I was studying the PROLOG programming language, testing some examples and reading documentations. I started then to do heavy research about lists in PROLOG. The idea is: Head and Tail. I then learned that lists can be expressed in PROLOG like this:
[Head | Tail]
The syntax is pretty simple, square brackets with a head and a tail, separated by a vertical slash |. I then asked myself what is the meaning (the semantics) of the vertical slash | in PROLOG. As I said, I had done research concerning lists and the vertical slash as well, but I was not able to find something helpful about the it.
So this is why I am a little bit confused. I suppose it is indeed a special character, but why does it necessarily have to be a vertical slash? Is it an operator? Is it used for system or language (meta) applications? What is its specific function in the language?
Yes, | is a right-associative infix operator of precedence 1105, right-associative meaning that an expression like
a|b|c|d
binds as
'|'( a , '|'( b , '|'( c , d ) ) )
rather than the left-associative binding
'|'( '|'( '|'( a , b ) , c ) , d )
It is part of Prolog's syntactic sugar for list notation. In Prolog, any non-empty list, has a single item that is denoted as its head, and the remainder of the list, itself another list (which may be empty), denoted as the tail. (A rather nice recursive definition, eh?)
So one can easily partition a list into its head and tail using |. So
[Head|Tail] = [a,b,c,d]
results in
Head = a
Tail = [b,c,d]
From my answer here,
Prolog's list notation is syntactic sugar on top of very simple prolog terms. Prolog lists are denoted thus:
The empty list is represented by the atom []. Why? Because that looks like the mathematical notation for an empty list. They could have used an atom like nil to denote the empty list but they didn't.
A non-empty list is represented by the term .\2, where the first (leftmost) argument is the head of the list and the second (rightmost) argument is the tail of the list, which is, recursively, itself a list.
Some examples:
An empty list: [] is represented as the atom it is:
[]
A list of one element, [a] is internally stored as
.(a,[])
A list of two elements [a,b] is internally stored as
.(a,.(b,[]))
A list of three elements, [a,b,c] is internally stored as
.(a,.(b,.(c,[])))
Examination of the head of the list is likewise syntactic sugar over the same ./2 notation:
[X|Xs] is identical to .(X,Xs)
[A,B|Xs] is identical to .(A,.(B,Xs))
[A,B] is (see above) identical to .(A,.(B,[]))
There seems to be a bit of confusion b/w the usage of vertical bar | generally used in list pattern matching and the |/2 operator.
I am not familiar with other prologs so this might be swi-prolog specific. Help for '|' states the following:
help('|').
:Goal1 | :Goal2
Equivalent to ;/2. Retained for compatibility only. New code should use ;/2.
So, the | used in list notation is not this operator.
?- X = '[|]'(1, []).
X = [1].
?- X = '|'(1, []).
X = (1| []).
?- [1] = '|'(1, []).
false.
?- [1] = '[|]'(1, []).
true.
As seen above using just | only creates a compound term and not a list.
Following uses Univ =.. and makes it more clear.
?- X = '[|]'(a, '[|]'(b, [])).
X = [a, b].
?- [a, b, c] =.. X.
X = ['[|]', a, [b, c]].
?- deep_univ([a, b, c, d], X).
X = ['[|]', a, ['[|]', b, ['[|]', c, ['[|]', d, []]]]].
I have used deep_univ/2 from here
I am trying to build a simple predicate which get as inputs two lists and the results is a third one consisting of the intersection of the first two.
I have decided to do using logical statement. I am pretty sure my logic is correct but my predicate is not working. Any ideas?:
element(X,[H|T]) :-
X=H
;
element(X,T).
intersection(L1,L2,R) :-
not((
element(A,L1),
not(element(A,L2))
)),
not((
element(A,L1),
not(element(A,R))
)).
Please do not post alternative methods I am wondering why this one returns FALSE every time.
Your definition is correct too general. It admits e.g. that [] is the intersection of any two lists which is too general. I.e. it incorrectly succeeds for intersection([],[a],[a]). It lacks a third "for all" idiom stating that all elements that are in both lists will be in the resulting list.
