kind

f

logical statement is an atomic fo rmula

eg
mantom

tom is a man womanmary ma ry is a w

man

marriedtommary tom and ma ry a re ma rried

Mo

re complex fo rmulas can b e built up using logical connectives

X
X
eg
smarttom
dumbtom

smarttom

talltom
dumbtom

X marriedtom X tom is ma rried to something X lovestom X tom loves everything X marriedtom X

femaleX
humanX

tom is ma rried to a human female

SLIDE 8 Logical Implication richtom

smarttom

This implies that if tom is sma rt then he must b e rich So w e

ften

write this as richtom

smarttom

In general P

Q

and Q

P

a re abb revia tions fo r P

Q

F

r

example X personX

smartX
richX

every p erson who is sma rt is also rich X motherjohnX john has a mother X motherjohnX

Y

motherjohnY

Y
X

john has exactly

ne

mother

SLIDE 9 Ho rn Rules Logic p rogramming is based

n

fo rmulas called Ho rn rules

These

have the fo rm x

x

k A

B
B
B

j

where

k

j
F
r

example XY AX

BXY
CY

X AX

BX

X AXd

BXe

Acd

Bde

X AX X AXd Acd Note that atomic fo rmulas a re also Ho rn rules

ften

called facts

A

set

f

Ho rn rules is called a Logic Program

SLIDE 10 Logical Inference with Ho rn Rules Logic Programming is based

n

a simple idea F rom rules and facts derive mo re facts Example

Given

the facts A B C D and these rules

E
A
B
F
C
D
G
E
F

F rom

derive

E F rom

derive

F F rom

derive

G Example

Given

these facts manplato plato is a man mansocrates so crates is a man and this rule X manX

mortalX

all men a re mo rtal derive mortalplato mortalsocrates

SLIDE 11 Recursive Inference Example Given X mortalX

mortalson
fX

mortalplato Derive mortalson

fplato

using X

plato

mortalson

fson
fplato

using X

son
fplato

mortalson

fson
fson
fplato

using X

son
fson
fplato
This

kind

f

inference simulates recursive p ro grams as w e shall see

SLIDE 12 Logic Programming Ho rn rules co rresp

nd

to p rograms and a fo rm

f

Ho rn inference co rresp

nds

to execution F

r

example consider the follo wing rule XY pX

qXY
rXY
sXY

Later w e shall see that this rule can b e inter p reted as a p rogram where p is the p rogram name qrs a re sub routine names X is a pa rameter

f

the p rogram and Y is a lo cal va riable

SLIDE 13 NonHo rn F

rmulas

The follo wing fo rmulas a re not Ho rn A

B

A

B

A

B
C

X AX

BX

A

B
C

X flagX

redX
whiteX

every ag is red

r

white X Y wifeX

marriedXY

every wife is ma rried to someone

SLIDE 14 NonHo rn Inference Inference with nonHo rn fo rmulas is mo re com plex than with Ho rn rules alone Example A

B

A

C

B

C

nonHo rn W e can infer A but must do case analysis either B

r

C is true if B then A if C then A Therefore A is true in all cases NonHo rn fo rmulas do not co rresp

nd

to p ro grams and nonHo rn inference do es not co r resp

nd

to execution

SLIDE 15 Logical Equivalence Many nonHo rn fo rmulas can b e put into Ho rn fo rm using t w

metho

ds

logical

equivalence

sk
lemization

Example

Logical

Equivalance A

B
A
B
A
B
B
A

Ho rn

B
A

Logical La ws

A
A
A
B
A
B

A

B
C
A
B
A
C

A

B
A
B

Example

Logical

Equivalance A

B
C
A
B
C
A
B
C
A
B
A
C

Ho rn

A
B
A
C

SLIDE 16 Example

Logical

Equivalence A

B
C
A
B
C
A
B
C
A
B
C
A
B
C
A
B
A
C

non Ho rn

A
B
A
C

In general rules

f

the follo wing fo rm cannot b e converted into Ho rn fo rm xA

A

n

B
B

m

F
r

example A

B
C
D

A

B
C

A

B

X AX

Bx
CX
DX

ie if it is p

ssible

to infer a nontrivial dis junction from a set

f

fo rmulas then the set is inherently nonHo rn A rule lik e p

q
q

infers a trivial dis junction since the rule is a logical tautology

Such

rules can simply b e igno red

SLIDE 17 Sk

lemization

NonHo rn fo rmulas lik e x Ax can b e con verted to Ho rn fo rm Example

Replace
X

motherjohnX nonHo rn with

motherjohnm

Ho rn Here m is a new constant symb

l

called a sk

lem

constant

that

stands fo r the unkno wn mother

f

john Note

but

they sa y almost the same thing In pa rticula r

can

sometimes b e replaced b y

during

inference as w e shall see

SLIDE 18 Sk

lemization

Contd Example

A

nonHo rn fo rmula

X

personX

Y

motherXY every p erson has a mother Let mx stand fo r the unkno wn mother

f

X Then w e can replace

b

y a Ho rn rule

X

personX

motherXmX

mX is called a sk

lem

function It is an a rticial name w e have created eg

mmary

denotes the mother

f

ma ry

mtom

denotes the mother

f

tom mjfk denotes the mother

f

jfk So w e

nly

need X b ecause w e dont have a name fo r X By creating a rtical names sk

lem

symb

ls

w e can eliminate many s and convert many fo rmulas to Ho rn rules which Prolog can then use Sk

lemization

is a technical device fo r doing inference

SLIDE 19 Inference with Sk

lemization
X

manX

personX

every man is a p erson

X

Y personX

motherXY

every p erson has a mothernon Ho rn

XY

motherXY

lovesYX

every mother loves her children

manplato

plato is a man Question

Y

lovesYplato do es someone love plato Step

Sk
lemize
to

get a Ho rn rule

X

personX

motherX

mX Step

Use

Ho rn inference personplato from

motherplatomplato

from

lovesmplatoplato

from

Thus
Y

lovesYplato ie Y

mplato

So answ er is YES

SLIDE 20 Sk

lem

Dep endencies

X

Y pXY sk

lemizes

to Y paY where a is a sk

lem

constant

Y

X pXY sk

lemizes

to Y pbYY where b is a sk

lem

function ie in

X

dep ends

n

Y But in

X

is indep endent

f

Y

X

Y Z qXYZ sk

lemizes

to X Y qXYcXY where c is a sk

lem

function

f

b

th

X and Y ie in

Z

dep ends

n

b

th

X and Y

SLIDE 21 Sk

lem

Dep endencies

Concrete

Examples X Y lovesXY someone loves everyb

dy
Y

lovespY p loves everyb

dy

X Y motherXY every

ne

has a mother

X

motherXmX mX is the mother

f

X X Y Z

wnsXY
documentZXY

if X

wns

Y then there is a do cument Z sa ying that X

wns

Y

X

Y

wnsXY
documentdXYXY

dXY is a do cument sa ying that X

wns

Y