Related
I'd like to test whether a term has only one solution.
(Understanding that this might be done in different ways) I've done the following and would like to understand why it doesn't work, if it can be made to work, and if not, what the appropriate implementation would be.
First, I have an "implies" operator (that has seemed to work elsewhere):
:- op(1050,xfy,'==>').
'==>'(A,B) :-·forall(call(A), call(B)).
next I have my singleSolution predicate:
singleSolution(G) :- copy_term(G,G2), (call(G), call(G2)) ==> (G = G2).
Here I'm trying to say: take a term G and make a copy of it, so I can solve them independently. Now if solving both independently implies they are equal, then there must be only one solution.
This works in some simple cases.
BUT.
I have a predicate foo(X,Y,Z) (too large to share) which solves things properly, and for which singleSolution can answer correctly. However, X,Y,Z are not fully ground after singleSolution(foo(X,Y,Z)) is called, even though they would be after directly calling foo(X,Y,Z).
I don't understand that. (As a sanity test: I've verified that I get the same results under swi-prolog and gprolog.)
EDIT: Here is an example of where this fails.
increasing([]).
increasing([_]).
increasing([X,Y|T]) :- X < Y, increasing([Y|T]).
increasingSublist(LL,L) :-·
sublist(L,LL),
length(L, Len),
Len > 1,
increasing(L).
then
| ?- findall(L, singleSolution(increasingSublist([1,2],L)),R).
R = [_]
yes
But we don't know what L is.
This seems to work, but I'm not sure if it's logically sound :)
It uses call_nth/2, a nonstandard but common predicate. It abuses throw to short-circuit the computation. Using bagof/3 instead of findall/3 lets us keep the Goal argument bound (and it will fail where findall/3 would succeed if it finds 0 solutions).
only_once(Goal) :-
catch(bagof(_, only_once_(Goal), _), too_many, fail).
only_once_(Goal) :-
call_nth(Goal, N),
( N > 1
-> throw(too_many)
; true
).
Testing it (on SWI):
?- only_once(member(X, [1])).
X = 1.
?- only_once(member(a, [a, b])).
true.
?- only_once(member(X, [a, b])).
false.
?- only_once(between(1,inf,X)).
false.
Unfortunately, I don't think call_nth/2 is supported in GNU Prolog.
Another possible solution:
single_solution(G) :-
copy_term(G, H),
call(G),
!,
( ground(H)
-> true
; \+ ( call(H), G \= H ) % There is no H different from G
).
p(a).
p(a).
q(b).
q(c).
Examples:
?- single_solution( p(X) ).
X = a.
?- single_solution( q(X) ).
false.
?- single_solution( member(X, [a,a,a]) ).
X = a.
?- single_solution( member(X, [a,b,c]) ).
false.
?- single_solution( repeat ).
true.
?- single_solution( between(1,inf,X) ).
false.
?- single_solution( between(1,inf,5) ).
true.
Here is an another approach I came up with after #gusbro commented that forall/2 doesn't bind variables from the calling goal.
single_solution(G) :-·
% duplicate the goal so we can solve independently
copy_term(G,G2),
% solve the first goal at least / at most once.
G, !,
% can we solve the duplicate differently?
% if so, cut & fail. Otherwise, succeed.
(G2, G2 \= G, !, fail; true).
I'm trying to trigger backtracking on a goal but in a dynamic way, if it's possible. To better exemplify my issue let's say we have the following PROLOG code:
num(1).
num(2).
num(3).
num(4).
num(5).
Then I head to SWI-Prolog and call: num(X). This triggers backtracking looking for all solutions, by typing ; .
What I would like is to remove those facts (num(1),num(2), etc) and replace that code with something thata generates those facts dynamically. Is there any way in which I can achieve this? Someting of the sorts,maybe?
num(X):- for X in 1..5
that yields the same solutions as the code above?
As far as I know, the findall predicate returns a list, which is not what I'm looking for. I would like to backtrack through all answers and look through them using ; in the console.
