So I have this mathematical language, it goes like this:
E -> number
[+,E,E,E] //e.g. [+,1,2,3] is 1+2+3 %we can put 2 to infinite Es here.
[-,E,E,E] //e.g. [-,1,2,3] is 1-2-3 %we can put 2 to infinite Es here.
[*,E,E,E] //e.g. [*,1,2,3] is 1*2*3 %we can put 2 to infinite Es here.
[^,E,E] //e.g. [^,2,3] is 2^3
[sin,E] //e.g. [sin,0] is sin 0
[cos,E] //e.g. [cos,0] is cos 0
and I want to write the set of rules that finds the numeric value of a mathematical expression written by this language in prolog.
I first wrote a function called "check", it checks to see if the list is written in a right way according to the language we have :
check1([]).
check1([L|Ls]):- number(L),check1(Ls).
check([L|Ls]):-atom(L),check1(Ls).
now I need to write the function "evaluate" that takes a list that is an expression written by this language, and a variable that is the numeric value corresponding to this language.
example:
?-evaluate([*,1,[^,2,2],[*,2,[+,[sin,0],5]]]],N) -> N = 40
so I wrote this:
sum([],0).
sum([L|Ls],N):- not(is_list(L)),sum(Ls,No),N is No + L.
min([],0).
min([L|Ls],N):-not(is_list(L)), min(Ls,No),N is No - L.
pro([],0).
pro([X],[X]).
pro([L|Ls],N):-not(is_list(L)), pro(Ls,No), N is No * L.
pow([L|Ls],N):-not(is_list(L)), N is L ^ Ls.
sin_(L,N):-not(is_list(L)), N is sin(L).
cos_(L,N):-not(is_list(L)), N is cos(L).
d([],0).
d([L|Ls],N):- L == '+' ,sum(Ls,N);
L == '-',min(Ls,N);
L == '*',pro(Ls,N);
L == '^',pow(Ls,N);
L == 'sin',sin_(Ls,N);
L == 'cos',cos_(Ls,N).
evaluate([],0).
evaluate([L|Ls],N):-
is_list(L) , check(L) , d(L,N),L is N,evaluate(Ls,N);
is_list(L), not(check(L)) , evaluate(Ls,N);
not(is_list(L)),not(is_list(Ls)),check([L|Ls]),d([L|Ls],N),
L is N,evaluate(Ls,N);
is_list(Ls),evaluate(Ls,N).
and it's working for just a list and returning the right answer , but not for multiple lists inside the main list, how should my code be?
The specification you work with looks like a production rule that describes that E (presumably short for Expression) might be a number or one of the 6 specified operations. That is the empty list [] is not an expression. So the fact
evaluate([],0).
should not be in your code. Your predicate sum/2 almost works the way you wrote it, except for the empty list and a list with a single element, that are not valid inputs according to your specification. But the predicates min/2 and pro/2 are not correct. Consider the following examples:
?- sum([1,2,3],X).
X = 6 % <- correct
?- sum([1],X).
X = 1 % <- incorrect
?- sum([],X).
X = 0 % <- incorrect
?- min([1,2,3],X).
X = -6 % <- incorrect
?- pro([1,2,3],X).
X = 6 ? ; % <- correct
X = 0 % <- incorrect
Mathematically speaking, addition and multiplication are associative but subtraction is not. In programming languages all three of these operations are usually left associative (see e.g. Operator associativity) to yield the mathematically correct result. That is, the sequence of subtractions in the above query would be calculated:
1-2-3 = (1-2)-3 = -4
The way you define a sequence of these operations resembles the following calculation:
[A,B,C]: ((0 op C) op B) op A
That works out fine for addition:
[1,2,3]: ((0 + 3) + 2) + 1 = 6
But it doesn't for subtraction:
[1,2,3]: ((0 - 3) - 2) - 1 = -6
And it is responsible for the second, incorrect solution when multiplying:
[1,2,3]: ((0 * 3) * 2) * 1 = 0
There are also some other issues with your code (see e.g. #lurker's comments), however, I won't go into further detail on that. Instead, I suggest a predicate that adheres closely to the specifying production rule. Since the grammar is describing expressions and you want to know the corresponding values, let's call it expr_val/2. Now let's describe top-down what an expression can be: It can be a number:
expr_val(X,X) :-
number(X).
It can be an arbitrarily long sequence of additions or subtractions or multiplications respectively. For the reasons above all three sequences should be evaluated in a left associative way. So it's tempting to use one rule for all of them:
expr_val([Op|Es],V) :-
sequenceoperator(Op), % Op is one of the 3 operations
exprseq_op_val(Es,Op,V). % V is the result of a sequence of Ops
The power function is given as a list with three elements, the first being ^ and the others being expressions. So that rule is pretty straightforward:
expr_val([^,E1,E2],V) :-
expr_val(E1,V1),
expr_val(E2,V2),
V is V1^V2.
