chương 3 tri thức và lập luận. nội dung chính chương 3 i.logic – ngôn ngữ của...

Chương 3

Tri thức và lập luận

Nội dung chính chương 3

I. Logic – ngôn ngữ của tư duy

II. Logic mệnh đề (cú pháp, ngữ nghĩa, sức mạnh biểu diễn, các thuật toán suy diễn)

III. Prolog (cú pháp, ngữ nghĩa, lập trình prolog, bài tập và thực hành)

IV. Logic cấp một (cú pháp, ngữ nghĩa, sức mạnh biểu diễn, các thuật toán suy diễn)

Lecture 1

Lecture 2

Lecture 3,4

Lecture 2:Prolog, Lập trình prolog

Lecture2-outline

I. Cơ bản về Prolog, lập trình prolog

II. Suy diễn trong Prolog (kiểm tra,Suy diễn lùi, đệ qui)

III. CUT và Phủ định

I. Cơ bản về ngôn ngữ Prolog, lập trình prolog

27/09/04 AIPP Lecture 2: Prolog Fundamentals 5

16:10 23/09/04 Lecture 1: An Introduction 6

Chương trình và thực hiện chương trình trong prolog

• Program is a database of facts and rules.– Some are always true (facts):

father( john, jim).– Some are dependent on others being true (rules):

parent( Person1, Person2 ) :- father( Person1, Person2 ).

• To run a program, we ask questions about the database.

parent(john,jim).parent(john,X).


Prolog in EnglishExample Database:

John is the father of Jim.Jane is the mother of Jim.

Jack is the father of John.

Person 1 is a parent of Person 2 if Person 1 is the father of Person 2 or Person 1 is the mother of Person 2.

Person 1 is a grandparent of Person 2 if some Person 3 is a parent of Person 2 and

Person 1 is a parent of Person 3.

Example questions:

Who is Jim's father?Is Jane the mother of Fred?

Is Jane the mother of Jim? Does Jack have a grandchild?


Prolog in PrologExample Database:

father( john, jim ).mother( jane, jim ).father( jack, john ).

parent( Person1, Person2 ) :- father( Person1, Person2 ).

parent( Person1, Person2 ) :- mother( Person1,

Person2 ).

grandparent( Person1, Person2 ) :- parent( Person3,

Person2 ), parent( Person1,

Person3 ).

Example questions:

?- father( Who, jim ).?- mother( jane, fred ).?- mother( jane, jim ).?- grandparent( jack, _ ).

Example Database:

John is the father of Jim.Jane is the mother of Jim.

Jack is the father of John.

Person 1 is a parent of Person 2 if Person 1 is the father of Person 2 or Person 1 is the mother of Person 2.

Person 1 is a grandparent of Person 2 if some Person 3 is a parent of Person 2 and

Person 1 is a parent of Person 3.

Example questions:

Who is Jim's father?Is Jane the mother of Fred?

Is Jane the mother of Jim? Does Jack have a grandchild?


Cú pháp• Prolog program consist of clauses.

• A clause has a head and a body (Rule) or just a head (Fact).

• A head consists of a predicate name and arguments.

• A clause body consists of a conjunction of terms.

• Terms can be constants, variables, or compound terms.


Clauses - Câu

• Prolog program consist of clauses.A clause = An individual definition (whether it be a

fact or rule).

e.g. mother(jane,alan). = Fact

parent(P1,P2):- mother(P1,P2). = Rule

• A clause consists of a head• and sometimes a body.

– Facts don’t have a body because they are always true.

head body


• A predicate denotes a property or relationship between objects

• ‘Predicate’ is the name given to the word occurring before the bracket in a fact or rule:

parent(jane,alan).

• By defining a predicate you are specifying which information needs to be known for the property denoted by the predicate to be true.

Predicate – Vị từ

Predicate name


Arguments – Các tham số của vị từ

• A predicate head consists of a predicate name and sometimes some arguments contained within brackets and separated by commas.

mother(jane,alan).

• A body can be made up of any number of subgoals (calls to other predicates) and terms.

• Arguments also consist of terms, which can be:– Constants e.g. jane,– Variables e.g. Person1, or– Compound terms (explained in later lectures).

