Standard ML

1
Prolog
2
Matching

• point(A,B) point(1,2)
• A1 , B2
• point(A,B) point(X,Y,Z)
• No!
• 22 4
• No!
• triangle(point(-1,0),P2,P3) triangle(P1,point(1,
  0),point(0,Y))
• P1point(-1,0) , P2point(1,0), P3point(0,Y)

3
Matching

• An operation on terms. Two terms match if
• they are identical, or
• the variables in both terms can be instantiated
  to objects in such a way that after the
  substitution of variables by these objects the
  terms become identical.
• date(D,M,1995) matches date(D1,may,Y1)
• date(D,M,1995) doesnt match date(D1,M1,1996)
• date(D) doesnt match day(D)
• If matching succeeds it always results in the
  most general instantiation possible.
• date(D,M,1995) date(D1,may,Y1).D D1
  MmayY11995

4
General rules for matching two terms S and T

• (1) If S and T are constants then S and T match
  only if they are the same object.
• (2) If S is a variable and T is anything, then
  they match, and S is instantiated to T. (or the
  other way around...)
• (3) If S and T are structures then they match
  only if
• (a) S and T have the same principal functor
  and the same number of components, and
• (b) all their corresponding components
  match.
• The resulting instantiation is determined
  by the matching of the components.

5
An Illustration

• Use structures to represent simple geometric
  shapes.
• point - two numbers representing X and Y
  coordinates.
• seg - a line defined by two points
• triangle - defined by three points.
• point(1,1)
• seg( point(1,1), point(2,3) )
• triangle( point(4,2), point(6,4), point(7,1) )
• In the same program we can also use three
  dimensional points
• point(1,3,5)
• This will result in a different relation with the
  same name.
• Match triangle(point(1,1), A,
  point(2,3)) triangle(X,
  point(4,Y),point(2,Z)).

6

• match by
• triangle triangle
• point(1,1) X
• A point(4,Y)
• point(2,3) point(2,Z)
• The resulting instantiation is
• X point(1,1)
• A point(4,Y)
• Z 3

7
Matching as means of Computation

• A program with two facts
• vertical( seg( point(X,Y), point(X, Y1) )
  ).horizontal( seg( point(X,Y), point(X1,Y) ) ).
• Conversation
• ?- vertical( seg( point(1,1), point(1,2) )).yes
• ?- vertical( seg( point(1,1), point(2,Y) )).no
• ?- vertical( seg( point(2,3), P)).P point(2,Y)
• When prolog has to invent a variable name (like
  the Y above) it will be in the form _n where n is
  an arbitrary number.

8
Regular Rectangle

• Both vertical and Horizontal, i.e.,
• regular(rectangle(point(X1,Y1),point(X1,Y2),
  point(X2,Y2),point(X2,Y1)).
• regular(rectangle(point(X1,Y2),point(X2,Y2),
  point(X2,Y1),point(X1,Y1)).

(x1,y2)
(x2,y2)
(x2,y1)
(x1,y1)
9
List 1

• Write a goal, using append, to delete the last
  three elements from a list L producing another
  list L1.
• Hint L is the concatenation of L1 and a
  three-element list.
• del3(L,L1) - append(L1,_,_,_,L).

10
List 2

• Write a goal to delete the first three elements
  and the last three elements from a list L
  producing list L2.
• del_3_3(L,L2) - del3(L,L1), _,_,_L2L1.
• del_3_3(L,L2) - del3(L,_,_,_L2).
• del_3_3(_,_,_L4,L2) - del3(L4,L2).

11
List 3

• Using append, define the relation last(Item,List)
  to test if Item is the last element of a list
  List.
• last(Item,List)- append(_,Item,List).
• The same without using append.
• last(Item,Item).
• last(Item,_L)- last(Item,L).

12
List 4

• Define two predicates evenlength(L) and
  oddlength(L) so that they are true if their
  argument is a list of even or odd length
  respectively.
• evenlength().
• oddlength(_L)-evenlength(L).
• evenlength(_L)- oddlength(L).

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14
Arithmetics

• Predefined operators for basic arithmetic
• , -, , /, mod
• This is a special case in which an operator may
  invoke an operation. It will be done only if it
  is explicitly indicated.
• Example
• X 1 2.X12
• The predefined operator is forces evaluation.
• X is 1 2.X3
• The mod operator associates to the right.All the
  others associates to the left.
• The comparison operators also force evaluation.
• 145 34 gt 100.Yes

15

• The comparison operators
• X gt Y X is greater than Y.
• X lt Y X is less than Y.
• X gt Y X is greater than or equal to Y.
• X lt Y X is less than or equal to Y.
• X Y the values of X and Y are equal.
• X \ Y the values of X and Y are not equal.

