Related
The following Prolog program defines a predicate rev/2 for reversing a list passed in first argument which results in the list passed in second argument:
rev([], []).
rev([XH|XT], Y) :-
rev(XT, Z),
append(Z, [XH], Y).
append([], Y, Y).
append([XH|XT], Y, [XH|ZT]) :-
append(XT, Y, ZT).
The following Prolog program is an alternative implementation of rev/2:
rev(X, Y) :-
revappend(X, [], Y).
revappend([], Y, Y).
revappend([XH|XT], Y, Z) :-
revappend(XT, [XH|Y], Z).
Both programs work as expected for queries in this argument mode:
?- rev([a, b, c], Y).
Y = [c, b, a]
; false.
But both programs exhaust resources for queries in this argument mode:
?- rev(X, [a, b, c]).
X = [c, b, a]
;
Time limit exceeded
Questions:
How to fix both programs?
Are both programs equivalent?
In both programs, the third argument has no influence on (universal) termination, as can be seen by the following failure slices:
rev([], []) :- false.
rev([XH|XT], Y) :-
rev(XT, Z), false,
append(Z, [XH], Y).
Y can be whatever it wants, it will never cause failure in this fragment. Thus it has no influence on termination. It is termination neutral.
rev(X, Y) :-
revappend(X, [], Y), false.
revappend([], Y, Y) :- false.
revappend([XH|XT], Y, Z) :-
revappend(XT, [XH|Y], Z), false.
Similarly, the third argument in revappend/3 is just handed over without any chance of causing failure and thus termination.
In order to fix the problem something has to be added to specialize the remaining visible part. One observation is that the list length of both the first and the last argument is the same. And thus adding an extra fourth argument for ensuring that both arguments are of same length will help to get the optimal termination condition:
rev(X, Y) :-
revappend(X, [], Y, Y).
revappend([], Y, Y, []).
revappend([XH|XT], Y, Z, [_|Ylen]) :-
revappend(XT, [XH|Y], Z, Ylen).
And here is a generalization of this program to better understand how termination is influenced by the arguments:
rev(X, Y) :-
revappend(X, _, _, Y).
revappend([], _, _, []).
revappend([_|XT], _, _, [_|Ylen]) :-
revappend(XT, _, _, Ylen).
So the 2nd and 3rd argument is just replaced by _. This generalization is now exactly same_length/2.
I've started to learn Prolog recently and I can't solve how to make union of three lists.
I was able to make union of 2 lists :
%element
element(X,[X|_]).
element(X,[_|Y]):-
element(X,Y).
%union
union([],M,M).
union([X|Y],L,S) :- element(X,L),union(Y,L,S).
union([X|Y],L,[X|S]) :- (not(element(X,L))),union(Y,L,S).
can anybody help me please ?
union(A, B, C, U) :-
union(A, B, V),
union(C, V, U).
Your definition of union/3 can be improved by replacing
... not(element(X,L)), ...
by
... maplist(dif(X),L), ...
or
... non_member(X, L), ....
non_member(_X, []).
non_member(X, [E|Es]) :-
dif(X, E),
non_member(X, Es).
Here is a case where the difference shows:
?- union([A],[B],[C,D]).
A = C, B = D, dif(C, D).
How must [A] and [B] look like such that their union contains 2 elements?
The answer is: they must be different.
Your original version fails for this query, yet, it succeeds for a specialized instance like:
?- A = 1, B = 2, union([A],[B],[C,D]).
So it succeeds for this, but fails for a generalization of it. Therefore it is not a pure, logical relation.
So is everything fine and perfect with dif/2? Unfortunately not. #TudorBerariu has good reason to go for a cut, since it reflects some of the intention we have about the relation. The cut effectively reflects two key intentions
that the alternative of not being a member is now excluded, which is true for certain modes, like Arg1 and Arg2 being both sufficiently instantiated terms. A safe approximation would be ground terms.
that there is no need to look at further elements in the list Arg2, which again is only true if Arg1 and Arg2 are sufficiently instantiated.
Problems only show when terms are not sufficiently instantiated..
The drawback of OP's definition and the one above, is that both are unnecessarily too general which can be observed with repeated elements in Arg2:
?- union([a,a],[a,a],Zs).