Predicate name Arguments


Terms: ConstantsHạng thức hằng

Constants can either be:• Numbers:

– integers are the usual form (e.g. 1, 0, -1, etc), but

– floating-point numbers can also be used (e.g. 3.0E7)

• Symbolic (non-numeric) constants:– always start with a lower case alphabetic character and

contain any mixture of letters, digits, and underscores (but no spaces, punctuation, or an initial capital).

• e.g. abc, big_long_constant, x4_3t).

• String constants:– are anything between single quotes e.g. ‘Like this’.


Terms: VariablesHạng thức biến

• Variables always start with an upper case alphabetic character or an underscore.

• Other than the first character they can be made up of any mixture of letters, digits, and underscores.

e.g. X, ABC, _89two5, _very_long_variable

• There are no “types” for variables (or constants) – a variable can take any value.

• All Prolog variables have a “local” scope:– they only keep the same value within a clause; the same

variable used outside of a clause does not inherit the value (this would be a “global” scope).


Naming tips – Một số qui ước khi đặt các tên

• Use real English when naming predicates, constants, and variables.e.g. “John wants to help Somebody.”

Could be: wants(john,to_help,Somebody).

Not: x87g(j,_789).

• Use a Verb Subject Object structure: wants(john,to_help).

• BUT do not assume Prolog Understands the meaning of your chosen names!– You create meaning by specifying the body (i.e.

preconditions) of a clause.


Using predicate definitions Xây dựng vị từ

• Command line programming is tediouse.g. | ?- write(‘What is your name?’), nl, read(X),

write(‘Hello ‘), write(X).

• We can define predicates to automate commands:

greetings:- write(‘What is your name?’),

nl,

read(X),

write(‘Hello ‘),

write(X).

| ?- greetings.What is your name?|: tim.Hello timX = tim ?yes

Prolog Code Terminal


Using multiple clauses (Định nghĩa vị từ bởi nhiều câu)

• Different clauses can be used to deal with different arguments.

greet(hamish):-

write(‘How are you doin, pal?’).

greet(amelia):-

write(‘Awfully nice to see you!’).

= “Greet Hamish or Amelia” = a disjunction of goals.

| ?- greet(hamish). | ?- greet(amelia).

How are you doin, pal? Awfully nice to see you!

yes yes

• Clauses are tried in order from the top of the file.• The first clause to match succeeds (= yes).


Re-trying Goals – Câu truy vấn có nhiều đáp số

• When a question is asked with a variable as an argument (e.g. greet(Anybody).) we can ask the Prolog interpreter for multiple answers using: ;

| ?- greet(Anybody).

How are you doin, pal?

Anybody = hamish? ; “Redo that match.”

Anybody = amelia? ; “Redo that match.”

no “Fail as no more matches.”

• This fails the last clause used and searches down the program for another that matches.

• RETURN = accept the match• ; = reject that match


Ordering of clauses (Thứ tự các câu là quan trọng)

• The order of multiple clauses is important.

greet(hamish):-

write('How are you doin, pal?').

greet(amelia):-

write('Awfully nice to see you!').

• The most specific clause should always be at the top.• General clauses (containing variables) at the bottom.

| ?- greet(hamish).Hullo hamish?yes

greet(Anybody):-write('Hullo '), write(Anybody).


Ordering of clauses (thứ tự các câu)

• The order of multiple clauses is important.

greet(hamish):-

write('How are you doin, pal?').

greet(amelia):-

write('Awfully nice to see you!').

• The most specific clause should always be at the top.• General clauses (containing variables) at the bottom.

| ?- greet(hamish).How are you doin, pal?.yes

greet(Anybody):-write('Hullo '), write(Anybody).


Unification (hợp nhất hai hạng thức)• When two terms match we say that they unify.

– Their structures and arguments are compatible.• This can be checked using =/2

|?- loves(john,X) = loves(Y,mary).