Yes there is, and you were already very close!
:- use_module(library(clpfd)).
num(X) :-
X in 1..5.
?- num(X).
X in 1..5.
?- num(X), X #>3.
X in 4..5.
?- num(X), labeling([], [X]).
X = 1
; X = 2
; X = 3
; X = 4
; X = 5.
SWI-Prolog has the (non-ISO) predicate between/3 for that:
num(X) :- between(1, 5, X).
You can implement the predicate (for other Prologs and for further tweaking) like this:
between2(A, A, A) :- !. % green cut
between2(A, B, A) :- A < B.
between2(A, B, C) :-
A < B,
A1 is A + 1,
between2(A1, B, C).
The signature for both between/3 and between2/3 is (+From,+To,?X). It means that the From and To must be bound and X can be either bound or not. Also note that From and To must be integers such that From <= To. (Oh, and these integers must be written using Arabic numerals with an optional plus or minus sign before. And using ASCII. Is something non-obvious still missed? And the integers must not be too large or too small, although SWI-Prolog is usually compiled with unbounded integer support, so both between(1, 100000000000000000000000000000000000000000000, X) and between2(1, 100000000000000000000000000000000000000000000, X) usually work.)
counter([],[]).
counter([H|T],[[H,C1]|R]) :- counter(T,[[H,C]|R]),!, C1 is C+1.
counter([H|T],[[H,1]|R]) :- counter(T,R).
What is the effect of the "!" as I'm getting the same output for an input in both the above and below code?
counter([],[]).
counter([H|T],[[H,C1]|R]) :- counter(T,[[H,C]|R]),C1 is C+1.
counter([H|T],[[H,1]|R]) :- counter(T,R).
I'm new to Prolog.
What is the effect of the "!"
The cut prunes the search space. That is, in an otherwise pure and monotonic program, the cut will remove some solutions or answers. As long as those are redundant that's fine. It sounds so innocent and useful, doesn't it? Let's have a look!
And lest I forget, using [E,Nr] to denote pairs is rather unusual, better use a pair E-Nr.
We will now compare counter_cut/2 and counter_sans/2.
| ?- counter_cut([a,a],Xs).
Xs = [[a,2]].
| ?- counter_sans([a,a],Xs).
Xs = [[a, 2]]
; Xs = [[a, 1], [a, 1]]. % <<< surprise !!!
So the cut-version has fewer solutions. Seems the solution counter_cut/2 retained is the right one. In this very particular case. Will it always take the right one? I will try a minimally more general query:
| ?- counter_cut([a,B],Xs).
B = a,
Xs = [[a, 2]].
| ?- counter_sans([a,B],Xs).
B = a,
Xs = [[a, 2]]
; Xs = [[a, 1], [B, 1]].
Again, _sans is chattier, and this time, it is even a bit right-er; for the last answer includes B = b. In other words,
| ?- counter_cut([a,B], Xs), B = b.
fails. % incomplete !
| ?- counter_sans([a,B], Xs), B = b.
B = b,
Xs = [[a,1],[b,1]].
So sometimes the _cut version is better, and sometimes _sans. Or to put more directly: Both are wrong somehow, but the _sans-version at least includes all solutions.
Here is a "purified" version, that simply rewrites the last rule into two different cases: One for the end of the list and the other for a further, different element.
counter_pure([],[]).
counter_pure([H|T],[[H,C1]|R]) :- counter_pure(T,[[H,C]|R]), C1 is C+1.
counter_pure([H],[[H,1]]).
counter_pure([H,D|T],[[H,1]|R]) :- dif(H,D), counter_pure([D|T],R).
From an efficiency viewpoint that is not too famous.
Here is a test case for efficiency for a system with rational tree unification:
?- Es = [e|Es], counter(Es, Dict).
resource_error(stack).
Instead, the implementation should loop smoothly, at least till the end of this universe. Strictly speaking, that query has to produce a resource error, but only after it has counted up to a number much larger than 10^100000000.
