SWI-Prolog -- Standard Order of Terms

4.6.1 Standard Order of Terms

Comparison and unification of arbitrary terms. Terms are ordered in the so-called “standard order''. This order is defined as follows:

Variables < Numbers < Strings < Atoms < Compound Terms
Variables are sorted by address.
Numbers are compared by value. Mixed rational/float are compared using cmpr/2.^{66Up
to 9.1.4, comparison was done as float.} NaN is considered smaller than all numbers, including -inf. If the comparison is equal, the float is considered the smaller value. If the Prolog flag iso is defined, all floating point numbers precede all rationals.
Strings are compared alphabetically.
Atoms are compared alphabetically.
Compound terms are first checked on their arity, then on their functor name (alphabetically) and finally recursively on their arguments, leftmost argument first.

Although variables are ordered, there are some unexpected properties one should keep in mind when relying on variable ordering. This applies to the predicates below as to predicate such as sort/2 as well as libraries that reply on ordering such as library library(assoc) and library library(ordsets). Obviously, an established relation A @< B no longer holds if A is unified with e.g., a number. Also unifying A with B invalidates the relation because they become equivalent (==/2) after unification.

As stated above, variables are sorted by address, which implies that they are sorted by‘age', where‘older' variables are ordered before‘newer' variables. If two variables are unified their‘shared' age is the age of oldest variable. This implies we can examine a list of sorted variables with‘newer' (fresh) variables without invalidating the order. Attaching an attribute, see section 8.1, turns an‘old' variable into a‘new' one as illustrated below. Note that the first always succeeds as the first argument of a term is always the oldest. This only applies for the first attribute, i.e., further manipulation of the attribute list does not change the‘age'.

?- T = f(A,B), A @< B.
T = f(A, B).

?- T = f(A,B), put_attr(A, name, value), A @< B.
false.

The above implies you can use e.g., an assoc (from library library(assoc), implemented as an AVL tree) to maintain information about a set of variables. You must be careful about what you do with the attributes though. In many cases it is more robust to use attributes to register information about variables.

Note that the standard order is not well defined on rational trees, also known as cyclic terms. This issue was identified by Mats Carlsson, quoted below. See also issue#1162 on GitHub.

Consider the terms A and B defined by the equations
[1] A = s(B,0).
[2] B = s(A,1).
Clearly, A and B are not identical, so either A @< B or A @> B must hold.

Assume A @< B. But then, s(A,1) @> s(B,0) i.e., B @< A. Contradicton.

Assume A @> B. But then, s(A,1) @< s(B,0) i.e., B @< A. Contradicton.

[ISO]@Term1 == @Term2: True if Term1 is equivalent to Term2. A variable is only identical to a sharing variable.
[ISO]@Term1 \== @Term2: Equivalent to \+Term1 == Term2.
[ISO]@Term1 @< @Term2: True if Term1 is before Term2 in the standard order of terms.
[ISO]@Term1 @=< @Term2: True if both terms are equal (==/2) or Term1 is before Term2 in the standard order of terms.
[ISO]@Term1 @> @Term2: True if Term1 is after Term2 in the standard order of terms.
[ISO]@Term1 @>= @Term2: True if both terms are equal (==/2) or Term1 is after Term2 in the standard order of terms.
[ISO]compare(?Order, @Term1, @Term2): Determine or test the Order between two terms in the standard order of terms. Order is one of <, > or =, with the obvious meaning.