X = mary, unification leads to instantiationY = john? yes Terms that unify Outcome

fred = fred. yes.‘Hey you’ = ‘Hey you’. yesfred=X. X=fred.X=Y. Y = X.foo(X) = foo(bar). X=bar.foo(N,N) = foo(bar,X). N=bar, X=bar.foo(foo(bar)) = foo(X) X = foo(bar)

Terms that don’t unifyfred = jim.‘Hey you’ = ‘Hey me’.frou(frou) = f(frou).foo(bar) = foo(bar,bar).foo(N,N) = foo(bar,rab).


Asking questions of the databaseTruy vấn cơ sở dữ liệu

We can ask about facts directly:

|?- mother(X,alan).

X = jane?

Yes

Or we can define rules that prove if a property or relationship holds given the facts currently in the database.

|?- parent(jane,X).

X = alan?

yes

mother(jane,alan).

father(john,alan).

parent(Mum,Child):-

mother(Mum,Child).

parent(Dad,Child):-

father(Dad,Child).


Summary• Prolog program consist of clauses.• A clause has a head and a body (Rule) or just a head (Fact).• A head consists of a predicate name and arguments.• A clause body consists of a conjunction of terms.• Terms can be constants, variables, or compound terms.• We can set our program goals by typing a command that unifies

with a clause head.• A goal unifies with clause heads in order (top down).• Unification leads to the instantiation of variables to values.• If any variables in the initial goal become instantiated this is

reported back to the user.

II. Suy diễn trong prolog (kiểm tra, suy diễn lùi, đệ qui)


30/09/04 AIPP Lecture 3: Rules, Results, and Backtracking 25

Tests – Kiểm tra• When we ask Prolog a question we are asking for

the interpreter to prove that the statement is true. ?- 5 < 7, integer(bob).yes = the statement can be proven.no = the proof failed because either

– the statement does not hold, or– the program is broken.

Error = there is a problem with the question or program. *nothing* = the program is in an infinite loop.

• We can ask about:– Properties of the database: mother(jane,alan).– Built-in properties of individual objects: integer(bob).– Absolute relationships between objects:

• Unification: =/2• Arithmetic relationships: <, >, =<, >=, =:=, +, -, *, /


Arithmetic Operators – Các phép toán số• Operators for arithmetic and value comparisons are

built-in to Prolog= always accessible / don’t need to be written

• Comparisons: <, >, =<, >=, =:= (equals), =\= (not equals)

= Infix operators: go between two terms.=</2 is used

• 5 =< 7. (infix) • =<(5,7). (prefix) all infix operators can also be prefixed

• Equality is different from unification=/2 checks if two terms unify

=:=/2 compares the arithmetic value of two expressions ?- X=Y. ?- X=:=Y. ?-X=4,Y=3, X+2 =:= Y+3.

yes Instantiation error X=4, Y=3? yes


| ?- X is 5+4*2.X = 13 ? yes

Arithmetic Operators (2)• Arithmetic Operators: +, -, *, /

= Infix operators but can also be used as prefix.– Need to use is/2 to access result of the arithmetic

expression otherwise it is treated as a term:

|?- X = 5+4. |?- X is 5+4.

X = 5+4 ? X = 9 ?

yes yes

(Can X unify with 5+4?) (What is the result of 5+4?)

• Mathematical precedence is preserved: /, *, before +,- • Can make compound sums using round brackets

– Impose new precedence– Inner-most brackets first

| ?- X is (5+4)*2.X = 18 ? yes


Tests within clauses – Các phép toán bool trong các câu

• These operators can be used within the body of a clause:– To manipulate values,sum(X,Y,Sum):-

Sum is X+Y.

– To distinguish between clauses of a predicate definition

bigger(N,M):-

N < M, write(‘The bigger number is ‘), write(M).

bigger(N,M):-

N > M, write(‘The bigger number is ‘), write(N).

bigger(N,M):-

N =:= M, write(‘Numbers are the same‘).


Backtracking – Suy diễn lùi|?- bigger(5,4).

bigger(N,M):-

N < M,

write(‘The bigger number is ‘), write(M).

bigger(N,M):-

N > M,

write(‘The bigger number is ‘), write(N).

bigger(N,M):-

N =:= M,

write(‘Numbers are the same‘).



bigger(5,4):-

5 < 4, fails


bigger(N,M):-

N > M,


bigger(N,M):-

N =:= M,


Backtrack



bigger(N,M):-

N < M,


bigger(5,4):-

5 > 4,


bigger(N,M):-

N =:= M,



|?- bigger(5,4).

bigger(N,M):-

N < M,


bigger(5,4):-

5 > 4, succeeds, go on with body.write(‘The bigger number is ‘), write(5).