Zs = [a, a]
; Zs = [a, a]
; Zs = [a, a]
; Zs = [a, a]
; false.
In fact, we get |Arg2||Arg1|-1 redundant answers. So the cut had some good reason to be there.
Another reason why union/3 as it stands is not very efficient is that for the (intended) ground case it leaves open unnecessary choice points. Again, #TudorBerariu's solution does not have this problem:
?- union([a],[a],Zs).
Zs = [a]
; false. % <--- Prolog does not know that there is nothing left
Eliminating redundancy
The actual culprit for that many redundant answers is the first rule. element(a,[a,a]) (commonly called member/2) will succeed twice.
union([X|Y],L,S) :- element(X,L), union(Y,L,S).
^^^^^^^^^^^^
Here is an improved definition:
memberd(X, [X|_Ys]).
memberd(X, [Y|Ys]) :-
dif(X,Y), % new!
memberd(X, Ys).
The recursive rule, reading it right-to-left, reads as follows:
Assume memberd(X, Ys) is true already for some X and Ys. Given that, and given that we have a fitting Y which is different from X. Thenwe can conclude that also memberd(X, [Y|Ys]) is true.
So this has eliminated the redundant solutions. But our definition is still not very efficient: it still has to visit Arg2 twice for each element, and then it is unable to conclude that no alternatives are left. In any case: resist to place a cut to remove this.
Introducing determinism via reification.
Compare the definitions of memberd/2 and non_member/2. Although they describe "the opposite" of each other, they look very similar:
non_member(_X, []).
non_member(X, [Y|Ys]) :-
dif(X,Y),
non_member(X, Ys).
memberd(X, [X|_Ys]).
memberd(X, [Y|Ys]) :-
dif(X,Y),
memberd(X, Ys).
The recursive rule is the same! Only the fact is a different one. Let's merge them into one definition - with an additional argument telling whether we mean memberd (true) or non_member (false):
memberd_t(_X, [], false).
memberd_t(X, [X|_Ys], true).
memberd_t(X, [Y|Ys], Truth) :-
dif(X, Y),
memberd_t(X, Ys, Truth).
Now, our definition gets a bit more compact:
unionp([], Ys, Ys).
unionp([X|Xs], Ys, Zs0) :-
if_( memberd_t(X, Ys), Zs0 = Zs, Zs0 = [X|Zs] ),
unionp(Xs, Ys, Zs).
memberd_t(_X, [], false). % see below
memberd_t(X, [Y|Ys], Truth) :-
if_( X = Y, Truth=true, memberd_t(X, Ys, Truth) ).
Note the difference between if_(If_1, Then_0, Else_0) and the if-then-else control construct ( If_0 -> Then_0 ; Else_0 ). While If_1 may succeed several times with different truth values (that is, it can be both true and false), the control construct makes If_0 succeed only once for being true only.
if_(If_1, Then_0, Else_0) :-
call(If_1, T),
( T == true -> call(Then_0)
; T == false -> call(Else_0)
; nonvar(T) -> throw(error(type_error(boolean,T),_))
; /* var(T) */ throw(error(instantiation_error,_))
).
=(X, Y, T) :-
( X == Y -> T = true
; X \= Y -> T = false
; T = true, X = Y
; T = false,
dif(X, Y) % ISO extension
% throw(error(instantiation_error,_)) % ISO strict
).
equal_t(X, Y, T) :-
=(X, Y, T).
To ensure that memberd_t/3 will always profit from first-argument indexing, rather use the following definition (thanks to #WillNess):
memberd_t(E, Xs, T) :-
i_memberd_t(Xs, E, T).
i_memberd_t([], _E, false).
i_memberd_t([X|Xs], E, T) :-
if_( X = E, T = true, i_memberd_t(Xs, E, T) ).
You can make the union of the first two lists and then the union between that result and the third:
union(L1, L2, L3, U):-union(L1, L2, U12), union(U12, L3, U).
You can improve union/3 with a cut operator:
union([],M,M).
union([X|Y],L,S) :- element(X,L), !, union(Y,L,S).
union([X|Y],L,[X|S]) :- union(Y,L,S).
Using only predicates with an extra argument such as memberd_t/3 leads only to weak reification. For strong reification we also need to generate constraints. Strong reification is a further approach to eliminate non-determinism.
