11.3.1. COMMUTATOR

The COMMUTATOR clause, if provided, names an operator that is the commutator of the operator being defined. We say that operator A is the commutator of operator B if (x A y) equals (y B x) for all possible input values x, y. Notice that B is also the commutator of A. For example, operators < and > for a particular data type are usually each others' commutators, and operator + is usually commutative with itself. But operator - is usually not commutative with anything.

The left operand type of a commuted operator is the same as the right operand type of its commutator, and vice versa. So the name of the commutator operator is all that PostgreSQL needs to be given to look up the commutator, and that's all that needs to be provided in the COMMUTATOR clause.

When you are defining a self-commutative operator, you just do it. When you are defining a pair of commutative operators, things are a little trickier: how can the first one to be defined refer to the other one, which you haven't defined yet? There are two solutions to this problem:

• One way is to omit the COMMUTATOR clause in the first operator that you define, and then provide one in the second operator's definition. Since PostgreSQL knows that commutative operators come in pairs, when it sees the second definition it will automatically go back and fill in the missing COMMUTATOR clause in the first definition.

• The other, more straightforward way is just to include COMMUTATOR clauses in both definitions. When PostgreSQL processes the first definition and realizes that COMMUTATOR refers to a non-existent operator, the system will make a dummy entry for that operator in the system catalog. This dummy entry will have valid data only for the operator name, left and right operand types, and result type, since that's all that PostgreSQL can deduce at this point. The first operator's catalog entry will link to this dummy entry. Later, when you define the second operator, the system updates the dummy entry with the additional information from the second definition. If you try to use the dummy operator before it's been filled in, you'll just get an error message. (Note: This procedure did not work reliably in PostgreSQL versions before 6.5, but it is now the recommended way to do things.)

11.3.2. NEGATOR

The NEGATOR clause, if provided, names an operator that is the negator of the operator being defined. We say that operator A is the negator of operator B if both return Boolean results and (x A y) equals NOT (x B y) for all possible inputs x, y. Notice that B is also the negator of A. For example, < and >= are a negator pair for most data types. An operator can never validly be its own negator.

Unlike commutators, a pair of unary operators could validly be marked as each others' negators; that would mean (A x) equals NOT (B x) for all x, or the equivalent for right unary operators.

An operator's negator must have the same left and/or right operand types as the operator itself, so just as with COMMUTATOR, only the operator name need be given in the NEGATOR clause.

Providing a negator is very helpful to the query optimizer since it allows expressions like NOT (x = y) to be simplified into x <> y. This comes up more often than you might think, because NOT operations can be inserted as a consequence of other rearrangements.

Pairs of negator operators can be defined using the same methods explained above for commutator pairs.

11.3.3. RESTRICT

The RESTRICT clause, if provided, names a restriction selectivity estimation function for the operator (note that this is a function name, not an operator name). RESTRICT clauses only make sense for binary operators that return boolean. The idea behind a restriction selectivity estimator is to guess what fraction of the rows in a table will satisfy a WHERE-clause condition of the form

column OP constant

for the current operator and a particular constant value. This assists the optimizer by giving it some idea of how many rows will be eliminated by WHERE clauses that have this form. (What happens if the constant is on the left, you may be wondering? Well, that's one of the things that COMMUTATOR is for...)

Writing new restriction selectivity estimation functions is far beyond the scope of this chapter, but fortunately you can usually just use one of the system's standard estimators for many of your own operators. These are the standard restriction estimators:

You can frequently get away with using either eqsel or neqsel for operators that have very high or very low selectivity, even if they aren't really equality or inequality. For example, the approximate-equality geometric operators use eqsel on the assumption that they'll usually only match a small fraction of the entries in a table.

You can use scalarltsel and scalargtsel for comparisons on data types that have some sensible means of being converted into numeric scalars for range comparisons. If possible, add the data type to those understood by the routine convert_to_scalar() in src/backend/utils/adt/selfuncs.c. (Eventually, this routine should be replaced by per-data-type functions identified through a column of the pg_type system catalog; but that hasn't happened yet.) If you do not do this, things will still work, but the optimizer's estimates won't be as good as they could be.

There are additional selectivity functions designed for geometric operators in src/backend/utils/adt/geo_selfuncs.c: areasel, positionsel, and contsel. At this writing these are just stubs, but you may want to use them (or even better, improve them) anyway.

11.3.4. JOIN

The JOIN clause, if provided, names a join selectivity estimation function for the operator (note that this is a function name, not an operator name). JOIN clauses only make sense for binary operators that return boolean. The idea behind a join selectivity estimator is to guess what fraction of the rows in a pair of tables will satisfy a WHERE-clause condition of the form

table1.column1 OP table2.column2

for the current operator. As with the RESTRICT clause, this helps the optimizer very substantially by letting it figure out which of several possible join sequences is likely to take the least work.

