579 lines
13 KiB
Prolog
579 lines
13 KiB
Prolog
%% -*- mode: prolog-*-
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%% vim: set softtabstop=4 shiftwidth=4 tabstop=4 expandtab:
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%% Follows 'Coding guidelines for Prolog' - Theory and Practice of Logic Programming
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%% https://doi.org/10.1017/S1471068411000391
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%% Import the Constraint Logic Programming over Finite Domains lybrary
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%% Essentially, this library improves the way Prolog deals with integers,
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%% allowing more predicates to be reversible.
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%% For instance, number(N) is always false, which prevents the
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%% reversing of a predicate.
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:- use_module(library(clpfd)).
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%% polynomial_variable_list(-List) is det
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%
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% List of possible polynomial variables
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%
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polynomial_variable_list([x, y, z]).
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%% polynomial_variable(?X:atom) is det
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%
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% Returns true if X is a polynomial variable, false otherwise.
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%
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polynomial_variable(X) :-
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polynomial_variable_list(V),
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member(X, V).
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%% Tests:
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%% ?- polynomial_variable(x).
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%@ true .
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%% ?- polynomial_variable(a).
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%@ false.
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%% power(+X:atom) is semidet
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%
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% Returns true if X is a power term, false otherwise.
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%
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power(P^N) :-
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(
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zcompare((<), 0, N),
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polynomial_variable(P)
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;
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fail
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).
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power(X) :-
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polynomial_variable(X).
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%% Tests:
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%% ?- power(x).
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%@ true .
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%% ?- power(a).
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%@ false.
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%% ?- power(x^1).
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%@ true .
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%% ?- power(x^3).
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%@ true .
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%% ?- power(x^(-3)).
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%@ false.
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%% ?- power(X).
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%@ X = x^_7334,
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%@ _7334 in 1..sup ;
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%@ X = y^_7334,
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%@ _7334 in 1..sup ;
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%@ X = z^_7334,
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%@ _7334 in 1..sup ;
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%@ X = x ;
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%@ X = y ;
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%@ X = z.
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%% term(+N:atom) is det
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%
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% Returns true if N is a term, false otherwise.
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%
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term(N) :-
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number(N).
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%% N in inf..sup.
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term(X) :-
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power(X).
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term(L * R) :-
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term(L),
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term(R).
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%% append_two_atoms_with_star(L, R, T).
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%% Tests:
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%% ?- term(2*x^3).
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%@ true .
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%% ?- term(x^(-3)).
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%@ false.
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%% ?- term(a).
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%@ false.
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%% ?- term(-1*x).
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%@ true .
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%% ?- term((-3)*x^2).
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%@ true .
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%% ?- term(3.2*x).
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%@ true .
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%% ?- term(X).
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%% Doesn't give all possible terms, because number(N) is not reversible
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%% The ic library seems to be able to help here, but it's not a part of
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%% SwiPL by default
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%% is_term_valid_in_predicate(+T, +F) is det
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%
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% Returns true if valid Term, fails with UI message otherwise.
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% The fail message reports which Term is invalid and in which
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% predicate the problem ocurred.
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%
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is_term_valid_in_predicate(T, F) :-
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(
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term(T)
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;
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write("Invalid term in "),
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write(F),
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write(": "),
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write(T),
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fail
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).
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%% Tests:
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%% ?- is_term_valid_in_predicate(1, "Test").
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%@ true .
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%% ?- is_term_valid_in_predicate(a, "Test").
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%% polynomial(+M:atom) is det
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%
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% Returns true if polynomial, false otherwise.
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%
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polynomial(M) :-
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term(M).
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polynomial(L + R) :-
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polynomial(L),
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term(R).
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%% Tests:
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%% ?- polynomial(x).
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%@ true .
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%% ?- polynomial(x^3).
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%@ true .
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%% ?- polynomial(3*x^7).
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%@ true .
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%% ?- polynomial(2 + 3*x + 4*x*y^3).
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%@ true .
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%% ?- polynomial(a).
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%@ false.
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%% ?- polynomial(x^(-3)).
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%@ false.
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%% power_to_canon(+T:atom, -T^N:atom) is det
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%
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% Returns a canon power term.