Here's my pure and hopefully efficient solution:
counter([X|L], C):- counter(L, X, 1, C).
counter([],X, Cnt, [[X,Cnt]]).
counter([Y|L], X, Cnt, [[X,Cnt]|C]):-
dif(X, Y),
counter(L, Y, 1, C).
counter([X|L],X, Cnt, [[X,XCnt]|C]):-
Cnt1 #= Cnt+1,
Cnt1 #=< XCnt,
counter(L, X, Cnt1, [[X,XCnt]|C]).
Using if_3 as suggested by #false:
counter([X|L], C):- counter(L, X, 1, C).
counter([],X, Cnt, [[X,Cnt]]).
counter([Y|L], X, Cnt, [[X,XCnt]|C]):-
if_(X=Y,
(
Cnt1 #= Cnt+1,
Cnt1 #=< XCnt,
counter(L, X, Cnt1, [[X,XCnt]|C])
),
(
XCnt=Cnt,
counter(L, Y, 1, C)
)
).
The cut operator ! commits to the current derivation path by pruning all choice points. Given some facts
fact(a).
fact(b).
you can compare the answers with and without cut:
?- fact(X).
X = a ;
X = b.
?- fact(X), !.
X = a.
As you can see, the general query now only reports its first success. Still, the query
?- fact(b), !.
true.
succeeds. This means, that cut violates the interpretation of , as logical conjunction:
?- X = b, fact(X), !.
X = b.
?- fact(X), !, X=b.
false.
but from our understanding of conjunction, A ∧ B should hold exactly when B ∧ A holds. So why do this at all?
Efficiency: cuts can be used such that they only change execution properties but not the answers of a predicate. These so called green cuts are for instance described in Richard O'Keefe's Craft of Prolog. As demonstrated above, maintaining correctness of a predicate with cut is much harder than one without, but obviously, correctness should come before efficiency.
It looks as if your problem was green, but I am not 100% sure if there is not a change in the answers.
Negation: logical negation according to the closed world assumption is expressed with cut. You can define neg(X) as:
neg(X) :-
call(X),
!,
false.
neg(_) :-
true.
So if call(X) succeeds, we cut the choice point for the second rule away and derive false. Otherwise, nothing is cut and we derive true. Please be aware that this is not negation in classical logic and that it suffers from the non-logical effects of cut. Suppose you define the predicate land/1 to be one of the continents:
land(africa).
land(america).
land(antarctica).
land(asia).
land(australia).
land(europe).
and then define water as everything not on land:
water(X) :-
neg(land(X)).
then you can correctly obtain:
?- water(pacific).
true.
?- water(africa).
false.
But you can also derive:
?- water(space).
true.
which should not hold. In particular, in classical logic:
land(africa) ∧
land(america) ∧
land(antarctica) ∧
land(asia) ∧
land(australia) ∧
land(europe) → ¬ land(space).
is not valid. Again, you should know well what you are doing if you use negation in Prolog.
Here is my attempt using if_/3:
counter([], []).
counter([H|T], [[H,C]|OutT] ):-
if_(
T=[],
(C = 1,OutT=[]),
(
[H|T] = [H,H1|T2],
if_(
H=H1,
(counter([H1|T2], [[H1,C1]|OutT]), C is C1+1),
(C = 1, counter([H1|T2], OutT))
)
)
).
I have two, slightly different, implementations of a predicate, unique_element/2, in Prolog. The predicate succeeds when given an element X and a list L, the element X appears only once in the list. Below are the implementations and the results:
Implementation 1:
%%% unique_element/2
unique_element(Elem, [Elem|T]) :-
not(member(Elem, T)).
unique_element(Elem, [H|T]) :-
member(Elem, T),
H\==Elem,
unique_element(Elem, T),
!.
Results:
?- unique_element(X, [a, a, b, c, c, b]).
false.
