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:

1. Variables < Numbers < Strings < Atoms < Compound Terms
2. Variables are sorted by address.
3. Numbers are compared by value. Mixed rational/float are compared using cmpr/2.^{70Up 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.
4. Strings are compared alphabetically.
5. Atoms are compared alphabetically.
6. Non-text blobs (see section 12.4.10) occupy a band of their own between Strings and Atoms. Two blobs of the same type are compared by that type; between types the order is the order in which the types were registered and is therefore not guaranteed to be the same in another run.
7. 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. The predicate partial_compare/3 can be used to detect that a comparison depends on a variable and is therefore not stable.

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. Contradiction.

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

By default, comparing such terms returns an arbitrary order, which breaks predicates that rely on a consistent order such as sort/2, library library(assoc) and library library(ordsets). Setting the Prolog flag incomparable to error raises an exception instead. The predicate partial_compare/3 reports such pairs as incomparable(Sub1, Sub2).

[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.

partial_compare(?Order, @Term1, @Term2)

As compare/3, but only decides the Order if this decision is stable, i.e., if it cannot be invalidated by binding a variable. This predicate is the building block for comparisons that must be sound on non-ground terms. If the order depends on a variable, Order is unified with undecided(Sub1, Sub2), where Sub1 and Sub2 are the first pair of subterms of Term1 and Term2 that could not be compared. Because the standard order compares terms lexicographically, this pair decides the entire comparison: after any further instantiation, compare(Order, Term1, Term2) gives the same result as compare(Order, Sub1, Sub2), unless Sub1 and Sub2 become equal (==/2), in which case the comparison must continue with the remaining subterms.

If the two terms are cyclic and have no order (see section 4.6.1), Order is unified with incomparable(Sub1, Sub2), where Sub1 and Sub2 are a pair of subterms at which the lexicographic comparison runs into a cycle. Unlike undecided(Sub1, Sub2), further instantiation cannot make the terms comparable.

?- partial_compare(Order, f(a,1), f(a,2)).
Order = (<).
?- partial_compare(Order, f(a,X), f(a,b)).
Order = undecided(X, b).
?- partial_compare(Order, f(X,1), f(X,2)).
Order = (<).
?- partial_compare(Order, f(_,1), g(_,2)).
Order = (<).
?- A = s(B,0), B = s(A,1), partial_compare(Order, A, B).
A = s(s(A, 1), 0),
B = s(A, 1),
Order = incomparable(s(s(A, 1), 0), s(A, 1)).

The last two queries illustrate that a variable only makes the comparison undecided if it is actually reached: the first of these compares the same variable to itself and continues with the second argument, while the second is decided by the functor name before the arguments are examined at all.