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Preparing Declarative Programming Exam -- Prolog

996worker
2022-10-25 / 0 评论 / 0 点赞 / 43 阅读 / 2,499 字
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Intro

Fast notes for preparing exam

links

https://kalabovi.org/pitel:flp:99pl
https://www.ic.unicamp.br/~meidanis/courses/mc336/2009s2/prolog/problemas/

Prolog

syntac

  • Atom: A string of characters, butch , big_kahuna_burger; An arbitrary sequence of characters enclosed in single quotes, ’ Vincent ’, ’ The Gimp ’; A string of special characters, @= and ====> and ; and :- are all atoms.
  • Numbers: 23 , 1001 , 0 , -365
  • Variables: X , Y , Variable , _tag , X_526 , List , List24 , _head , Tail , _input and Output are all Prolog variables.
  • Compound terms: A compound term begins with a functor, which is an atom, and follows with one or more terms as arguments, enclosed within parentheses and separated by commas: card(clubs, 3).
  • Facts: The clauses simply say some relationship holds.
  • Rules: The rule specifies that the relationship in the head holds if the body holds: classmates(X, Y) :- enrolled(X, Class), enrolled(Y, Class).
  • List: [E|Es], [E1,E2,E3|Es].
  • Import package: :- ensure_loaded(borders).

logic

  • And: enrolled(S, maths), enrolled(S, logic).
  • Or: enrolled(S, logic) ; enrolled(S, maths).
  • Negation: enrolled(S, logic), \+ enrolled(S, maths).
  • Disequality: Prolog also defines a not equal predicate, spelled =, for convenience. X \= Y always behaves the same as \+ X = Y
    • Note: Negations, including \=, should be written after goals that bind all the variables used in the negation: X = bob, Y = alice, X \= Y.

Unification

Two terms can be unified roughly like this:

  1. If one of the terms to be unified is a variable, it is bound to the other term; unification succeeds. 如果一个项是变量, 绑定至另一个任意项,则可继续unify, 否则挂.
  2. If the two terms are identical atomic terms, the unification succeeds. 如果两项是相同的atom, 则可继续unify;
  3. If the two terms are both compound, with the same functor, the arguments of the two terms are unified pairwise; unification succeeds if and only if all the pairs of arguments can be unified. 如果两项是compound, 并且两项functor相同, 当且仅当两项的argument也必须能两两对应unify, 则可继续unify,否则挂.
  4. otherwise, the unification fails. 其他情况均挂.

算数 Arithmetic

Arithmetic in Prolog works through the special is/2 predicate, which evaluates expressions:

?- X is 6*7.
X = 42.

?- X = 6*7.
X = 6*7.

Note: the right argument of is/2 must be a ground term.

Predicates Desc
V is Expr Unify V with the value of expression Expr
Expr1 =:= Expr2 Succeeds if the values of Expr1 and Expr2 are equal
Expr1 == Expr2 Succeeds if the values of Expr1 and Expr2 are different
Expr1 < Expr2 <
Expr1 =< Expr2 =<
Expr1 > Expr2 >
Expr1 >= Expr2 >=

In all of these cases, arithmetic expressions must be ground at the time of the predicate call.

image

Terms Comparation

  • Numbers are always considered smaller than atoms, and both are smaller than compound terms.
  • A compound term A is smaller than compound term B if the arity of A is smaller than the arity of B.
  • If their arities are the same, the compound term with the smaller functor is smaller.
  • If their arities and functors are the same, then the term with the smaller first argument is smaller.
  • If arities, functors and first arguments are the same, the second argument is considered, and so on.
  • Variables are smaller than any bound term.

image-1666696340440

?- a @< b.
true.

?- a @>= 3.
true.

?- f(2) @< f(1,2).
true.

?- f(2,1) @< f(1,2).
false.

?- X @< a.
true.

?- X @< a, X = b.
X = b.

Prolog has predicates to test if two terms are identical, including the variables, without binding any variables.

X == Y
X \== Y

These check for indentity without binding variables

?- X == Y.
false.

?- X = Y, X == Y.
X = Y.

?- X \== Y.
true.