As before, this chapter will make no attempt to explain how to write a join selectivity estimator function, but will just suggest that you use one of the standard estimators if one is applicable:

11.3.5. HASHES

The HASHES clause, if present, tells the system that it is permissible to use the hash join method for a join based on this operator. HASHES only makes sense for binary operators that return boolean, and in practice the operator had better be equality for some data type.

The assumption underlying hash join is that the join operator can only return true for pairs of left and right values that hash to the same hash code. If two values get put in different hash buckets, the join will never compare them at all, implicitly assuming that the result of the join operator must be false. So it never makes sense to specify HASHES for operators that do not represent equality.

In fact, logical equality is not good enough either; the operator had better represent pure bitwise equality, because the hash function will be computed on the memory representation of the values regardless of what the bits mean. For example, equality of time intervals is not bitwise equality; the interval equality operator considers two time intervals equal if they have the same duration, whether or not their endpoints are identical. What this means is that a join using = between interval fields would yield different results if implemented as a hash join than if implemented another way, because a large fraction of the pairs that should match will hash to different values and will never be compared by the hash join. But if the optimizer chose to use a different kind of join, all the pairs that the equality operator says are equal will be found. We don't want that kind of inconsistency, so we don't mark interval equality as hashable.

There are also machine-dependent ways in which a hash join might fail to do the right thing. For example, if your data type is a structure in which there may be uninteresting pad bits, it's unsafe to mark the equality operator HASHES. (Unless, perhaps, you write your other operators to ensure that the unused bits are always zero.) Another example is that the floating-point data types are unsafe for hash joins. On machines that meet the IEEE floating-point standard, minus zero and plus zero are different values (different bit patterns) but they are defined to compare equal. So, if the equality operator on floating-point data types were marked HASHES, a minus zero and a plus zero would probably not be matched up by a hash join, but they would be matched up by any other join process.

The bottom line is that you should probably only use HASHES for equality operators that are (or could be) implemented by memcmp().

11.3.6. MERGES (SORT1, SORT2, LTCMP, GTCMP)

The MERGES clause, if present, tells the system that it is permissible to use the merge join method for a join based on this operator. MERGES only makes sense for binary operators that return boolean, and in practice the operator must represent equality for some data type or pair of data types.

Merge join is based on the idea of sorting the left- and right-hand tables into order and then scanning them in parallel. So, both data types must be capable of being fully ordered, and the join operator must be one that can only succeed for pairs of values that fall at the "same place" in the sort order. In practice this means that the join operator must behave like equality. But unlike hash join, where the left and right data types had better be the same (or at least bitwise equivalent), it is possible to merge-join two distinct data types so long as they are logically compatible. For example, the int2-versus-int4 equality operator is merge-joinable. We only need sorting operators that will bring both data types into a logically compatible sequence.

Execution of a merge join requires that the system be able to identify four operators related to the merge-join equality operator: less-than comparison for the left input data type, less-than comparison for the right input data type, less-than comparison between the two data types, and greater-than comparison between the two data types. (These are actually four distinct operators if the merge-joinable operator has two different input data types; but when the input types are the same the three less-than operators are all the same operator.) It is possible to specify these operators individually by name, as the SORT1, SORT2, LTCMP, and GTCMP options respectively. The system will fill in the default names <, <, <, > respectively if any of these are omitted when MERGES is specified. Also, MERGES will be assumed to be implied if any of these four operator options appear, so it is possible to specify just some of them and let the system fill in the rest.

The input data types of the four comparison operators can be deduced from the input types of the merge-joinable operator, so just as with COMMUTATOR, only the operator names need be given in these clauses. Unless you are using peculiar choices of operator names, it's sufficient to write MERGES and let the system fill in the details. (As with COMMUTATOR and NEGATOR, the system is able to make dummy operator entries if you happen to define the equality operator before the other ones.)

There are additional restrictions on operators that you mark merge-joinable. These restrictions are not currently checked by CREATE OPERATOR, but errors may occur when the operator is used if any are not true:

• A merge-joinable equality operator must have a merge-joinable commutator (itself if the two data types are the same, or a related equality operator if they are different).

• If there is a merge-joinable operator relating any two data types A and B, and another merge-joinable operator relating B to any third data type C, then A and C must also have a merge-joinable operator; in other words, having a merge-joinable operator must be transitive.

• Bizarre results will ensue at runtime if the four comparison operators you name do not sort the data values compatibly.