问题
Lets assume there is pure_2 Prolog with dif/2 and pure_1 Prolog without dif/2. Can we realize Peano apartness for values, i.e. Peano numbers, without using dif/2? Thus lets assume we have Peano apartness like this in pure_2 Prolog:
/* pure_2 Prolog */
neq(X, Y) :- dif(X, Y).
Can we replace neq(X,Y) by a more pure definition, namely from pure_1 Prolog that doesn't use dif/2? So that we have a terminating neq/2 predicate that can decide inequality for Peano numbers? So what would be its definition?
/* pure_1 Prolog */
neq(X, Y) :- ??
回答1:
Using less
from this comment:
less(0, s(_)).
less(s(X), s(Y)) :- less(X, Y).
neq(X, Y) :- less(X, Y); less(Y, X).
回答2:
I had something else in mind, which is derived from two of the Peano Axioms, which is also part of Robinson Arithmetic. The first axiom is already a Horn clause talking about apartness:
∀x(0 ≠ S(x))
∀x∀y(S(x) = S(y) ⇒ x = y)
Applying contraposition to the second axiom gives.
The axiom is now a Horn clause talking about apartness:
∀x∀y(x ≠ y ⇒ S(x) ≠ S(y))
Now we have everything to write some Prolog code.
Adding some symmetry we get:
neq(0, s(_)).
neq(s(_), 0).
neq(s(X), s(Y)) :- neq(X, Y).
Here are some example queries. Whether the predicate leaves a choice
point depends on the Prolog system. I get:
SWI-Prolog 8.3.15 (some choice point):
?- neq(s(s(0)), s(s(0))).
false.
?- neq(s(s(0)), s(0)).
true ;
false.
Jekejeke Prolog 1.4.6 (no choice point):
?- neq(s(s(0)), s(s(0))).
No
?- neq(s(s(0)), s(0)).
Yes
来源:https://stackoverflow.com/questions/65427391/pure-prolog-peano-number-apartness