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%
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power_to_canon(T^N, T^N) :-
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polynomial_variable(T),
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N #\= 1.
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power_to_canon(T, T^1) :-
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polynomial_variable(T).
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%% Tests:
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%% ?- power_to_canon(x, X).
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%@ X = x^1 .
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%% ?- power_to_canon(X, x^1).
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%@ X = x .
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%% ?- power_to_canon(X, x^4).
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%@ X = x^4 .
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%% ?- power_to_canon(X, a^1).
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%@ false.
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%% ?- power_to_canon(X, x^(-3)).
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%@ X = x^ -3 .
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%% term_to_list(?T, ?List) is det
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%
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% Converts a term to a list and vice versa.
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% Can verify if term and list are compatible.
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%
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term_to_list(L * N, [N | TS]) :-
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number(N),
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term_to_list(L, TS).
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term_to_list(L * P, [P2 | TS]) :-
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power(P),
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power_to_canon(P, P2),
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term_to_list(L, TS).
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term_to_list(N, [N]) :-
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number(N).
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term_to_list(P, [P2]) :-
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power(P),
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power_to_canon(P, P2).
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%% Tests:
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%% ?- term_to_list(1, X).
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%@ X = [1] .
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%% ?- term_to_list(-1, X).
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%@ X = [-1] .
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%% ?- term_to_list(1*2*y*z*23*x*y*x^3*x, X).
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%@ X = [x^1, x^3, y^1, x^1, 23, z^1, y^1, 2, 1] .
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%% ?- term_to_list(X, [-1]).
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%@ X = -1 .
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%% ?- term_to_list(X, [x^1, -1]).
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%@ X = -1*x .
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%% ?- term_to_list(X, [- 1, x^1]).
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%@ false.
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%@ X = x* -1 .
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%% ?- term_to_list(X, [y^1, x^1]).
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%@ X = x*y .
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%% ?- term_to_list(X, [x^4]).
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%@ X = x^4 .
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%% ?- term_to_list(X, [y^6, z^2, x^4]).
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%@ X = x^4*z^2*y^6 .
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%% simplify_term(+Term_In:term, ?Term_Out:term) is det
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%
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% Simplifies a term.
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%
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simplify_term(Term_In, Term_Out) :-
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term_to_list(Term_In, L),
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sort(0, @=<, L, L2),
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(
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member(0, L2),
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Term_Out = 0
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;
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(
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length(L2, 1),
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Term_Out = Term_In
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);
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exclude(==(1), L2, L3),
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join_similar_parts_of_term(L3, L4),
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sort(0, @>=, L4, L5),
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term_to_list(Term_Out, L5)
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),
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% First result is always the most simplified form.
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!.
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%% Tests:
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%% ?- simplify_term(1, X).
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%@ X = 1.
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%% ?- simplify_term(x, X).
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%@ X = x.
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%% ?- simplify_term(2*y*z*x^3*x, X).
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%@ X = 2*x^4*y*z.
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%% ?- simplify_term(1*y*z*x^3*x, X).
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%@ X = x^4*y*z.
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%% ?- simplify_term(0*y*z*x^3*x, X).
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%@ X = 0.
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%% ?- simplify_term(6*y*z*7*x*y*x^3*x, X).
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%@ X = 42*x^2*x^3*y^2*z.
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%% ?- simplify_term(a, X).
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%@ false.
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%% ?- simplify_term(x^(-3), X).
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%@ false.
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%% join_similar_parts_of_term(+List, -List)
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%
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% Combine powers of the same variable in the given list
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%
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join_similar_parts_of_term([P1, P2 | L], L2) :-
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power(P1),
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power(P2),
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B^N1 = P1,
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B^N2 = P2,
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N is N1 + N2,
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join_similar_parts_of_term([B^N | L], L2).
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join_similar_parts_of_term([N1, N2 | L], L2) :-
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number(N1),
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number(N2),
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N is N1 * N2,
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join_similar_parts_of_term([N | L], L2).
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join_similar_parts_of_term([X | L], [X | L2]) :-
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join_similar_parts_of_term(L, L2).
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join_similar_parts_of_term([], []).
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%% Tests:
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%% ?- join_similar_parts_of_term([3], T).
