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3. Lists, Operators, Arithmetic

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Contents

• Representation of lists• Some operations on lists• Operator notation• Arithmetic

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Representation of Lists

• The list is a simple data structure used in non-numeric programming.

• A list is a sequence of any number of terms.– [ann, tennis, tom, skiing]

• An empty list is written as a Prolog atom [ ].

• A list can be viewed as consisting of two things:– the first item, called the head of the list;– the remaining part of the list, called the tail.

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Representation of Lists

• For [ann, tennis, tom, skiing], the head is ann and the tail is the list: [tennis, tom, skiing].

• In general, the head can be anything; the tail has to be a list.

• The head and tail are then combined into a structure by a special functor (depending upon the Prolog implementation).– .(Head, Tail)– .(ann,.(tennis,.(tom,.(skiing,[]))))

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Representation of Lists

.

.

.

.

ann

tennis

tom

skiing [ ]

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Representation of Lists

?-List1=[a,b,c], List2=.(a,.(b,.(c,.[]))).List1=[a,b,c]List2=[a,b,c]?-Hobbies1=.(tennis,.(music,[])), Hobbies2=[skiing,food], L=[ann,Hobbies1,tom,Hobbies2].Hobies1=[tennis,music]Hobbies2=[skiing,food]L=[ann,[tennis,music],tom,[skiing,food]]

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Representation of Lists

L=[a,b,c]L=[a|Tail]Tail=[b,c][a,b,c]=[a,|[b,c]] =[a,b|[c]] =[a,b,c|[]]

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Some Operations on Lists

• Membershipmember(b,[a,b,c]) is truemember(b,[a,[b,c]]) is not true

X is a member of L if either(1) X is the head of L, or(2) X is a member of the tail of L.

member(X,[X|Tail]).member(X,[Head|Tail]):- member(X,Tail).

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Some Operations on Lists

• Concatenationconc([a,b],[c,d],[a,b,c,d]) is true, butconc([a,b],[c,d],[a,b,a,c,d]) is false.

(1) If the first argument is the empty list then the second and the third arguments must be the same list.

conc([],L,L).(2) If the first argument of conc is a non-empty list then it has a head and a tail and must look like this:

[X|L1]conc([X|L1],L2,[X|L3]):-

conc(L1,L2,L3).L1X L2

L3X

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Some Operations on Lists?-conc([a,b,c],[1,2,3],L).L=[a,b,c,1,2,3]?-conc([a,[b,c],d],[a,[],b]).?-conc(L1,L2,[a,b,c]).L1=[]L2=[a,b,c];

L1=[a]L2=[b,c];

L1=[a,b]L2=[c];

L1=[a,b,c]L2=[];no

?-conc(Before,[may}After], [jan,feb,mar,apr,may,jun, jul,aug,sep,oct,nov,dec]).Before=[jan,feb,mar,apr]After=[jun,jul,aug,sep,oct, nov,dec]

?-conc(_,[Month1,may,Month2|_], [jan,feb,mar,apr,may,jun, jul,aug,sep,oct,nov,dec]).Month1=aprMonth2=jun

?-L1=[a,b,z,z,c,z,z,z,d,e], conc(L2,[z,z,z|_],L1).L1=[a,b,z,z,c,z,z,z,d,e]L2=[a,b,z,z,c]

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Some Operations on Lists

member1(X,L):- conc(_, [X|_],L).

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Some Operations on Lists

• Adding an itemadd(X,L,[X|L]).

• Deleting an item:– If X is the head of the list then the result

after the deletion is the tail of the list.– If X is in the tail then it is deleted from

there.delete( X,[X|Tail,Tail).

delete(X,[Y|Tail],[Y|Tail1]):-

delete(X,Tail,Tail1).

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Some Operations on Lists

?-delete(a,[a,b,a,a],L).L=[b,a,a];L=[a,b,a];L=[a,b,a];no?-delete(a,L,[1,2,3]).L=[a,1,2,3];L=[1,a,2,3];L=[1,2,a,3];L=[1,2,3,a];no

insert(X,List,BiggerList):- delete(X,BiggerList,List).

Member2(X,List):- delete(X,List,_).

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Some Operations on Lists• Sublist

sublist([c,d,e],[a,b,c,d,e,f]) is true, butsublist([c,e],[a,b,c,d,e]) is not.

• The Prolog program for sublist can be based on the same idea as member1.– S is a sublist of L if

(1) L can be decomposed into two lists, L1 and L2, and(2) L2 can be decomposed into two lists, S and L3.

sublist(S,L):- conc(L1,L2,L), conc(S,L3,L2). L1 S L3

L

L2

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Some Operations on Lists

• Permutations?-permutation([a,b,c],P).

P=[a,b,c];

P=[a,c,b];

P=[b,a,c];

…

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Some Operations on Lists

• The program– If the first list is empty then the second list must

also be empty– If the first list is not empty and it has the form

[X|L], then a permutation can be constructed by first permute L obtaining L1 and then insert X at any position into L1.

permutation([],[]).permutation([X|L],P);- permutation(L,L1), insert(X,L1,P).

