Difference in implementation of gcd between logic and functional programming - functional-programming

Question

I'm currently learning programming language concepts and pragmatics, hence I feel like I need help in differentiating two subbranches of declarative language family.
Consider the following code snippets which are written in Scheme and Prolog, respectively:
;Scheme
(define gcd
(lambda (a b)
(cond ((= a b) a)
((> a b) (gcd (- a b) b))
(else (gcd (- b a) a)))))
%Prolog
gcd(A, B, G) :- A = B, G = A.
gcd(A, B, G) :- A > B, C is A-B, gcd(C, B, G).
gcd(A, B, G) :- B > A, C is B-A, gcd(C, A, G).
The thing that I didn't understand is:
How do these two different programming languages behave
differently?
Where do we make the difference so that they are categorized either
Functional or Logic-based programming language?
As far as I'm concerned, they do exactly the same thing, calling recursive functions until it terminates.

Answer 1

Since you are using very low-level predicates in your logic programming version, you cannot easily see the increased generality that logic programming gives you over functional programming.
Consider this slightly edited version of your code, which uses CLP(FD) constraints for declarative integer arithmetic instead of the low-level arithmetic you are currently using:
gcd(A, A, A).
gcd(A, B, G) :- A #> B, C #= A - B, gcd(C, B, G).
gcd(A, B, G) :- B #> A, C #= B - A, gcd(C, A, G).
Importantly, we can use this as a true relation, which makes sense in all directions.
For example, we can ask:
Are there two integers X and Y such that their GCD is 3?
That is, we can use this relation in the other direction too! Not only can we, given two integers, compute their GCD. No! We can also ask, using the same program:
?- gcd(X, Y, 3).
X = Y, Y = 3 ;
X = 6,
Y = 3 ;
X = 9,
Y = 3 ;
X = 12,
Y = 3 ;
etc.
We can also post even more general queries and still obtain answers:
?- gcd(X, Y, Z).
X = Y, Y = Z ;
Y = Z,
Z#=>X+ -1,
2*Z#=X ;
Y = Z,
_1712+Z#=X,
Z#=>X+ -1,
Z#=>_1712+ -1,
2*Z#=_1712 ;
etc.
That's a true relation, which is more general than a function of two arguments!
See clpfd for more information.

Answer 2

The GCD example only lightly touches on the differences between logic programming and functional programming as they are much closer to each other than to imperative programming. I will concentrate on Prolog and OCaml, but I believe it is quite representative.
Logical Variables and Unification:
Prolog allows to express partial datastructures e.g. in the term node(24,Left,Right) we don't need to specify what Left and Right stand for, they might be any term. A functional language might insert a lazy function or a thunk which is evaluated later on, but at the creation of the term, we need to know what to insert.
Logical variables can also be unified (i.e. made equal). A search function in OCaml might look like:
let rec find v = function
| [] -> false
| x::_ when v = x -> true
| _::xs (* otherwise *) -> find v xs
While the Prolog implementation can use unification instead of v=x:
member_of(X,[X|_]).
member_of(X,[_|Xs]) :-
member_of(X,Xs).
For the sake of simplicity, the Prolog version has some drawbacks (see below in backtracking).
Backtracking:
Prolog's strength lies in successively instantiating variables which can be easily undone. If you try the above program with variables, Prolog will return you all possible values for them:
?- member_of(X,[1,2,3,1]).
X = 1 ;
X = 2 ;
X = 3 ;
X = 1 ;
false.
This is particularly handy when you need to explore search trees but it comes at a price. If we did not specify the size of the list, we will successively create all lists fulfilling our property - in this case infinitely many:
?- member_of(X,Xs).
Xs = [X|_3836] ;
Xs = [_3834, X|_3842] ;
Xs = [_3834, _3840, X|_3848] ;
Xs = [_3834, _3840, _3846, X|_3854] ;
Xs = [_3834, _3840, _3846, _3852, X|_3860] ;
Xs = [_3834, _3840, _3846, _3852, _3858, X|_3866] ;
Xs = [_3834, _3840, _3846, _3852, _3858, _3864, X|_3872]
[etc etc etc]
This means that you need to be more careful using Prolog, because termination is harder to control. In particular, the old-style ways (the cut operator !) to do that are pretty hard to use correctly and there's still some discussion about the merits of recent approaches (deferring goals (with e.g. dif), constraint arithmetic or a reified if). In a functional programming language, backtracking is usually implemented by using a stack or a backtracking state monad.
Invertible Programs:
Perhaps one more appetizer for using Prolog: functional programming has a direction of evaluation. We can use the find function only to check if some v is a member of a list, but we can not ask which lists fulfill this. In Prolog, this is possible:
?- Xs = [A,B,C], member_of(1,Xs).
Xs = [1, B, C],
A = 1 ;
Xs = [A, 1, C],
B = 1 ;
Xs = [A, B, 1],
C = 1 ;
false.
These are exactly the lists with three elements which contain (at least) one element 1. Unfortunately the standard arithmetic predicates are not invertible and together with the fact that the GCD of two numbers is always unique is the reason why you could not find too much of a difference between functional and logic programming.
To summarize: logic programming has variables which allow for easier pattern matching, invertibility and exploring multiple solutions of the search tree. This comes at the cost of complicated flow control. Depending on the problem it is easier to have a backtracking execution which is sometimes restricted or to add backtracking to a functional language.