The expressions for sine and cosine are both lists with two elements, the first being sin or cos and the second being an expression. Note that the argument of sin and cos is the angle in radians. If the second argument of the list yields the angle in radians you can use sin/1 and cos/2 as you did in your code. However, if you get the angle in degrees, you need to convert it to radians first. I include the latter case as an example, use the one that fits your application.
expr_val([sin,E],V) :-
expr_val(E,V1),
V is sin(V1*pi/180). % radians = degrees*pi/180
expr_val([cos,E],V) :-
expr_val(E,V1),
V is cos(V1*pi/180). % radians = degrees*pi/180
For the second rule of expr_val/2 you need to define the three possible sequence operators:
sequenceoperator(+).
sequenceoperator(-).
sequenceoperator(*).
And subsequently the predicate exprseq_op_val/3. As the leading operator has already been removed from the list in expr_val/2, the list has to have at least two elements according to your specification. In order to evaluate the sequence in a left associative way the value of the head of the list is passed as an accumulator to another predicate exprseq_op_val_/4
exprseq_op_val([E1,E2|Es],Op,V) :-
expr_val(E1,V1),
exprseq_op_val_([E2|Es],Op,V,V1).
that is describing the actual evaluation. There are basically two cases: If the list is empty then, regardless of the operator, the accumulator holds the result. Otherwise the list has at least one element. In that case another predicate, op_val_args/4, delivers the result of the respective operation (Acc1) that is then recursively passed as an accumulator to exprseq_op_val_/4 alongside with the tail of the list (Es):
exprseq_op_val_([],_Op,V,V).
exprseq_op_val_([E1|Es],Op,V,Acc0) :-
expr_val(E1,V1),
op_val_args(Op,Acc1,Acc0,V1),
exprseq_op_val_(Es,Op,V,Acc1).
At last you have to define op_val_args/4, that is again pretty straightforward:
op_val_args(+,V,V1,V2) :-
V is V1+V2.
op_val_args(-,V,V1,V2) :-
V is V1-V2.
op_val_args(*,V,V1,V2) :-
V is V1*V2.
Now let's see how this works. First your example query:
?- expr_val([*,1,[^,2,2],[*,2,[+,[sin,0],5]]],V).
V = 40.0 ? ;
no
The simplest expression according to your specification is a number:
?- expr_val(-3.14,V).
V = -3.14 ? ;
no
The empty list is not an expression:
?- expr_val([],V).
no
The operators +, - and * need at least 2 arguments:
?- expr_val([-],V).
no
?- expr_val([+,1],V).
no
?- expr_val([*,1,2],V).
V = 2 ? ;
no
?- expr_val([-,1,2,3],V).
V = -4 ? ;
no
The power function has exactly two arguments:
?- expr_val([^,1,2,3],V).
no
?- expr_val([^,2,3],V).
V = 8 ? ;
no
?- expr_val([^,2],V).
no
?- expr_val([^],V).
no
And so on...
I'm new to prolog so the concept of unifying is still new, so bear with me.
I want something like:
pair(X,Y)
That works like this:
?- pair(4,Y).
Y = 4,
?- pair(3,Y).
Y = 0,
But I don't really know how to do this. I know you can use something like X mod 2 =:= 0 which is true or false, but I don't know how to make "if this than that" code in prolog's way.
There are several ways to express a conditional statement in Prolog, besides using CLPFD (Constraint Logic Programming over Finite Domains):
1) make several statements, one for each case you are considering, and let Prolog execution mechanism find the one that is appropriate:
pair(X,Y) :- even(X), !, Y = X. % 'if X is even then Y = X'
pair(_,0). % 'else Y = 0'
In the second case here the unification is implicit, but in the first case it is explicit.
The _ symbol is a syntactic sugar for an 'anonymous variable', that is a variable for whose bindings we do not care and that we do not use.
The cut operator ! is used in the first clause to tell Prolog that it should not try to find other solutions for the Y if the first clause succeeds (like, backtracking and trying the second clause):
% without cut | % with cut
?- pair(4,Y). | ?- pair3(4,Y).
|
Y = 4 ? ; | Y = 4 ? ;
|
Y = 0 ? ; % <- wrong behaviour! | no.
|
no |
2) use the (Condition -> ThenStatement ; ElseStatement) if-then-else construct:
pair(X,Y) :- (even(X) -> Y = X ; Y = 0).
No cuts are allowed in the Condition argument. The first solution of the Condition is used.
3) some systems have the if/3 predicate, that is evaluated as if(Condition,ThenStatement,ElseStatement):
pair(X,Y) :- if( even(X), Y = X, Y = 0).
No cuts are allowed in the Condition argument.
I'd like to assert facts about all members of a List in prolog, and have any resulting unification retained. As an example, I'd like to assert that each list member is equal to five, but none of the below constructs does this:
?- L=[X,Y,Z], forall(member(E,L), E=5).
L = [_h27057686,_h27057704,_h27057722]
X = _h27057686
Y = _h27057704
Z = _h27057722
yes
?- L=[X,Y,Z], foreach(member(E,L), E=5).
L = [_h27057686,_h27057704,_h27057722]
X = _h27057686
Y = _h27057704
Z = _h27057722
yes
I would like a way to pose the query such that X=5,Y=5, and Z=5.