But otherwise your definition is fine. For the ground case. What is a bit unusual is that the intersection is the first and not the last argument. Quite irritating to me are the variable names. I believe that R means "result", thus the intersection. And L1 and L2 are the two sets to build the intersection.
It is a bit too general, though - like many Prolog predicates - think of append([], non_list, non_list). Apart from lists, your definition admits also terms that are neither lists nor partial lists:
?- intersection(non_list1,[1,2|non_list2],[3,4|non_list3]).
To make it really useful safe, use it like so:
?- when(ground(intersection(I, A, B)), intersection(I, A, B)).
or so:
?- ( ground(intersection(I, A, B))
-> intersection(I, A, B)
; throw(error(instantiation_error, intersection(I, A, B)))
).
Or, using iwhen/2:
?- iwhen(ground(intersection(I, A, B)), intersection(I, A, B) ).
As a minor remark, rather write (\+)/1 in place of not/1.
The problem is that not/1 merely negates the outcome of your element/2. It doesn't cause element/2 to backtrack to find other instantiations for which the enclosing not/1 will be true.
Consider the following program.
a(1).
a(2).
b(1).
b(2).
b(3).
And the following queries:
b(X), not(a(X)).
not(a(X)), b(X).
The first one yields X = 3 while the second one yields false. That is because the first query first instantiates X with 1, then with 2, then with 3, until finally not(a(X)) succeeds.
The second query first instantiates X with 1, a(1) succeeds, so not(a(1)) fails. There is no backtracking done!
The lack of backtracking due to negation as pointed out by #SQB is actually not the only problem with your code. If you play around a little with ground queries you find that non-lists and the empty list as pointed out by #false are also not the only issue. Consider the following queries:
?- intersection([2,3],[1,2,3],[2,3,4]).
yes
?- intersection([2],[1,2,3],[2,3,4]).
yes
?- intersection([3],[1,2,3],[2,3,4]).
yes
?- intersection([],[1,2,3],[2,3,4]).
yes
The first is what usually is understood as intersection. The other three are all sublists of the intersection including the trivial sublist []. This is due to the way the predicate describes what an intersection is: In an intersection is not the case that an element is in the first but not the second list AND that said element is in the first but not the third list. This description clearly fits the three above queries hence they succeed. Fooling around a little more with this description in mind there are some other noteworthy ground queries that succeed:
?- intersection([2,2,3],[1,2,3],[2,3,4]).
yes
The question whether the presence of duplicates in the solution is acceptable or not is in fact quite a matter of debate. The lists [2,2,3] and [2,3] although different represent the same set {2,3}. There is this recent answer to a question on Prolog union that is elaborating on such aspects of answers. And of course the sublists of the intersection mentioned above can also contain duplicates or multiples:
?- intersection([2,2,2],[1,2,3],[2,3,4]).
yes
But why is this? For the empty list this is quite easy to see. The query
?- element(A,[]).
no
fails hence the conjunction element(A,L1), not(element(A,L2)) also fails for L1=[]. Therefore the negation wrapped around it succeeds. The same is true for the second negation, consequently [] can be derived as intersection. To see why [2] and [3] succeed as intersection it is helpful to write your predicate as logic formula with the universal quantifiers written down explicitly:
∀L1∀L2∀R∀A (intersection(L1,L2,R) ← ¬ (element(A,L1) ∧ ¬ element(A,L2)) ∧ ¬ (element(A,L1) ∧ ¬ element(A,R)))
If you consult a textbook on logic or one on logic programming that also shows Prolog code as logic formulas you'll find that the universal quantifiers for variables that do not occur in the head of the rule can be moved into the body as existential quantifiers. In this case for A:
∀L1∀L2∀R (intersection(L1,L2,R) ← ∃A ( ¬ (element(A,L1) ∧ ¬ element(A,L2)) ∧ ¬ (element(A,L1) ∧ ¬ element(A,R))))
So for all arguments L1,L2,R there is some A that satisfies the goals. Which explains the derivation of the sublists of the intersection and the multiple occurrences of elements.
However, it is much more annoying that the query
?- intersection(L1,[1,2,3],[2,3,4]).
loops instead of producing solutions. If you consider that L1 is not instantiated and look at the results for the following query
?- element(A,L1).