?- X \== Y, X = Y.
X = Y.

?- X = f(A,B,3), Y = f(A,B,3), X == Y.
X = Y, Y = f(A, B, 3).

?- X = f(A,B,3), Y = f(p,B,3), X == Y.
false.

Type Testing

image-1666696893187
As usual, these goals reflect the state of the argument when they are executed. This means they are sensitive to goal order

?- X = 42, integer(X).
X = 42.

?- integer(X), X = 42.
false.

image-1666696923600

IO

image-1666697369622

?- write('Hello, world!').
Hello, world!
true.

?- writeq('Hello, world!').
'Hello, world!'
true.

?- write(f(X,g(X),'a b c')).
f(_18534,g(_18534),a b c)
true.

?- read(X).
|: f(_18534,g(_18534),'a b c').

X = f(_20192, g(_20192), 'a b c').

image-1666697425605

常用函数/算法

append

append/3, 已集成.

append([], Ys, Ys).
append([X|Xs], Ys, [X|Zs]) :-
        append(Xs, Ys, Zs).

使用

?- append([1,2], [3,4,5], List).
List = [1, 2, 3, 4, 5].

?- append([1,2], List, [1,2,3,4,5]).
List = [3, 4, 5].

?- append([3,4], List, [1,2,3,4,5]).
false.

?- Front = [_,_], append(Front, Back, [1,2,3,4,5]).
Front = [1, 2],
Back = [3, 4, 5].

sort

已集成
sort(List0, List)
List contains all the elements of List0 in sorted order with duplicates removed. Because duplicates are removed, this is not usually the sorting predicate you want.

msort(List0, List)
List contains all the elements of List0 in sorted order without removing duplicates. This is usually the sorting predicate you want.

keysort(List0, List)
List0 is a list of Key-Value pairs, and List is a list of the same pairs sorted by the Keys, but not by the Values. This is a stable sort, so pairs with the same value will appear in List in the same relative order they appear in List0. For example:

?- keysort([a-2,b-3,a-1], L).
L = [a-2, a-1, b-3].

length

已集成

?- length([a,b,c], Len).
Len = 3.

?- length(List, 3).
List = [_7224, _7230, _7236].

take

取出前几个元素

take(N, List, Front) :-
        length(Front,N),
        append(Front, _, List).
        
% mode-sensitive的实现方法
take(N, List, Front) :-
    (   nonvar(N)
    ->  length(Front, N),
        append(Front, _, List)
    ;   append(Front, _, List),
        length(Front, N)
    ).

member

已集成, member(E, List) holds when E is one of the elements of List.

?- member(c, [a,b,c,d,e]).
true ;
false.

?- member(X, [a,b,c]).
X = a ;
X = b ;
X = c.

select

已集成, select(Elem, List1, List2) is like member(Elem, List1), except that List2 is everything in List1 other than Elem. You can use this to remove a single occurrence of Elem from List1, or to insert Elem in any place in List2, or to select an element of List1, producing one element plus the rest of the list.

?- select(c, [a,b,c,d,e], List).
List = [a, b, d, e] ;
false.

?- select(x, List, [a,b,c]).
List = [x, a, b, c] ;
List = [a, x, b, c] ;
List = [a, b, x, c] ;
List = [a, b, c, x] ;
false.

?- select(Elem, [a,b,c], Rest).
Elem = a,
Rest = [b, c] ;
Elem = b,
Rest = [a, c] ;
Elem = c,
Rest = [a, b].

nth0(Index, List, Elem)

已集成, inds the nth element of List, where n counts from 0. But it can also determine at what position Elem appears in List, and can even produce the elements of List together with their positions.

?- nth0(3, [a,b,c,d,e], Elem).
Elem = d.

?- nth0(N, [a,b,c,d,e], d).
N = 3 ;
false.

?- nth0(N, [a,b,c], Elem).
N = 0,
Elem = a ;
N = 1,
Elem = b ;
N = 2,
Elem = c.

nth0(N, List, Elem, Rest)

It can be used to remove an element from a list by position, or by value while providing the position, or to insert an element into a list in a particular position.