Answer 3

The difference is not very clear from one example. Programming language are categorized to logic,functional,... based on some characteristics that they support and as a result they are designed in order to be more easy for programmers in each field (logic,functional...). As an example imperative programming languages (like c) are very different from object oriented (like java,C++) and here the differences are more obvious.
More specifically, in your question the Prolog programming language has adopted he philosophy of logic programming and this is obvious for someone who knows a little bit about mathematical logic. Prolog has predicates (rather than functions-basically almost the same) which return true or false based on the "world" we have defined which is for example what facts and clauses do we have already defined, what mathematical facts are defined and more....All these things are inherited by mathematical logic (propositional and first order logic). So we could say that Prolog is used as a model to logic which makes logical problems (like games,puzzles...) more easy to solve. Moreover Prolog has some features that general-purpose languages have. For example you could write a program in your example to calculate gcd:
gcd(A, B, G) :- A = B, G = A.
gcd(A, B, G) :- A > B, C is A-B, gcd(C, B, G).
gcd(A, B, G) :- B > A, C is B-A, gcd(C, A, G).
In your program you use a predicate gcd in returns TRUE if G unifies with GCD of A,B, and you use multiple clauses to match all cases. When you query gcd(2,5,1). will return True (NOTE that in other languages like shceme you can't give the result as parameter), while if you query gcd(2,5,G). it unifies G with gcd of A,B and returns 1, it is like asking Prolog what should be G in order gcd(2,5,G). be true. So you can understand that it is all about when the predicate succeeds and for that reason you can have more than one solutions, while in functional programming languages you can't.
Functional languages are based in functions so always return the SAME
TYPE of result. This doesn't stand always in Prolog you could have a predicate predicate_example(Number,List). and query predicate_example(5,List). which returns List=... (a list) and also query
predicate_example(Number,[1,2,3]). and return N=... (a number).
The result should be unique, In mathematics, a function is a relation
between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output
Should be clear what parameter is the variable that will be returned
for example gcd function is of type : N * N -> R so gets A,B parameters which belong to N (natural numbers) and returns gcd. But prolog (with some changes in your program) could return the parameter A,so querying gcd(A,5,1). would give all possible A such that predicate gcd succeeds,A=1,2,3,4,5 .
Prolog in order to find gcd tries every possible way with choice
points so in every step it will try all of you three clauses and will
find every possible solutions. Functional programming languages on
the other hand, like functions should have well unique defined steps
to find the solution.
So you can understand that the difference between Functional and logic languages may not be always visible but they are based on different philosophy-way of thinking.
Imagine how hard would be to solve tic-tac-toe or N queens problem or man-goat-wolf-cabbage problem in Scheme.