L1 = [A|_A] ? ;
L1 = [_A,A|_B] ? ;
L1 = [_A,_B,A|_C] ? ;
...
it becomes clear that the query
?- element(A,L1),not(element(A,[1,2,3])).
has to loop due to the infinitely many lists L1, that contain A, described by the first goal. Hence the corresponding conjunction in your predicate has to loop as well. Additionally to producing results it would also be nice if such a predicate mirrored the relational nature of Prolog and worked the other way around too (2nd or 3rd arguments variable). Let's compare your code with such a solution. (For the sake of comparison the following predicate describes sublists of the intersection just as your code does, for a different definition see further below.)
To reflect its declarative nature lets call it list_list_intersection/3:
list_list_intersection(_,_,[]).
list_list_intersection(L1,L2,[A|As]) :-
list_element_removed(L1,A,L1noA),
list_element_removed(L2,A,L2noA),
list_list_intersection(L1noA,L2noA,As).
list_element_removed([X|Xs],X,Xs).
list_element_removed([X|Xs],Y,[X|Ys]) :-
dif(X,Y),
list_element_removed(Xs,Y,Ys).
Like your predicate this version is also using the elements of the intersection to describe the relation. Hence it's producing the same sublists (including []):
?- list_list_intersection([1,2,3],[2,3,4],I).
I = [] ? ;
I = [2] ? ;
I = [2,3] ? ;
I = [3] ? ;
I = [3,2] ? ;
no
but without looping. However, multiple occurrences are not produced anymore as already matched elements are removed by list_element_removed/3. But multiple occurrences in both of the first lists are matched correctly:
?- list_list_intersection([1,2,2,3],[2,2,3,4],[2,2,3]).
yes
This predicate also works in the other directions:
?- list_list_intersection([1,2,3],L,[2,3]).
L = [2,3|_A] ? ;
L = [2,_A,3|_B],
dif(_A,3) ? ;
L = [2,_A,_B,3|_C],
dif(_A,3),
dif(_B,3) ? ;
...
?- list_list_intersection(L,[2,3,4],[2,3]).
L = [2,3|_A] ? ;
L = [2,_A,3|_B],
dif(_A,3) ? ;
L = [2,_A,_B,3|_C],
dif(_A,3),
dif(_B,3) ? ;
...
So this version corresponds to your code without the duplicates. Note how the element A of the intersection explicitly appears in the head of the rule where all elements of the intersection are walked through recursively. Which I believe is what you tried to achieve by utilizing the implicit universal quantifiers in front of Prolog rules.
To come back to a point in the beginning of my answer, this is not what is commonly understood as the intersection. Among all the results list_list_intersection/3 describes for the arguments [1,2,3] and [2,3,4] only [2,3] is the intersection. Here another issue with your code comes to light: If you use the elements of the intersection to describe the relation, how do you make sure you cover all intersecting elements? After all, all elements of [2] occur in [1,2,3] and [2,3,4]. An obvious idea would be to walk through the elements of one of the other lists and describe those occurring in both as also being in the intersection. Here is a variant using if_/3 and memberd_t/3:
list_list_intersection([],_L2,[]).
list_list_intersection([X|Xs],L2,I) :-
if_(memberd_t(X,L2),
(I=[X|Is],list_element_removed(L2,X,L2noX)),
(I=Is,L2noX=L2)),
list_list_intersection(Xs,L2noX,Is).
Note that it is also possible to walk through the arguments of the second list instead of the first one. The predicate memberd_t/3 is a reified variant of your predicate element/2 and list_element_removed/3 is again used in the description to avoid duplicates in the solution. Now the solution is unique
?- list_list_intersection([1,2,3],[2,3,4],L).
L = [2,3] ? ;
no
and the "problem queries" from above fail as expected:
?- list_list_intersection([1,2,3],[2,3,4],[]).
no
?- list_list_intersection([1,2,3],[2,3,4],[2]).
no
?- list_list_intersection([1,2,3],[2,3,4],[3]).
no
?- list_list_intersection([1,2,3],[2,3,4],[2,2,3]).
no
?- list_list_intersection([1,2,3],[2,3,4],[2,2,2]).
no
And of course you can also use the predicate in the other directions:
?- list_list_intersection([1,2,3],L,[2,3]).