?- nth0(3, [a,b,c,d,e], Elem, Rest).
Elem = d,
Rest = [a, b, c, e].

?- nth0(N, [a,b,c,d,e], d, Rest).
N = 3,
Rest = [a, b, c, e] ;
false.

all_in

all_in([], _).
all_in([E|Es], L) :-
    member(E, L),
    all_in(Es, L).

same_elements

ll elements of list L1 are in L2 and vice versa, though **they may be in different orders and have different numbers of occurrences of the elements. **

same_elements(L1, L2) :-
    all_in(L1, L2),
    all_in(L2, L1).

all_in([], _).
all_in([E|Es], L) :-
    member(E, L),
    all_in(Es, L).
    
% n log(n) time implementation
same_elements(L1, L2) :-
    sort(L1, Sorted),
    sort(L2, Sorted).

all_same

listof(_, []).
listof(Elt, [Elt|List]) :-
	listof(Elt, List).

all_same(List) :-
	listof(_, List).

adacent

Implement the predicate adjacent(E1, E2, List) such that E1 appears immediately before E2 in List.

adjacent(E1, E2, List) :- 
    append(_, [E1, E2|_], List).
    
% 递归解法
adjacent(E1, E2, [E1,E2|_]).
adjacent(E1, E2, [_|Tail]) :- 
    adjacent(E1, E2, Tail).

before

E1 and E2 are both elements of List, where E2 occurrs after E1 on List.

before(E1, E2, [E1|List]) :-
	member(E2, List).

% 递归解法
before(E1, E2, [_|List]) :-
	before(E1, E2, List).

sumlist

sumlist([], 0).
sumlist([X|Xs], Sum) :-
    sumlist(Xs, S1),
    Sum is X + S1.
    
% 尾递归
% tail recursive
sumlist(List, Sum) :-
	sumlist(List, 0, Sum).

sumlist([], Sum, Sum).
sumlist([N|Ns], Sum0, Sum) :-
	Sum1 is Sum0 + N,
	sumlist(Ns, Sum1, Sum).

replace

this holds when list L1 is the same as L2, except that in one place where L1 has the value E1, L2 has E2.

replace(2,[1,2,3,4],5,X) should have only the solution X = [1,5,3,4].

replace(2,[1,2,3,2,1],5,X) should backtrack over the solutions X = [1,5,3,2,1] and X = [1,2,3,5,1].

replace(E1,[E1|Xs],E2,[E2|Xs]).

replace(E1,[H|Hs],E2,[H|Ts]):-
  replace(E1,Hs,E2,Ts).

zip

zip(As, Bs, ABs)

zip([1,2,3,4],[a,b,c,d],L) should have only the solution L=[1-a,2-b,3-c,4-d]].

zip(X,Y,[1-a,2-b,3-c,4-d]) should have only the solution X=[1,2,3,4], Y=[a,b,c,d].

zip([],[],[]).
zip([H|T],[H1|T1],[H-H1|T2]) :- 
    zip(T,T1,T2).

setof(Template, Goal, List)

setof(Template, Goal, List)
List is a list of all the instances of Template that satisfy Goal. List is sorted (using @=</2 order), with duplicates removed; thus List is really a set of solutions in a canonical order.

?- setof(S, enrolled(alice, S), Subjects).
Subjects = [logic, maths].

?- setof(D-T, class(logic, D, T), Lectures).
Lectures = [monday-10, wednesday-10].

?- setof(S, enrolled(Student, S), Subjects).
Student = alice,
Subjects = [logic, maths] ;
Student = bob,
Subjects = [maths] ;
Student = claire,
Subjects = [physics] ;
Student = don,
Subjects = [art_history, logic].

bagof

which is just like setof/3, except that it does not sort and remove duplicates. The order Prolog returns solutions and the number of times it returns a solution are not supposed to be significant, and bagof/3 is sensitive to these things, so setof/3 is a more logical predicate.

findall

indall/3 which is similar to bagof/3, except that any variable appearing in the Goal but not the Template is implicitly existentially quantified. Also findall/3 produces an empty list if there are no solutions to the Goal

?- findall(S, enrolled(Student, S), Subjects).
Subjects = [logic, maths, maths, physics, logic, art_history].