The bigger number is 5

yes

|?-

Backtracking – Suy diễn lùi

Reaches full-stop = clause succeeds


Backtracking – Suy diễn lùi|?- bigger(5,5). If our query only matches the final clause

bigger(N,M):-

N < M,


bigger(N,M):-

N > M,


bigger(5,5):-

5 =:= 5,


Is already known as the first two clauses failed.


Backtracking – Suy diễn lùi|?- bigger(5,5). If our query only matches the final clause

bigger(N,M):-

N < M,


bigger(N,M):-

N > M,


bigger(5,5):-


Numbers are the same

yes

Satisfies the same conditions.

Clauses should be ordered according to specificityMost specific at top Universally applicable at bottom


Satisfying Subgoals – Đích trung gian

• Most rules contain calls to other predicates in their body. These are known as Subgoals.

• These subgoals can match:– facts,– other rules, or– the same rule = a recursive call

1) drinks(alan,beer).2) likes(alan,coffee).3) likes(heather,coffee).

4) likes(Person,Drink):-drinks(Person,Drink). a different subgoal

5) likes(Person,Somebody):-likes(Person,Drink), recursive subgoalslikes(Somebody,Drink).


Representing Proof using TreesCây biểu diễn chứng minh

To help us understand Prolog’s proof strategy we can represent its behaviour using AND/OR trees.

1. Query is the top-most point (node) of the tree.

2. Tree grows downwards (looks more like roots!).

3. Each branch denotes a subgoal.1. The branch is labelled with the number of the matching clause and

2. any variables instantiated when matching the clause head.

4. Each branch ends with either:1. A successful match ,

2. A failed match , or

3. Another subgoal.

|?- likes(alan,X).

2 X/coffee

X = coffee

1st solution= “Alan likes coffee.”


Representing Proof using Trees (2)

|?- likes(alan,X).

X/coffee

X = coffee

• Using the tree we can see what happens when we ask for another match ( ; )

2

4

drinks(alan,X).

1 X/beer

X = beer

2nd solution = “Alan likes beer because Alan drinks beer.”

1st match is failed and forgotten

Backtracking


Recursion using TreesCây đệ qui

|?- likes(alan,X).

X/coffee

X = coffee

• When a predicate calls itself within its body we say the clause is recursing

2

4

1 X/beer

X = beer

5

likes(alan,Drink)

Conjoined subgoals

likes(Somebody,Drink)

drinks(alan,X).

X/coffee 2

X = coffee


|?- likes(alan,X).

X/coffee

X = coffee

2

4

1 X/beer

X = beer

3rd solution = “Alan likes Alan because Alan likes coffee.”

5

likes(alan,coffee)

likes(Somebody,coffee)

drinks(alan,X).

X/coffee 2

X = coffee

Somebody/alan

2

Somebody = alan

Recursion using Trees (2)• When a predicate calls itself within its body we say the clause is recursing


|?- likes(alan,X).

X/coffee

X = coffee

2

4

1 X/beer

X = beer

4th solution = “Alan likes Heather

because Heather likes coffee.”

5

likes(alan,coffee) likes(Somebody,coffee)

drinks(alan,X).

X/coffee 2

X = coffee

Somebody/alan

2

Somebody = alan

Somebody / heather3

Somebody = heather

• When a predicate calls itself within its body we say the clause is recursing

Recursion using Trees (3)


Infitite Recursive Loop• If a recursive clause is called with an incorrect goal it will loop as it can neither prove it nor disprove it.

likes(Someb,coffee)

2Somebody = alan

3

Somebody = heather

5likes(Someb,coffee)

Someb = alan

2 likes(coffee,coffee)

likes(coffee,X) likes(coffee,X)

likes(coffee,X2)

likes(coffee,X3)

likes(X,X2)

likes(X2,X3)

II. CUT và phủ định và findall


CUTa(X, Y) :- b(X), !, c(Y).

b(1). b(2). b(3).

c(1). c(2). c(3).