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%@ T = [3].
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%% ?- join_similar_parts_of_term([x^2], T).
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%@ T = [x^2].
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%% ?- join_similar_parts_of_term([x^1, x^1, x^1, x^1], T).
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%@ T = [x^4] .
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%% ?- join_similar_parts_of_term([2, 3, x^1, x^2], T).
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%@ T = [6, x^3] .
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%% ?- join_similar_parts_of_term([2, 3, x^1, x^2, y^1, y^6], T).
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%@ T = [6, x^3, y^7] .
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%% simplify_polynomial(+P:atom, -P2:atom) is det
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%
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% Simplifies a polynomial.
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% TODO: not everything is a +, there are -
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%
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simplify_polynomial(0, 0) :-
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!.
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simplify_polynomial(P, P2) :-
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polynomial_to_list(P, L),
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maplist(term_to_list, L, L2),
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maplist(sort(0, @>=), L2, L3),
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sort(0, @>=, L3, L4),
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maplist(join_similar_parts_of_term, L4, L5),
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maplist(sort(0, @=<), L5, L6),
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join_similar_terms(L6, L7),
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maplist(reverse, L7, L8),
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maplist(term_to_list, L9, L8),
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delete(L9, 0, L10),
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sort(0, @=<, L10, L11),
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polynomial_to_list(P2, L11),
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!.
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%% Tests:
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%% ?- simplify_polynomial(1, X).
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%@ X = 1.
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%% ?- simplify_polynomial(0, X).
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%@ X = 0.
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%% ?- simplify_polynomial(x, X).
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%@ X = x.
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%% ?- simplify_polynomial(x*x, X).
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%@ X = x^2.
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%% ?- simplify_polynomial(2 + 2, X).
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%@ X = 2*2.
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%% ?- simplify_polynomial(x + x, X).
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%@ X = 2*x.
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%% ?- simplify_polynomial(0 + x*x, X).
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%@ X = x^2.
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%% ?- simplify_polynomial(x^2*x + 3*x^3, X).
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%@ X = 4*x^3.
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%% ?- simplify_polynomial(x^2*x + 3*x^3 + x^3 + x*x*x, X).
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%@ X = 6*x^3.
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%% ?- simplify_polynomial(x^2*x + 3*x^3 + x^3 + x*x*4 + z, X).
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%@ X = 5*x^3+4*x^2+z.
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%% ?- simplify_polynomial(x + 1 + x, X).
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%@ X = 2*x+1.
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%% ?- simplify_polynomial(x + 1 + x + 1 + x + 1 + x, X).
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%@ X = 4*x+3*1.
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%% join_similar_terms(+P:ListList, -P2:ListList) is det
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%
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% Joins similar sublists representing terms by using
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% `add_terms` to check if they can be merged and perform
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% the addition. Requires the list of list be sorted with
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% `maplist(sort(0, @>=), L, L2),
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% sort(0, @>=, L2, L3)`
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% and that the sublists to be sorted with
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% `sort(0, @=<)` since that is inherited from `add_terms`
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%
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join_similar_terms([TL, TR | L], L2) :-
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%% Check if terms can be added and add them
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add_terms(TL, TR, T2),
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%% Recurse, accumulation on the first element
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join_similar_terms([T2 | L], L2),
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%% Give only first result. Red cut
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!.
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join_similar_terms([X | L], [X | L2]) :-
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%% If a pair of elements can't be added, skip one
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%% and recurse
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join_similar_terms(L, L2),
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%% Give only first result. Red cut
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!.
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join_similar_terms([], []).
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%% Tests:
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%% ?- join_similar_terms([[2, x^3], [3, x^3], [x^3]], L).
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%@ L = [[6, x^3]].
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%% term_to_canon(+T:List, -T2:List) is det
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%
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% Adds a 1 if there's no number in the list
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% Requires the list to be sorted such that the
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% numbers come first. For instance with
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% `sort(0, @=<)`
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%
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term_to_canon([T | TS], [1, T | TS]) :-
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%% Since the list is sorted, if the first element
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%% is not a number, then we need to add the 1
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not(number(T)),
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%% Give only first result. Red cut
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!.
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term_to_canon(L, L).