There is a lot of terminology that you might be getting wrong, or I am misunderstanding you.
"Equal to" is not the same as "could unify", or "unify", but it depends how you mean it.
With SWI-Prolog, from the top level:
?- X == 5.
false. % the free variable X is not the integer 5
?- unifiable(X, 5, U).
U = [X=5]. % you could unify X with 5, then X will be 5
?- X = 5.
X = 5. % X unifies with 5 (and is now bound to the integer 5)
The comment by CapelliC already has the answer that you are most likely after: given a list of variables (either free or not), make so that each variable in the list is bound to the integer 5. This is best done by unification (the third query above). The maplist simply applies the unification to each element of the list.
PS. In case you are wondering how to read the maplist(=(5), L):
These three are equivalent:
maplist(=(5), [X,Y,Z])
maplist(=, [5,5,5], [X,Y,Z])
X=5, Y=5, Z=5
And of course X=5 is the same as =(X,5).
I have this scenario wherein I get a linear equation in the Prolog query like below:
?- myquery( 3X + 5Y = 10, Result).
So my query has an equation 3X + 5Y = 10, which in general assumes the form AX + BY = C, where A=3, B=5 and C=10.
Now, in my prolog program, I am trying to define a predicate that can take in the expression mentioned in the query above. That is, I somehow want to get A, B and C values and also the operator involved (in the above case the plus operator) stored and then used on the logic that I define withing the program. I am wondering how this can be done.
To be more generic, the question is how do I identify the constants and the operator involved in an equation that is passed on through the goal/query?
SWI-Prolog has a constraint library clp(Q,R) that solve at symbolic level these equations:
[debug] ?- [library(clpq)].
% library(clpq) compiled into clpq 0,27 sec, 992 clauses
true.
?- {3 * X + 5 * Y = 10}.
{Y=2-3 rdiv 5*X}.
Eclipse will surely have something more advanced. These libraries aren't simple, tough...
Of interest to you, the Prolog syntax is used, as a host language, so the usual builtins could be applied for identify vars, constants, and the like.
The following transcript may prove illuminating:
32 ?- Term = (3*_X + 5*_Y = 10), functor(Term,F,A).
Term = 3*_G527+5*_G530=10
F = =
A = 2
33 ?- Term = (3*_X + 5*_Y = 10), arg(Arg,Term,Val).
Term = 3*_G459+5*_G462=10
Arg = 1
Val = 3*_G459+5*_G462 ; % user pressed ';' interactively
Term = 3*_G459+5*_G462=10
Arg = 2
Val = 10 ; % user pressed ';' interactively
No
35 ?- Term = (3*_X + 5*_Y = 10), arg(1,Term,Val1), functor(Val1,F1,A1),
arg(2,Val1,Val12).
Term = 3*_G693+5*_G696=10
Val1 = 3*_G693+5*_G696
F1 = +
A1 = 2
Val12 = 5*_G696
The last query reads: for Term as given, 1st arg of Term is Val1, the functor of Val1 is F1 with arity A1 (meaning, it has A1 args - subparts - itself), and 2nd arg of the term in Val1 is stored under Val12 name. To clarify, any symbolic data in Prolog is in the form of fff(aa,bb,cc,...) where fff is some name, called functor, and the "arguments" in that expression can be accessed through the arg call.
That means that the original expression (3*_X + 5*_Y = 10) is actually stored in Prolog as '='( '+'( '*'(3,_X), '*'(5,_Y)), 10). When you get to the atomic parts (functors with arity 0), you can check them further:
47 ?- arg(1,(3*X),V), functor(V,F,A), number(V).
X = _G441
V = 3
F = 3
A = 0
Yes
EDIT: to answer your other question (from the comments):
1 ?- (3*_X + 5*_Y = 10) = (A*X + B*Y = C).
A = 3
X = _G412
B = 5
Y = _G415
C = 10
Yes
If you insist on not writing out the multiplication sign * explicitly, you will have to represent your terms as strings, and to analyze that string. That would be a much more involved task.
EDIT: another thing to try is =.. predicate, called "Univ":
4 ?- (3*_X + 5*_Y = 10) =.. X.
X = [=, 3*_G454+5*_G457, 10]
Yes
5 ?- (3*_X + 5*_Y = 10) =.. X, X=[X1,X2,X3], X2 =.. Y.
X = [=, 3*_G545+5*_G548, 10]
X1 = =
X2 = 3*_G545+5*_G548
X3 = 10
Y = [+, 3*_G545, 5*_G548]
Yes
You can for example use term inspection predicates: arg/3, functor/3, var/1, (=..)/2 etc.
You might want to take a look at examples of symbolic differentiation implemented using term rewrite rules; they handle such expressions.
Here's a chapter (minus 1 page) from the book Clause and Effect that you might find useful:
Clause and Effect - Chapter Six: Term Rewriting
Another from The art of Prolog: advanced programming techniques
23 An equation solver
Programming in Prolog also has a section (7.11) on symbolic differentiation.