But strong reification is difficult, a possible way to archive this is to use a CLP(*) instance which has also reified logical operators. Here is an example if using CLP(FD) for the union problem. Unfortunately this covers only the domain Z:
Strong Reification Code:
member(_, [], 0).
member(X, [Y|Z], B) :-
(X #= Y) #\/ C #<==> B,
member(X, Z, C).
union([], X, X).
union([X|Y], Z, T) :-
freeze(B, (B==1 -> T=R; T=[X|R])),
member(X, Z, B),
union(Y, Z, R).
The above doesn't suffer from unnecessary choice points. Here are some example that show that this isn't happening anymore:
Running a Ground Example:
?- union([1,2],[2,3],X).
X = [1, 2, 3].
Also the above example even doesn't create choice points, if we use variables somewhere. But we might see a lot of constraints:
Running a Non-Ground Example:
?- union([1,X],[X,3],Y).
X#=3#<==>_G316,
1#=X#<==>_G322,
_G316 in 0..1,
freeze(_G322, (_G322==1->Y=[X, 3];Y=[1, X, 3])),
_G322 in 0..1.
?- union([1,X],[X,3],Y), X=2.
X = 2,
Y = [1, 2, 3].
Since we didn't formulate some input invariants, the interpreter isn't able to see that producing constraints in the above case doesn't make any sense. We can use the all_different/1 constraint to help the interpreter a little bit:
Providing Invariants:
?- all_different([1,X]), all_different([X,3]), union([1,X],[X,3],Y).
Y = [1, X, 3],
X in inf..0\/2\/4..sup,
all_different([X, 3]),
all_different([1, X]).
But we shouldn't expect too much from this singular example. Since the CLP(FD) and the freeze/2 is only an incomplete decision procedure for propositions and Z equations, the approach might not work as smooth as here in every situation.
Bye
I need to implement y = 1/x on a list of numbers.
I.e.
inv (List1, ResultingList).
inv ([2 , 1 , 0 , 0.25 ] , R).
R = [ 0.5, 1, inf, 4.0 ].
I've try with a recursive function but it doesn't work.
this is my "solution":
inv([], []).
inv(list, R):- list == [H|T], T \== [], Y is (1/H),
append(R, Y, R), inv(T);
list = [H|T], T == [], R = T;
list = [H|T], H == [0], append(R, "inf", R).
I know that there are a lot of mistakes but I don't find the way.
Variable names start with a capital letter, and (==)/2 is only used for very rare cases. Further append/3 is rarely used for such predicates. Briefly:
inv([], []).
inv([X|Xs], [Y|Ys]) :-
Y is 1/X,
inv(Xs, Ys).
Or:
reciprocal(X, Y) :-
Y is 1/X.
inv(Xs, Ys) :-
maplist(reciprocal, Xs, Ys)
or using library(lambda)
inv(Xs, Ys) :-
maplist(\X^Y^(Y is 1/X), Xs, Ys).
note that most commonly maplist/3 is called directly without the auxiliary definition.
There is, however, no standard for the usage of a continuation value inf. In case you really need it, you have have to make that extra handling yourself.
How can I replace a list with another list that contain the variable to be replaced. for example
rep([x, d, e, z, x, z, p], [x=z, z=x, d=c], R).
R = [z, c, e, x, z, x, p]
the x to z and z doesn't change after it has been replaced.
so far I did only the one without the list
rep([], _, []).
rep(L1, H1=H2, L2) :-
rep(L1, H1, H2, L2).
rep([],_,_,[]).
rep([H|T], X1, X2, [X2|L]) :-
H=X1,
rep(T,X1,X2,L),
!.
rep([H|T],X1,X2,[H|L]) :-
rep(T,X1,X2,L).
If you use SWI-Prolog, with module lambda.pl found there : http://www.complang.tuwien.ac.at/ulrich/Prolog-inedit/lambda.pl you can write :
:- use_module(library(lambda)).
rep(L, Rep, New_L) :-
maplist(\X^Y^(member(X=Z, Rep)
-> Y = Z
; Y = X), L, New_L).