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%% Tests:
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%% ?- term_to_canon([2], T).
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%@ T = [2].
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%% ?- term_to_canon([x^3], T).
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%@ T = [1, x^3].
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%% ?- term_to_canon([x^3, z], T).
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%@ T = [1, x^3, z].
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%% ?- term_to_canon([2, x^3], T).
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%@ T = [2, x^3].
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%% add_terms(+L:List, +R:List, -Result:List) is det
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%
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% Adds two terms represented as list by adding
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% the coeficients if the power is the same.
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% Requires the list of terms to be simplified.
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%
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add_terms([NL | TL], [NR | TR], [N2 | TL2]) :-
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term_to_canon([NL | TL], [NL2 | TL2]),
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term_to_canon([NR | TR], [NR2 | TR2]),
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TL2 == TR2,
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N2 is NL2 + NR2.
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%% Tests
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%% ?- add_terms([1], [1], R).
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%@ R = [2].
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%% ?- add_terms([x], [x], R).
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%@ R = [2, x].
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%% ?- add_terms([2, x^3], [x^3], R).
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%@ R = [3, x^3].
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%% ?- add_terms([2, x^3], [3, x^3], R).
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%@ R = [5, x^3].
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%% simplify_polynomial_list(+L1,-L3) is det
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%
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% Simplifies a list of polynomials
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%
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simplify_polynomial_list(L, L2) :-
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maplist(simplify_polynomial, L, L2).
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%% polynomial_to_list(+P:polynomial, -L:List)
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%
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% Converts a polynomial in a list.
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% TODO: not everything is a +, there are -
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%
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polynomial_to_list(L - T, [T2 | LS]) :-
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term(T),
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negate_term(T, T2),
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polynomial_to_list(L, LS).
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polynomial_to_list(L + T, [T | LS]) :-
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term(T),
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polynomial_to_list(L, LS).
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polynomial_to_list(T, [T]) :-
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term(T).
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%% Tests:
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%% ?- polynomial_to_list(2, S).
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%@ S = [2] .
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%% ?- polynomial_to_list(x^2, S).
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%@ S = [x^2] .
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%% ?- polynomial_to_list(x^2 + x^2, S).
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%@ S = [x^2, x^2] .
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%% ?- polynomial_to_list(2*x^2+5+y*2, S).
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%@ S = [y*2, 5, 2*x^2] .
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%% ?- polynomial_to_list(2*x^2+5-y*2, S).
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%@ S = [-2*y, 5, 2*x^2] .
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%% ?- polynomial_to_list(2*x^2-5-y*2, S).
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%@ S = [-2*y, -5, 2*x^2] .
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%% ?- polynomial_to_list(P, [2]).
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%@ P = 2 .
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%% ?- polynomial_to_list(P, [x]).
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%@ P = x .
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%% ?- polynomial_to_list(P, [x^2, x, 2.3]).
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%@ Action (h for help) ? abort
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%@ % Execution Aborted
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%@ P = -2.3+x+x^2 .
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%% list_to_polynomial(+P:polynomial, -L:List)
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%
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% Converts a list in a polynomial.
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% TODO: not everything is a +, there are -
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%
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list_to_polynomial([T1|T2], P) :-
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list_to_polynomial(T2, L1),
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(
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not(L1 = []),
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P = L1+T1
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;
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P = T1
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),
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% The others computations are semantically meaningless
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!.
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list_to_polynomial(T, P) :-
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P = T.
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%% Tests:
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%% TODO
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|
|
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%% negate_term(T, T2) is det
|
|
%
|
|
% Negate the coeficient of a term and return the negated term
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%
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negate_term(T, T2) :-
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term_to_list(T, L),
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sort(0, @=<, L, L2),
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term_to_canon(L2, L3),
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[N | R] = L3,
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%% (-)/1 is an operator, needs to be evaluated, otherwise
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%% it gives a symbolic result, which messes with further processing
|
|
N2 is -N,
|
|
reverse([N2 | R], L4),
|
|
term_to_list(T2, L4),
|
|
!.
|
|
%% Tests:
|
|
%% ?- negate_term(1, R).
|
|
%@ R = -1.
|
|
%% ?- negate_term(x, R).
