Using Prolog to compute the GCD of a polynomial

Question

The title kind of says it all. I\'m looking to compute the GCD of two polynomials. Is there any way this can be done in Prolog? If so, what\'s a good starting point? Specificall

Answer 1

This answer is meant as a push in the right direction.

First, forget for a moment that you need to parse an expression like x^2 + 7x + 6; this isn't even a proper term in Prolog yet. If you tried to write it on the top level, you will get an error:

?- Expr = x^2 + 7x + 6.
ERROR: Syntax error: Operator expected
ERROR: Expr = x^2 + 
ERROR: ** here **
ERROR: 7x + 6 . 

Prolog doesn't know how to deal with the 7x you have there. Parsing the expression is a question of its own, and maybe it is easier if you assumed you have already parsed it and gotten a representation that looks for example like this:

[6, 7, 1]

Similarly, x^2 − 5x − 6 becomes:

[-6, -5, 1]

and to represent 0 you would use the empty list:

[]

Now, take a look at the algorithm at the Wikipedia page. It uses deg for the degree and lc for the leading coefficient. With the list representation above, you can define those as:

The degree is one less then the length of the list holding the coefficients.

poly_deg(F, D) :-
    length(F, N),
    D is N - 1.

The leading coefficient is the last element of the list.

poly_lc(F, C) :-
    last(F, C).

You also need to be able to do simple arithmetic with polynomials. Using the definitions on the Wikipedia page, we see that for example adding [] and [1] should give you [1], multiplying [-2, 2] with [1, -3, 1] should give you [-2, 8, -8, 2]. A precursory search gave me this question here on Stackoverflow. Using the predicates defined there:

?- poly_prod([-2,2], [1, -3, 1], P).
P = [-2.0, 8.0, -8.0, 2] .

?- poly_sum([], [1], S).
S = [1].

From here on, it should be possible for you to try and implement polynomial division as outlined in the Wiki article I linked above. If you get into more trouble, you should edit your question or ask a new one.