You should attempt to keep the code simpler than possible:
rep([], _, []).
rep([X|Xs], Vs, [Y|Ys]) :-
( memberchk(X=V, Vs) -> Y = V ; Y = X ),
rep(Xs, Vs, Ys).
Of course, note the idiomatic way (thru memberchk/2) to check for a variable value.
Still yet a more idiomatic way to do: transforming lists it's a basic building block in several languages, and Prolog is no exception:
rep(Xs, Vs, Ys) :- maplist(repv(Vs), Xs, Ys).
repv(Vs, X, Y) :- memberchk(X=V, Vs) -> Y = V ; Y = X .
Here's how you could proceed using if_/3 and (=)/3.
First, we try to find a single Key in a list of pairs K-V.
An extra argument reifies search success.
pairs_key_firstvalue_t([] ,_ ,_ ,false).
pairs_key_firstvalue_t([K-V|KVs],Key,Value,Truth) :-
if_(K=Key,
(V=Value, Truth=true),
pairs_key_firstvalue_t(KVs,Key,Value,Truth)).
Next, we need to handle "not found" cases:
assoc_key_mapped(Assoc,Key,Value) :-
if_(pairs_key_firstvalue_t(Assoc,Key,Value),
true,
Key=Value).
Last, we put it all together using the meta-predicate maplist/3:
?- maplist(assoc_key_mapped([x-z,z-x,d-c]), [x,d,e,z,a,z,p], Rs).
Rs = [z,c,e,x,a,x,p]. % OK, succeeds deterministically
Let's improve this answer by moving the "recursive part" into meta-predicate find_first_in_t/4:
:- meta_predicate find_first_in_t(2,?,?,?).
find_first_in_t(P_2,X,Xs,Truth) :-
list_first_suchthat_t(Xs,X,P_2,Truth).
list_first_suchthat_t([] ,_, _ ,false).
list_first_suchthat_t([E|Es],X,P_2,Truth) :-
if_(call(P_2,E),
(E=X,Truth=true),
list_first_suchthat_t(Es,X,P_2,Truth)).
To fill in the "missing bits and pieces", we define key_pair_t/3:
key_pair_t(Key,K-_,Truth) :-
=(Key,K,Truth).
Based on find_first_in_t/4 and key_pair_t/3, we can write assoc_key_mapped/3 like this:
assoc_key_mapped(Assoc,Key,Value) :-
if_(find_first_in_t(key_pair_t(Key),_-Value,Assoc),
true,
Key=Value).
So, does the OP's use-case still work?
?- maplist(assoc_key_mapped([x-z,z-x,d-c]), [x,d,e,z,a,z,p], Rs).
Rs = [z,c,e,x,a,x,p]. % OK. same result as before
Building on find_first_in_t/4
memberd_t(X,Xs,Truth) :- % memberd_t/3
find_first_in_t(=(X),_,Xs,Truth).
:- meta_predicate exists_in_t(2,?,?). % exists_in_t/3
exists_in_t(P_2,Xs,Truth) :-
find_first_in_t(P_2,_,Xs,Truth).
I find your code rather confused. For one thing, you have rep/3 and rep/4, but none of them have a list in the second position where you're passing the list of variable bindings. H1=H2 cannot possibly match a list, and that's the only rep/3 clause that examines the second argument. If this is a class assignment, it looks like you're a little bit behind and I'd suggest you spend some time on the previous material.
The solution is simpler than you'd think:
rep([], _, []).
rep([X|Xs], Vars, [Y|Rest]) :- member(X=Y, Vars), rep(Xs, Vars, Rest).
rep([X|Xs], Vars, [X|Rest]) :- \+ member(X=_, Vars), rep(Xs, Vars, Rest).
We're using member/2 to find a "variable binding" in the list (in quotes because these are atoms and not true Prolog variables). If it's in the list, Y is the replacement, otherwise we keep using X. And you see this has the desired effect:
?- rep([x, d, e, z, x, z, p], [x=z, z=x, d=c], R).
R = [z, c, e, x, z, x, p] ;
false.
This could be made somewhat more efficient using "or" directly (and save us a choice point):
rep([], _, []).
rep([X|Xs], Vars, [Y|Ys]) :-
(member(X=Y, Vars), ! ; X=Y),
rep(Xs, Vars, Ys).