|
|
%@ R = -1*x.
|
|
%% ?- negate_term(x^2, R).
|
|
%@ R = -1*x^2.
|
|
%% ?- negate_term(3*x*y^2, R).
|
|
%@ R = -3*x*y^2.
|
|
|
|
%% append_two_atoms_with_star(+V1, +V2, -R) is det
|
|
%
|
|
% Returns R = V1 * V2
|
|
%
|
|
append_two_atoms_with_star(V1, V2, R) :-
|
|
% Convert term V2 into a string V3
|
|
term_string(V2, V3),
|
|
% Concat atom V1 with * into a compound V4
|
|
atom_concat(V1, *, V4),
|
|
% Concat atom V4 with V3 into a compound S
|
|
atom_concat(V4, V3, S),
|
|
% Convert compound S into a term R
|
|
term_string(R, S).
|
|
%% Tests:
|
|
% ?- append_two_atoms_with_star(2, x^2, R).
|
|
%@ R = 2*x^2.
|
|
%@ R = 2*x^2.
|
|
%@ R = 2*3.
|
|
|
|
%% scale_polynomial(+P:polynomial,+C:constant,-S:polynomial) is det
|
|
%
|
|
% Scales a polynomial with a constant
|
|
%
|
|
scale_polynomial(P, C, S) :-
|
|
polynomial_to_list(P, L),
|
|
maplist(append_two_atoms_with_star(C), L, L2),
|
|
list_to_polynomial(L2, S).
|
|
%simplify_polynomial(S1, S).
|
|
%% Tests:
|
|
%% ?- scale_polynomial(3*x^2, 2, S).
|
|
%@ S = 2*3*x^2.
|
|
%@ S = 2*(3*x^2).
|
|
|
|
%% monomial_parts(X, Y, Z)
|
|
%
|
|
% TODO Maybe remove
|
|
% Separate monomial into it's parts. Given K*X^N, gives K and N
|
|
%
|
|
monomial_parts(X, 1, X) :-
|
|
power(X),
|
|
!.
|
|
monomial_parts(X^N, 1, X^N) :-
|
|
power(X^N),
|
|
!.
|
|
monomial_parts(K * M, K, M) :-
|
|
number(K),
|
|
!.
|
|
monomial_parts(K, K, indep) :-
|
|
number(K),
|
|
!.
|
|
|
|
|
|
delete_monomial(M, X, M, 0) :-
|
|
term(M),
|
|
monomial_parts(M, _, X),
|
|
!.
|
|
delete_monomial(M + M2, X, M, M2) :-
|
|
term(M2),
|
|
term(M),
|
|
monomial_parts(M, _, X),
|
|
!.
|
|
delete_monomial(P + M, X, M, P) :-
|
|
term(M),
|
|
monomial_parts(M, _, X),
|
|
!.
|
|
delete_monomial(P + M2, X, M, P2 + M2) :-
|
|
delete_monomial(P, X, M, P2).
|
|
|
|
add_monomial(K1, K2, K3) :-
|
|
number(K1),
|
|
number(K2), !,
|
|
K3 is K1 + K2.
|
|
add_monomial(M1, M2, M3) :-
|
|
monomial_parts(M1, K1, XExp),
|
|
monomial_parts(M2, K2, XExp),
|
|
K3 is K1 + K2,
|
|
p_aux_add_monomial(K3, XExp, M3).
|
|
|
|
p_aux_add_monomial(K, indep, K) :-
|
|
!.
|
|
p_aux_add_monomial(0, _, 0) :-
|
|
!.
|
|
p_aux_add_monomial(1, XExp, XExp) :-
|
|
!.
|
|
p_aux_add_monomial(K, XExp, K * XExp).
|
|
|
|
closure_simplify_polynomial(P, P) :-
|
|
simplify_polynomial(P, P2),
|
|
P==P2,
|
|
!.
|
|
closure_simplify_polynomial(P, P3) :-
|
|
simplify_polynomial(P, P2),
|
|
closure_simplify_polynomial(P2, P3),
|
|
!.
|
|
|
|
list_to_term([N | NS], N * L) :-
|
|
number(N),
|
|
term_to_list(L, NS).
|
|
|