?- unique_element(X, [a, b, c, c, b, d]).
X = a ;
X = d.
Implementation 2:
%%% unique_element/2
unique_element(Elem, [Elem|T]) :-
not(member(Elem, T)).
unique_element(Elem, [H|T]) :-
H\==Elem,
member(Elem, T),
unique_element(Elem, T),
!.
In case you didn't notice at first sight: H\==Elem and member(Elem, T) are flipped on the 2nd impl, rule 2.
Results:
?- unique_element(X, [a, a, b, c, c, b]).
X = a.
?- unique_element(X, [a, b, c, c, b, d]).
X = a ;
X = d.
Question: How does the order, in this case, affect the result? I realize that the order of the rules/facts/etc matters. The two specific rules that are flipped though, don't seem to be "connected" or affect each other somehow (e.g. a cut in the wrong place/order).
Note: We are talking about SWI-Prolog here.
Note 2: I am aware of, probably different and better implementations. My question here is about the order of sub-goals being changed.
H\==Elem is testing for syntactic inequality at the point in time when the goal is executed. But later unification might make variables identical:
?- H\==Elem, H = Elem.
H = Elem.
?- H\==Elem, H = Elem, H\==Elem.
false.
So here we test if they are (syntactically) different, and then they are unified nevertheless and thus are no longer different. It is thus just a temporary test.
The goal member(Elem, T) on the other hand is true if that Elem is actually an element of T. Consider:
?- member(Elem, [X]).
Elem = X.
Which can be read as
(When) does it hold that Elem is an element of the list [X]?
and the answer is
It holds under certain circumstances, namely when Elem = X.
If you now mix those different kinds of goals in your programs you get odd results that can only explained by inspecting your program in detail.
As a beginner, it is best to stick to the pure parts of Prolog only. In your case:
use dif/2 in place of \==
do not use cuts - in your case it limits the number of answers to two. As in
unique_element(X, [a,b,c])
do not use not/1 nor (\+)/1. It produces even more incorrectness. Consider unique_element(a,[a,X]),X=b. which incorrectly fails while X=b,unique_element(a,[a,X]) correctly succeeds.
Here is a directly purified version of your program. There is still room for improvement!
non_member(_X, []).
non_member(X, [E|Es]) :-
dif(X, E),
non_member(X, Es).
unique_element(Elem, [Elem|T]) :-
non_member(Elem, T).
unique_element(Elem, [H|T]) :-
dif(H,Elem),
% member(Elem, T), % makes unique_element(a,[b,a,a|Xs]) loop
unique_element(Elem, T).
?- unique_element(a,[a,X]).
dif(X, a)
; false. % superfluous
?- unique_element(X,[E1,E2,E3]).
X = E1, dif(E1, E3), dif(E1, E2)
; X = E2, dif(E2, E3), dif(E1, E2)
; X = E3, dif(E2, E3), dif(E1, E3)
; false.
Note how the last query reads?
When is X a unique element of (any) list [E1,E2,E3]?
The answer is threefold. Considering one element after the other:
X is E1 but only if it is different to E2 and E3
etc.
TL;DR: Read the documentation and figure out why:
?- X = a, X \== a.
false.
?- X \== a, X = a.
X = a.
I wonder why you stop so close from figuring it out yourself ;-)
There are too many ways to compare things in Prolog. At the very least, you have unification, which sometimes can compare, and sometimes does more; than you have equvalence, and its negation, the one you are using. So what does it do:
?- a \== b. % two different ground terms
true.
?- a \== a. % the same ground term
false.
Now it gets interesting:
?- X \== a. % a free variable and a ground term
true.
?- X \== X. % the same free variable
false.
?- X \== Y. % two different free variables
true.
I would suggest that you do the following: figure out how member/2 does its thing (does it use unification? equivalence? something else?) then replace whatever member/2 is using in all the examples above and see if the results are any different.