See:
?- rep([x, d, e, z, x, z, p], [x=z, z=x, d=c], R).
R = [z, c, e, x, z, x, p].
I am completely new to Prolog and trying some exercises. One of them is:
Write a predicate set(InList,OutList)
which takes as input an arbitrary
list, and returns a list in which each
element of the input list appears only
once.
Here is my solution:
member(X,[X|_]).
member(X,[_|T]) :- member(X,T).
set([],[]).
set([H|T],[H|Out]) :-
not(member(H,T)),
set(T,Out).
set([H|T],Out) :-
member(H,T),
set(T,Out).
I'm not allowed to use any of built-in predicates (It would be better even do not use not/1). The problem is, that set/2 gives multiple same solutions. The more repetitions in the input list, the more solutions will result. What am I doing wrong? Thanks in advance.
You are getting multiple solutions due to Prolog's backtracking. Technically, each solution provided is correct, which is why it is being generated. If you want just one solution to be generated, you are going to have to stop backtracking at some point. This is what the Prolog cut is used for. You might find that reading up on that will help you with this problem.
Update: Right. Your member() predicate is evaluating as true in several different ways if the first variable is in multiple positions in the second variable.
I've used the name mymember() for this predicate, so as not to conflict with GNU Prolog's builtin member() predicate. My knowledge base now looks like this:
mymember(X,[X|_]).
mymember(X,[_|T]) :- mymember(X,T).
not(A) :- \+ call(A).
set([],[]).
set([H|T],[H|Out]) :-
not(mymember(H,T)),
set(T,Out).
set([H|T],Out) :-
mymember(H,T),
set(T,Out).
So, mymember(1, [1, 1, 1]). evaluates as true in three different ways:
| ?- mymember(1, [1, 1, 1]).
true ? a
true
true
no
If you want to have only one answer, you're going to have to use a cut. Changing the first definition of mymember() to this:
mymember(X,[X|_]) :- !.
Solves your problem.
Furthermore, you can avoid not() altogether, if you wish, by defining a notamember() predicate yourself. The choice is yours.
A simpler (and likely faster) solution is to use library predicate sort/2 which remove duplicates in O(n log n). Definitely works in Yap prolog and SWIPL
You are on the right track... Stay pure---it's easy!
Use reified equality predicates =/3 and dif/3 in combination with if_/3, as implemented in Prolog union for A U B U C:
=(X, Y, R) :- X == Y, !, R = true.
=(X, Y, R) :- ?=(X, Y), !, R = false. % syntactically different
=(X, Y, R) :- X \= Y, !, R = false. % semantically different
=(X, Y, R) :- R == true, !, X = Y.
=(X, X, true).
=(X, Y, false) :-
dif(X, Y).
% dif/3 is defined like (=)/3
dif(X, Y, R) :- X == Y, !, R = false.
dif(X, Y, R) :- ?=(X, Y), !, R = true. % syntactically different
dif(X, Y, R) :- X \= Y, !, R = true. % semantically different
dif(X, Y, R) :- R == true, !, X \= Y.
dif(X, Y, true) :- % succeed first!
dif(X, Y).
dif(X, X, false).
if_(C_1, Then_0, Else_0) :-
call(C_1, Truth),
functor(Truth,_,0), % safety check
( Truth == true -> Then_0 ; Truth == false, Else_0 ).
Based on these predicates we build a reified membership predicate list_item_isMember/3. It is semantically equivalent with memberd_truth/3 by #false. We rearrange the argument order so the list is the 1st argument. This enables first-argument indexing which prevents leaving useless choice-points behind as memberd_truth/3 would create.
list_item_isMember([],_,false).
list_item_isMember([X|Xs],E,Truth) :-
if_(E = X, Truth = true, list_item_isMember(Xs,E,Truth)).
list_set([],[]).
list_set([X|Xs],Ys) :-
if_(list_item_isMember(Xs,X), Ys = Ys0, Ys = [X|Ys0]),
list_set(Xs,Ys0).
A simple query shows that all redundant answers have been eliminated and that the goal succeeds without leaving any choice-points behind:
?- list_set([1,2,3,4,1,2,3,4,1,2,3,1,2,1],Xs).