L = [2,3] ? ;
L = [3,2] ? ;
L = [2,3,_A],
dif(_A,1) ? ;
...
?- list_list_intersection(L,[2,3,4],[2,3]).
L = [2,3] ? ;
L = [2,3,_A],
dif(4,_A) ? ;
...
I am a newbie to prolog and am trying to write a program which returns the atoms in a well formed propositional formula. For instance the query ats(and(q, imp(or(p, q), neg(p))), As). should return [p,q] for As. Below is my code which returns the formula as As. I dont know what to do to split the single F in ats in the F1 and F2 in wff so wff/2 never gets called. Please I need help to proceed from here. Thanks.
CODE
logical_atom( A ) :-
atom( A ),
atom_codes( A, [AH|_] ),
AH >= 97,
AH =< 122.
wff(A):- ground(A),
logical_atom(A).
wff(neg(A)) :- ground(A),wff(A).
wff(or(F1,F2)) :-
wff(F1),
wff(F2).
wff(and(F1,F2)) :-
wff(F1),
wff(F2).
wff(imp(F1,F2)) :-
wff(F1),
wff(F2).
ats(F, As):- wff(F), setof(F, logical_atom(F), As).
First, consider using a cleaner representation: Currently, you cannot distinguish atoms by a common functor. So, wrap them for example in a(Atom).
Second, use a DCG to describe the relation between a well-formed formula and the list of its atoms, like in:
wff_atoms(a(A)) --> [A].
wff_atoms(neg(F)) --> wff_atoms(F).
wff_atoms(or(F1,F2)) --> wff_atoms(F1), wff_atoms(F2).
wff_atoms(and(F1,F2)) --> wff_atoms(F1), wff_atoms(F2).
wff_atoms(imp(F1,F2)) --> wff_atoms(F1), wff_atoms(F2).
Example query and its result:
?- phrase(wff_atoms(and(a(q), imp(or(a(p), a(q)), neg(a(p))))), As).
As = [q, p, q, p].
This should do what you want. It extracts the unique set of atoms found in any arbitrary prolog term.
I'll leave it up to you, though, to determine what constitutes a "well formed propositional formula", as you put it in your problem statement (You might want to take a look at DCG's for parsing and validation).
The bulk of the work is done by this "worker predicate". It simply extracts, one at a time via backtracking, any atoms found in the parse tree and discards anything else:
expression_atom( [T|_] , T ) :- % Case #1: head of list is an ordinary atom
atom(T) , % - verify that the head of the list is an atom.
T \= [] % - and not an empty list
. %
expression_atom( [T|_] , A ) :- % Case #2: head of listl is a compound term
compound(T) , % - verify that the head of the list is a compound term
T =.. [_|Ts] , % - decompose it, discarding the functor and keeping the arguments
expression_atom(Ts,A) % - recurse down on the term's arguments
. %
expression_atom( [_|Ts] , A ) :- % Finally, on backtracking,
expression_atom(Ts,A) % - we simply discard the head and recurse down on the tail
. %
Then, at the top level, we have this simple predicate that accepts any [compound] prolog term and extracts the unique set of atoms found within by the worker predicate via setof/3:
expression_atoms( T , As ) :- % To get the set of unique atoms in an arbitrary term,
compound(T) , % - ensure that's its a compound term,
T =.. [_|Ts] , % - decompose it, discarding the functor and keeping the arguments
setof(A,expression_atom(Ts,A),As) % - invoke the worker predicate via setof/3
. % Easy!
I'd approach this problem using the "univ" operator =../2 and explicit recursion. Note that this solution will not generate and is not "logically correct" in that it will not process a structure with holes generously, so it will produce different results if conditions are reordered. Please see #mat's comments below.
I'm using cuts instead of if statements for personal aesthetics; you would certainly find better performance with a large explicit conditional tree. I'm not sure you'd want a predicate such as this to generate in the first place.