And since you are trying to make sure that things are different, try out what dif/2 does. As in:
?- dif(a, b).
or
?- dif(X, X).
or
?- dif(X, a).
and so on.
See also this question and answers: I think the answers are relevant to your question.
Hope that helps.
Here is another possibility do define unique_element/2 using if_/3 and maplist/2:
:- use_module(library(apply)).
unique_element(Y,[X|Xs]) :-
if_(Y=X,maplist(dif(Y),Xs),unique_element(Y,Xs)).
In contrast to #user27815's very elegant solution (+s(0)) this version does not build on clpfd (used by tcount/3). The example queries given by the OP work as expected:
?- unique_element(a,[a, a, b, c, c, b]).
no
?- unique_element(X,[a, b, c, c, b, d]).
X = a ? ;
X = d ? ;
no
The example provided by #false now succeeds without leaving a superfluous choicepoint:
?- unique_element(a,[a,X]).
dif(a,X)
The other more general query yields the same results:
?- unique_element(X,[E1,E2,E3]).
E1 = X,
dif(X,E3),
dif(X,E2) ? ;
E2 = X,
dif(X,E3),
dif(X,E1) ? ;
E3 = X,
dif(X,E2),
dif(X,E1) ? ;
no
Can you not define unique_element like tcount Prolog - count repetitions in list
unique_element(X, List):- tcount(=(X),List,1).
I am studying Prolog for an university exam and I have problems with this exercise:
Implement the predicate not_member(X,L) that is TRUE if the element X does not belong to the list L.
If my reasoning is correct, I have found a solution:
% FACT (BASE CASE): It is TRUE that X is not in the list if the list is empty.
not_member(_,[]).
% RULE (GENERAL CASE): If the list is non-empty, I can divide it in its Head
% element and the sublist Tail. X does not belong to the list if it is different
% from the current Head element and if it does not belong to the sublist Tail.
not_member(X,[Head|Tail]) :-
X =\= Head,
not_member(X,Tail).
This code works well with lists of numbers, as the following queries show:
2 ?- not_member(4, [1,2,3]).
true.
3 ?- not_member(1, [1,2,3]).
false.
With lists having some non-numerical elements, however,
it does not work and reports an error:
4 ?- not_member(a, [a,b,c]).
ERROR: =\=/2: Arithmetic: `a/0' is not a function
Why?
Let's check the documentation!
(=\=)/2 is an arithmetic operator.
+Expr1 =\= +Expr2
True if expression Expr1 evaluates to a number non-equal to Expr2.
You have to use (\=)/2 to compare two generic terms:
not_member(_, []) :- !.
not_member(X, [Head|Tail]) :-
X \= Head,
not_member(X, Tail).
and:
?- not_member(d, [a,b,c]).
true.
Use prolog-dif to get logically sound answers—for both ground and non-ground cases!
Just like in this answer, we define non_member(E,Xs) as maplist(dif(E),Xs).
Let's put maplist(dif(E),Xs) and not_member(E,Xs) by #Haile to the test!
?- not_member(E,[1,2,3]).
false. % wrong! What about `E=4`?
?- maplist(dif(E),[1,2,3]).
dif(E,1), dif(E,2), dif(E,3). % success with pending goals
Is it steadfast? (For more info on this important issue, read
this, this, this, and this answer.)
?- E=d, not_member(E,[a,b,c]).
E = d.
?- not_member(E,[a,b,c]), E=d.
false. % not steadfast
?- E=d, maplist(dif(E),[a,b,c]).
E = d.
?- maplist(dif(E),[a,b,c]), E=d. % steadfast
E = d.
Let's not forget about the most general use!
?- not_member(E,Xs).
Xs = []. % a lot of solutions are missing!
?- maplist(dif(E),Xs).
Xs = []
; Xs = [_A] , dif(E,_A)
; Xs = [_A,_B] , dif(E,_A), dif(E,_B)
; Xs = [_A,_B,_C], dif(E,_A), dif(E,_B), dif(E,_C)
...