?- findall(S, enrolled(ted, S), Subjects).
Subjects = [].

sublist

sublist(Xs, Ys).
this holds when Xs is a list containing some of the elements of Ys, in the same order they appear in the list Ys.

sublist([], _).
sublist([H| Rest1], [H| Rest2]) :- 
    sublist(Rest1, Rest2).
sublist(H, [_ | Rest2]) :- 
    sublist(H, Rest2).

reverse

reverse([], Acc, Acc).
reverse([X|Rest], Acc0, Acc1) :- 
    reverse(Rest, [X|Acc0], Acc1).

ints_between

ints_between(N, N, [N]).
ints_between(N0, N, [N0|List]) :-
        N0 < N,
        N1 is N0 + 1,
        ints_between(N1, N, List).
        
%  使用if else
ints_between(N0, N, List) :-
        (   N0 < N
        ->  List = [N0|List1],
            N1 is N0 + 1,
            ints_between(N1, N, List1)
        ;   N0 = N,
            List = [N]
        ).

maplist

已集成
maplist(+Pred, +List)
holds if Pred succeeds for each element of List.

?- maplist(student, [bob,claire,don]).
true.

?- maplist(enrolled(Student), [X,Y]), X\=Y.
Student = alice,
X = logic,
Y = maths ;
Student = alice,
X = maths,
Y = logic ;
Student = don,
X = logic,
Y = art_history ;
Student = don,
X = art_history,
Y = logic ;
false.

maplist(+Pred, ?List1, ?List2)
holds if Pred succeeds for corresponding elements of List1 and List2. This is similar to the Haskell map function. For example:

?- maplist(enrolled, [alice,claire,don], Subjects).
Subjects = [logic, physics, logic] ;
Subjects = [logic, physics, art_history] ;
Subjects = [maths, physics, logic] ;
Subjects = [maths, physics, art_history].

?- maplist(enrolled, Students, [logic,art_history]).
Students = [alice, don] ;
Students = [don, don].

tree

二叉搜索树相关
intset_member(N, Set) such that N is a member of integer set Set.
intset_insert(N, Set0, Set) such that Set is the same as Set0, except that Set has N as a member.

intset_member(N, tree(_,N,_)).
intset_member(N, tree(L,N0,_)) :-
	N < N0,
	intset_member(N, L).
intset_member(N, tree(_,N0,R)) :-
	N > N0,
	intset_member(N, R).

intset_insert(N, empty, tree(empty,N,empty)).
intset_insert(N, tree(L,N,R), tree(L,N,R)).
intset_insert(N, tree(L0,N0,R), tree(L,N0,R)) :-
	N < N0,
	intset_insert(N, L0, L).
intset_insert(N, tree(L,N0,R0), tree(L,N0,R)) :-
	N > N0,
	intset_insert(N, R0, R).

more tree

Given a binary tree defined to satisfy the predicate tree/1

tree(empty).
tree(node(Left,_,Right)) :-
    tree(Left),
    tree(Right).

二叉树中序遍历

tree_list(empty, []).
tree_list(node(Left,Elt,Right), List) :-
	tree_list(Left, List1),
	tree_list(Right, List2),
	append(List1, [Elt|List2], List).
    
% 尾递归版本
% implement your tree_list(Tree, List) predicate
tree_list(Tree, List) :-
	tree_list(Tree, List, []).

tree_list(empty, List, List).
tree_list(node(Left,Elt,Right), List, List0) :-
	tree_list(Left, List, List1),
	List1 = [Elt|List2],
	tree_list(Right, List2, List0).

平衡树
Write a predicate list_tree(List, Tree) that holds when Tree is a balanced tree whose elements are the elements of List, in the order they appear in List.

list_tree([], empty).
list_tree([E|List], node(Left,Elt,Right)) :-
	length(List, Len),
	Len2 is Len // 2,
	length(Front, Len2),
	append(Front, [Elt|Back], [E|List]),
	list_tree(Front, Left),
	list_tree(Back, Right).

More built-ins, cheatlist

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