------------------

Kết quả truy vấn thế nào?

?- a(Q, R).

Q = 1,

R = 1 ;

Q = 1,

R = 2 ;

Q = 1,

R = 3.

Tai sao?


CUT

a(X) :- b(X), !, c(X).

b(1). b(2). b(3).

c(2).

----------------------

?- a(Q).

false.

?- a(2).

true.

Tại sao?


14/10/04 AIPP Lecture 7: The Cut 45

Green Cuts ! (Sử dụng CUT khi nào?)

f(X,0):- X < 3, !.

f(X,1):- 3 =< X, X < 6, !.

f(X,2):- 6 =< X.

|?- trace, f(2,N).

1 1 Call: f(2,_487) ?

2 2 Call: 2<3 ?

2 2 Exit: 2<3 ? ?

1 1 Exit: f(2,0) ?

N = 0 ? ;

noIf you reach this point don’t bother trying any other clause.

• Notice that the answer is still the same, with or without the cut.– This is because the cut does not alter the logical behaviour of the

program.

– It only alters the procedural behaviour: specifying which goals get checked when.

• This is called a green cut. It is the correct usage of a cut.• Be careful to ensure that your clauses are actually mutually

exclusive when using green cuts!

14/10/04 AIPP Lecture 7: The Cut 46

• Because the clauses are mutually exclusive and ordered we know that once the clause above fails certain conditions must hold.

• We might want to make our code more efficient by removing superfluous tests.

Red Cuts ! (Không sử dụng CUT khi nào?)

| ?- f(7,N).

1 1 Call: f(7,_475) ? 2 2 Call: 7<3 ? 2 2 Fail: 7<3 ? 3 2 Call: 3=<7 ? 3 2 Exit: 3=<7 ? 4 2 Call: 7<6 ? 4 2 Fail: 7<6 ? 5 2 Call: 6=<7 ? 5 2 Exit: 6=<7 ? 1 1 Exit: f(7,2) ?

N = 2 ?

yes

f(X,0):- X < 3, !.

f(X,1):- 3 =< X, X < 6, !.

f(X,2):- 6 =< X.

Redundant?

Phủ địnhparents(a,b).

parents(c,d).

parents(b,c).

parents(a,e).

nochild(X):- \+ parents(X,_).

------------------------

?- parents(f,X).

false.

?- nochild(a).

false.

?- nochild(e).

true.


\+ X means not(X) that is the way to implement negation in Prolog; however not(X) does not mean that X is false, it means that X can't be proved true from the database.

Phủ định (tiếp)• Kiểm tra một số có là nguyên tố?.

is_prime(2).

is_prime(3).

is_prime(P):-integer(P), P>3, P mod 2 =\= 0, \+ has_factor(P,3).

has_factor(P,N):- P mod N =:=0.

has_factor(P,N):- L is N+2, L*L<P, has_factor(P,L).

------------------------

?- is_prime(97).

true.

?- is_prime(18).

false.


findallparents(a,b).

parents(c,d).

parents(b,c).

parents(a,e).

---------------

?- parents(a,X).

X = b ;

X = e.

?- findall(X,parents(a,X),F).

F = [b, e]

?- findall([X,Y],parents(X,Y),F).

F = [[a, b], [c, d], [b, c], [a, e]].


Sử dụng dấu “;” nếu muốn Prolog in các kết quả khác

Vị từ xây dựng sẵn trong Prolog cho phép tìm tập F gồm tất cả các X thỏa mãn parents(a,X).

Bài tập chữa trên lớp

1. Tim phan tu cuoi cung cua mot danh sach.

Chu y: [X|R]: X la 1 phan tu, R la 1 list.

2. Hoanvi

3. Sap xep danh sach

4. Giai phuong trinh bac nhat

5. Hop va giao 2 tap hop.

6. Thay the 1 ky tu boi 1 ky tu khac trong danh sach

7. Trich xau con tu vi tri K den L cua mot xau.

8. Tim thua so chung lon nhat 2 so

9. Do thi va tim duong di trong do thi.


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