Xs = [4,3,2,1]. % succeeds deterministically
Edit 2015-04-23
I was inspired by #Ludwig's answer of set/2, which goes like this:
set([],[]).
set([H|T],[H|T1]) :- subtract(T,[H],T2), set(T2,T1).
SWI-Prolog's builtin predicate subtract/3 can be non-monotone, which may restrict its use. list_item_subtracted/3 is a monotone variant of it:
list_item_subtracted([],_,[]).
list_item_subtracted([A|As],E,Bs1) :-
if_(dif(A,E), Bs1 = [A|Bs], Bs = Bs1),
list_item_subtracted(As,E,Bs).
list_setB/2 is like set/2, but is based on list_item_subtracted/3---not subtract/3:
list_setB([],[]).
list_setB([X|Xs1],[X|Ys]) :-
list_item_subtracted(Xs1,X,Xs),
list_setB(Xs,Ys).
The following queries compare list_set/2 and list_setB/2:
?- list_set([1,2,3,4,1,2,3,4,1,2,3,1,2,1], Xs).
Xs = [4,3,2,1]. % succeeds deterministically
?- list_setB([1,2,3,4,1,2,3,4,1,2,3,1,2,1],Xs).
Xs = [1,2,3,4]. % succeeds deterministically
?- list_set(Xs,[a,b]).
Xs = [a,b]
; Xs = [a,b,b]
; Xs = [a,b,b,b]
... % does not terminate universally
?- list_setB(Xs,[a,b]).
Xs = [a,b]
; Xs = [a,b,b]
; Xs = [a,b,b,b]
... % does not terminate universally
I think that a better way to do this would be:
set([], []).
set([H|T], [H|T1]) :- subtract(T, [H], T2), set(T2, T1).
So, for example ?- set([1,4,1,1,3,4],S) give you as output:
S = [1, 4, 3]
Adding my answer to this old thread:
notmember(_,[]).
notmember(X,[H|T]):-X\=H,notmember(X,T).
set([],[]).
set([H|T],S):-set(T,S),member(H,S).
set([H|T],[H|S]):-set(T,S),not(member(H,S)).
The only virtue of this solution is that it uses only those predicates that have been introduced by the point where this exercise appears in the original text.
This works without cut, but it needs more lines and another argument.
If I change the [H2|T2] to S on line three, it will produce multiple results. I don't understand why.
setb([],[],_).
setb([H|T],[H|T2],A) :- not(member(H,A)),setb(T,T2,[H|A]).
setb([H|T],[H2|T2],A) :- member(H,A),setb(T,[H2|T2],A).
setb([H|T],[],A) :- member(H,A),setb(T,[],A).
set(L,S) :- setb(L,S,[]).
You just have to stop the backtracking of Prolog.
enter code here
member(X,[X|_]):- !.
member(X,[_|T]) :- member(X,T).
set([],[]).
set([H|T],[H|Out]) :-
not(member(H,T)),
!,
set(T,Out).
set([H|T],Out) :-
member(H,T),
set(T,Out).
Using the support function mymember of Tim, you can do this if the order of elements in the set isn't important:
mymember(X,[X|_]).
mymember(X,[_|T]) :- mymember(X,T).
mkset([],[]).
mkset([T|C], S) :- mymember(T,C),!, mkset(C,S).
mkset([T|C], S) :- mkset(C,Z), S=[T|Z].
So, for example ?- mkset([1,4,1,1,3,4],S) give you as output:
S = [1, 3, 4]
but, if you want a set with the elements ordered like in the list you can use:
mkset2([],[], _).
mkset2([T|C], S, D) :- mkset2(C,Z,[T|D]), ((mymember(T,D), S=Z,!) ; S=[T|Z]).
mkset(L, S) :- mkset2(L,S,[]).
This solution, with the same input of the previous example, give to you:
S = [1, 4, 3]
This time the elements are in the same order as they appear in the input list.
/* Remove duplicates from a list without accumulator */
our_member(A,[A|Rest]).
our_member(A, [_|Rest]):-
our_member(A, Rest).
remove_dup([],[]):-!.
remove_dup([X|Rest],L):-
our_member(X,Rest),!,
remove_dup(Rest,L).
remove_dup([X|Rest],[X|L]):-
remove_dup(Rest,L).