Univ is handy because it lets you treat Prolog terms similarly to how you would treat a complex s-expression in Lisp: it converts terms to lists of atoms. This lets you traverse Prolog terms as lists, which is handy if you aren't sure exactly what you'll be processing. It saves me from having to look for your boolean operators explicitly.
atoms_of_prop(Prop, Atoms) :-
% discard the head of the term ('and', 'imp', etc.)
Prop =.. [_|PropItems],
collect_atoms(PropItems, AtomsUnsorted),
% sorting makes the list unique in Prolog
sort(AtomsUnsorted, Atoms).
The helper predicate collect_atoms/2 processes lists of terms (univ only dismantles the outermost layer) and is mutually recursive with atoms_of_prop/2 when it finds terms. If it finds atoms, it just adds them to the result.
% base case
collect_atoms([], []).
% handle atoms
collect_atoms([A|Ps], [A|Rest]) :-
% you could replace the next test with logical_atom/1
atom(A), !,
collect_atoms(Ps, Rest).
% handle terms
collect_atoms([P|Ps], Rest) :-
compound(P), !, % compound/1 tests for terms
atoms_of_prop(P, PAtoms),
collect_atoms(Ps, PsAtoms),
append(PAtoms, PsAtoms, Rest).
% ignore everything else
collect_atoms([_|Ps], Rest) :- atoms_of_prop(Ps, Rest).
This works for your example as-is:
?- atoms_of_prop(ats(and(q, imp(or(p, q), neg(p))), As), Atoms).
Atoms = [p, q].
I starting to study for my upcoming exam and I'm stuck on a trivial prolog practice question which is not a good sign lol.
It should be really easy, but for some reason I cant figure it out right now.
The task is to simply count the number of odd numbers in a list of Int in prolog.
I did it easily in haskell, but my prolog is terrible. Could someone show me an easy way to do this, and briefly explain what you did?
So far I have:
odd(X):- 1 is X mod 2.
countOdds([],0).
countOdds(X|Xs],Y):-
?????
Your definition of odd/1 is fine.
The fact for the empty list is also fine.
IN the recursive clause you need to distinguish between odd numbers and even numbers. If the number is odd, the counter should be increased:
countOdds([X|Xs],Y1) :- odd(X), countOdds(Xs,Y), Y1 is Y+1.
If the number is not odd (=even) the counter should not be increased.
countOdds([X|Xs],Y) :- \+ odd(X), countOdds(Xs,Y).
where \+ denotes negation as failure.
Alternatively, you can use ! in the first recursive clause and drop the condition in the second one:
countOdds([X|Xs],Y1) :- odd(X), !, countOdds(Xs,Y), Y1 is Y+1.
countOdds([X|Xs],Y) :- countOdds(Xs,Y).
In Prolog you use recursion to inspect elements of recursive data structs, as lists are.
Pattern matching allows selecting the right rule to apply.
The trivial way to do your task:
You have a list = [X|Xs], for each each element X, if is odd(X) return countOdds(Xs)+1 else return countOdds(Xs).
countOdds([], 0).
countOdds([X|Xs], C) :-
odd(X),
!, % this cut is required, as rightly evidenced by Alexander Serebrenik
countOdds(Xs, Cs),
C is Cs + 1.
countOdds([_|Xs], Cs) :-
countOdds(Xs, Cs).
Note the if, is handled with a different rule with same pattern: when Prolog find a non odd element, it backtracks to the last rule.
ISO Prolog has syntax sugar for If Then Else, with that you can write
countOdds([], 0).
countOdds([X|Xs], C) :-
countOdds(Xs, Cs),
( odd(X)
-> C is Cs + 1
; C is Cs
).
In the first version, the recursive call follows the test odd(X), to avoid an useless visit of list'tail that should be repeated on backtracking.
edit Without the cut, we get multiple execution path, and so possibly incorrect results under 'all solution' predicates (findall, setof, etc...)
This last version put in evidence that the procedure isn't tail recursive. To get a tail recursive procedure add an accumulator:
countOdds(L, C) :- countOdds(L, 0, C).
countOdds([], A, A).
countOdds([X|Xs], A, Cs) :-
( odd(X)
-> A1 is A + 1
; A1 is A
),
countOdds(Xs, A1, Cs).