16
and

• X Y causes the matching of X and Y and possibly
  instantiation of variables.
• X Y causes an arithmetic evaluation of X and
  Y, and cannot cause any instantiation of
  variables.
• 1 2 2 1.
• gt yes
• 1 2 2 1.
• gt no
• 1 A B 2.
• gt A 2
• gt B 1
• 1 A B 2.
• gt WARNING Unbound variable in arithmetic
  expression
• gt Fail ( 6) 1 _G149 _G151 2 ?
• gt No

17
Example The Greatest Common Devisor

• given X and Y, the gcd D can be found by
• (1) If X and Y are equal then D is equal to
  X.(2) If X lt Y then D is equal to the gcd of X
  and (Y-X).(3) If Y lt X then do the same as in
  (2) with X and Y interchanged.
• gcd(X,X,X).
• gcd(X,Y,D) -
• XltY,
• Y1 is Y - X,
• gcd(X,Y1,D).
• gcd(X,Y,D) -
• Y lt X,
• gcd(Y,X,D).

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19
Lists

• A sequence of any number of items.
• Structure of lists .( Head, Tail ) .(a,
  .(b, )) eq.
• Shorthand
• tom, jerry is the same as .(tom, .(jerry,
  ))
• a tail is the same as .(a, tail)
• a,b,c a b,c a,b c a,b,c
• Elements can be lists and structures
• a, 1, 2, 3, tom, 1995, date(1,may,1995)

20
Operations on Lists

• Membership
• member( X, L) if X is a member of the list L.
• member(X, X Tail).
• member(X, Head Tail) -
• member(X, Tail).
• Concatenation
• conc(L1, L2, L3) if L3 is the concatenation of L1
  and L2.
• conc(, L, L).
• conc(XL1, L2, XL3) -
• conc(L1, L2, L3).

21

• conc( a,b,c, 1,2,3, L).
• gt L a,b,c,1,2,3
• conc( L1, L2, a,b,c ).
• gt L1
• L2 a,b,c
• gt L1 a
• L2 b,c
• gt L1 a,b
• L2 c
• gt L1 a,b,c
• L2
• gt no
• conc( Before, 4After, 1,2,3,4,5,6,7).
• gt Before 1,2,3
• After 5,6,7
• conc(_, Pred, 4, Succ _, 1,2,3,4,5,6,7).
• gt Pred 3
• Succ 5

22

• Redefining member using conc
• member1(X, L) - conc(_, X_, L).
• Adding an Item in the front
• add(X, L, XL).
• Deleting an item
• del(X, XTail, Tail).del(X, YTail,
  YTail1) - del(X, Tail, Tail1).
• If there are several occurrences of X in the list
  then del will be able to delete only one of them.
• To insert an item at any place in the
  listdel(a, L, 1,2,3).gt L a,1,2,3gt L
  1,a,2,3gt L 1,2,a,3gt L 1,2,3,agt no

23

• We can define insert using del
• insert(X,List,BiggerList) - del(X,
  BiggerList, List).
• The sublist relation
• sublist(S, L) - conc(L1, L2, L), conc(S,
  L3, L2).
• sublist(S, a,b,c).gt S gt S a...gt S
  b,c ...

24

• Permutations
• permutation(, ).permutation(XL, P) -
  permutation(L, L1), insert(X, L1, P).
• permutation( a,b,c, P).gt P a,b,cgt P
  a,c,bgt P b,a,c...
• permutation2(, ).permutation2(L, XP) -
  del(X, L, L1), permutation2(L1, P).

25
Length

• The length of a list can be calculated in the
  following way
• if the list is empty then its length is 0.
• if the list is not empty then List Head
  Tail. In this case the length is equal to 1 plus
  the length of the tail Tail.
• length is built in. If you want to try defining
  it, change the name...
• length(, 0).length(_Tail,N) -
  length(Tail, N1), N is 1 N1.
• length(a,b,c,d,e, N).gt N 4length(L,4).gt
  L _5, _10, _15, _20 ..... ?

what happens if the order of these clauses is
changed?