LX

L1

permute L

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Some Operations on Lists

permutation2([],[]).permutation(L,[X|P]):- delete(X,L,L1), permutation2(L1,P).

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Operator Notation• An infix expression, for example, 2*a+b*c, can be

written in Prolog as: +(*(2,a),*(b,c)).• Since we normally prefer to have expressions written in

infix style, Prolog caters for this notational covenience.• Prolog will therefore accept the expression as:

2*a+b*c.• This will be, however, only the external representation

of the object, which will be automatically converted into the usual form of Prolog terms.

• Thus operators in Prolog are merely a notational extension.

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Operator Notation

• In order that Prolog properly understands expressions such as a+b*c, Prolog has to know that * binds stronger than +.

• The precedence of operators decides what is the correct interpretation of expressions.

• For example, a+b*c can be understood either as +(a,*(b,c)) or as *(+(a,b),c).

• The general rule is that the operator with highest precedence is the principal functor of the term.

• If expression containing + and * are to be understood according to our normal convention, then + has to have a higher precedence than *.

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Operator Notation

• A programmer can define her own operators.• For example, we can define the atoms has

and support as infix operators and then write in the program facts like

peter has information.floor supports table.

• These facts are exactly equivalent to:has(peter, information).Supports(floor, table).

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Operator Notation

• A programmer can define new operators by inserting the program special kinds of clauses, sometimes called directives, which act as operator definitions.

• An operator definition must appear in the program before any expression containing that operator.:- op(600, xfx,has).This tells Prolog that we want to user ‘has’ as an operator,

whose precedence is 600 and its type is ‘xfx’, which is a kind of infix operator.

The form specifier ‘xfx’ suggests that the operator, denoted by ‘f’, is between the two arguments denoted by ‘x’.

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Operator Notation

• Operators are normally used, as functor, only to combine objects into structures and not to invoke action on data.

• Operators are atoms, and their precedence must be in some range which depends on the implementation. (Assume [1,1200] here.)

• There are three groups of operator types :– infix operators: xfx, xfy, yfx– prefix operators: fx, fy– postfix operators: xf, yf

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Operator Notation

• There is a difference between ‘x’ and ‘y’.• We need to introduce the concept of the

precedence of argument.– If an argument is enclosed in parentheses or it is an

unstructured object then its precedence is 0;– If an argument is a structure, then its precedence is

equal to the precedence of its principal functor.

• ‘x’ represents an argument whose precedence must be strictly lower than that of the operator. ‘y’ represents an argument whose precedence is lower or equal to that of the operator.

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Operator Notation

• These rules help to disambiguate expression with several operators of the same precedence.

• For example, the expression, a-b-c, is normally understood as (a-b)-c, and not a-(b-c).– To achieve the normal interpretation the operator

‘–’ has to be defined as yfx.–

– c

a b

Precedence 500

Prec. 0

–

–

b c

Precedence 500

aPrec. 0

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Operator Notation

• Consider the prefix operator not. – If not is defined as fy then the expression

not not pis legal.

But if not is defined as fx then this expression is illegal because the argument to the first not is not p.

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Operator Notation

• Predefined operators in the Prolog system:

:-op(1200,xfx,’:-’).:-op(1200,fx,[:-,?-]).:-op(1100,xfy,’;’).:-op(1000,xfy,’,’).:-op(700,xfx,[=,is,<,>,=<,==,=\=,\==,=:=]).:-op(500,,yfx,[+,-]).:-op(500,fx,[+,-,not]).:-op(400,yfx,[*,/,div]).:-op(300,xfx,mod).

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Operator Notation

• The use of operators can greatly improve the readability of programs.

~(A&B)<===> ~A v ~B

equivalent(not(and(A,B)), or(not(A),not(B))).

:-op(800,xfx,<===>).:-op(700,xfy,v).:-op(600,xfy,&).:-op(500,fy,~).

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Arithmetic

• The means for numerical computing in Prolog are rather simple:– +, –, *, /, mod

?-X=1+2.X=1+2

?-X is 1+2.X=3

?-X is 3/2,Y is 3 div 2.X=1.5Y=1

?-277*37>10000.Yes

?-born(Name,Year), Year>=1950,Year=<1960.

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Arithmetic

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Arithmetic

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Arithmetic

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Arithmetic

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Arithmetic?-1+2=:=2+1.yes

?-1+2=2+1.no

?-1+A=B=2.A=2B=1

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Arithmetic

• Given two positive integers, X and Y, their greatest common divisor, D, can be found according to three cases:– If X and Y are equal then D is equal

to X.– If X<Y then D is equal to the greatest

common divisor of X and Y–X.

– If Y<X then do the same as in the preceding case with X and Y interchanged.

gcd(X,X,X).gcd(X,Y,D):- X<Y,Y1 is Y-X, gcd(X,Y1).

gcd(X,Y,D):- Y<X, gcd(Y,X,D).

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Arithmetic

length([],0).length([_|Tail],N):- length(Tail,N1), N is N1+1.