PostgreSQL 支持自定义操作符，本质上是调用函数来实现的。
语法如下：
例如创建一个求两个值的平均值的操作符：
首选要创建函数
postgres=# create function f_avg(numeric,numeric) returns numeric as $$
postgres$# select ($1+$2)/2;
postgres$# $$ language sql strict;
CREATE FUNCTION
验证函数
postgres=# select f_avg(1,null);
f_avg
-------
(1 row)
postgres=# select f_avg(1,2);
f_avg
--------------------
1.5000000000000000
(1 row)
创建操作符，指定左右参数类型，调用的函数名，commutator是一个和优化器相关的选项，我后面会重点介绍：
postgres=# create operator ## (procedure=f_avg, leftarg=numeric, rightarg=numeric, commutator='##');
CREATE OPERATOR
postgres=# select 1 ## 2;
?column?
--------------------
1.5000000000000000
(1 row)
注意到在创建操作符的语法中有6个和优化器有关的关键字：
[, COMMUTATOR = com_op ] [, NEGATOR = neg_op ]
[, RESTRICT = res_proc ] [, JOIN = join_proc ]
[, HASHES ] [, MERGES ]
介绍如下：
假设x表示操作符左侧的参数，y表示操作符右侧的参数
1. commutator，指明x op1 y等效于y op2 x，即操作数调换，返回的值一样。例如2>1 和1<2结果是一致的。那么>就是<的commutator或者反之。又例如1+2和2+1是等价的，那么+就是+的commutator。commutator只需要在创建其中一个操作符时指定，创建另一个对应的操作符时可以不需要指定，PostgreSQL会自动建立这个关系。例如创建>操作符时指定了它的commutator是<，那么在创建<操作符时可以不需要指定>是它的commutator。
另外需要注意，有commutator操作符的操作符的左右两侧的参数类型必须一致，这样才能满足x op1 y等价于y op2 x。
优化器如何利用commutator呢？例如索引扫描，必须列在操作符的左侧才能使用索引。1 > tbl.c这个条件，如果>没有commutator的话，是不能使用索引的。
例子，以int4的>和<操作符为例，实验一下：
>和<在PostgreSQL中是一对commutator
postgres=# select oprcom::regoper from pg_operator where oprname='>' and oprcode='int4gt'::regproc;
oprcom
--------------
pg_catalog.<
(1 row)
postgres=# select oprcom::regoper from pg_operator where oprname='<' and oprcode='int4lt'::regproc;
oprcom
--------------
pg_catalog.>
(1 row)
记录他们的oprcom对应的OID
postgres=# select * from pg_operator where oprname='>' and oprcode='int4gt'::regproc;
oprname | oprnamespace | oprowner | oprkind | oprcanmerge | oprcanhash | oprleft | oprright | oprresult | oprcom | oprnegate | oprc
ode | oprrest | oprjoin
---------+--------------+----------+---------+-------------+------------+---------+----------+-----------+--------+-----------+-----
----+-------------+-----------------
> | 11 | 10 | b | f | f | 23 | 23 | 16 | 97 | 523 | int4
gt | scalargtsel | scalargtjoinsel
(1 row)
postgres=# select * from pg_operator where oprname='<' and oprcode='int4lt'::regproc;
oprname | oprnamespace | oprowner | oprkind | oprcanmerge | oprcanhash | oprleft | oprright | oprresult | oprcom | oprnegate | oprc
ode | oprrest | oprjoin
---------+--------------+----------+---------+-------------+------------+---------+----------+-----------+--------+-----------+-----
----+-------------+-----------------
< | 11 | 10 | b | f | f | 23 | 23 | 16 | 521 | 525 | int4
lt | scalarltsel | scalarltjoinsel
(1 row)
接下来我要通过更新pg_operator解除他们的commutator关系，设置为0即可。
postgres=# update pg_operator set oprcom=0 where oprname='>' and oprcode='int4gt'::regproc;
UPDATE 1
postgres=# update pg_operator set oprcom=0 where oprname='<' and oprcode='int4lt'::regproc;
UPDATE 1
创建测试表，插入测试数据，创建索引：
postgres=# create table tbl(id int);
CREATE TABLE
postgres=# insert into tbl select generate_series(1,100000);
INSERT 0 100000
postgres=# create index idx_tbl_id on tbl(id);
CREATE INDEX
将列放在条件的左边可以走索引，但是放在右边不走索引。因为优化器不能决定>,<是否为commutator
postgres=# explain select * from tbl where id<10;
QUERY PLAN
---------------------------------------------------------------------------
Index Only Scan using idx_tbl_id on tbl (cost=0.29..8.45 rows=9 width=4)
Index Cond: (id < 10)
(2 rows)
postgres=# explain select * from tbl where 10>id;
QUERY PLAN
----------------------------------------------------------
Seq Scan on tbl (cost=0.00..1361.00 rows=33333 width=4)
Filter: (10 > id)
(2 rows)
重新建立这两个 operator的commutator关系后，优化器会自动将10>id转换为id<10，并且走索引了：
postgres=# update pg_operator set oprcom=521 where oprname='<' and oprcode='int4lt'::regproc;
UPDATE 1
postgres=# update pg_operator set oprcom=97 where oprname='>' and oprcode='int4gt'::regproc;
UPDATE 1
postgres=# explain select * from tbl where 10>id;
QUERY PLAN
---------------------------------------------------------------------------
Index Only Scan using idx_tbl_id on tbl (cost=0.29..8.45 rows=9 width=4)
Index Cond: (id < 10)
(2 rows)
2. negator，指x op1 y 等价于 not(y op2 x)，或者x op1等价于not( y op2)，或者op1 x 等价于not(op2 y)，因此negator支持一元和二元操作符。
例子:
如果=和<>是一对negator操作符，NOT (x = y) 可以简化为 x <> y。
postgres=# explain select * from tbl where 10=id;
QUERY PLAN
---------------------------------------------------------------------------
Index Only Scan using idx_tbl_id on tbl (cost=0.29..8.31 rows=1 width=4)
Index Cond: (id = 10)
(2 rows)
postgres=# explain select * from tbl where not(10<>id);
QUERY PLAN
---------------------------------------------------------------------------
Index Only Scan using idx_tbl_id on tbl (cost=0.29..8.31 rows=1 width=4)
Index Cond: (id = 10)
(2 rows)
同样，操作符两侧参数x,y的类型必须一致。并且仅适用于返回布尔逻辑类型的操作符。
3. restrict，是用于评估选择性的函数，仅适用于二元操作符，例如where col>100，这个查询条件，如何评估选择性呢？是通过操作符的restrict来指定的，选择性乘以pg_class.reltuples就可以评估得到这个查询条件的行数。
选择性函数的代码在 src/backend/utils/adt/
包括
-rw-r--r--. 1 1107 1107 33191 Jun 10 03:29 array_selfuncs.c
-rw-r--r--. 1 1107 1107 2316 Jun 10 03:29 geo_selfuncs.c
-rw-r--r--. 1 1107 1107 720 Jun 10 03:29 network_selfuncs.c
-rw-r--r--. 1 1107 1107 33895 Jun 10 03:29 rangetypes_selfuncs.c
-rw-r--r--. 1 1107 1107 218809 Jun 10 03:29 selfuncs.c
选择性函数，还需要依赖数据库的统计信息，从而计算选择性，常见的选择性计算函数有：
postgres=# select distinct oprrest from pg_operator order by 1;
oprrest
--------------
-
eqsel 相等
neqsel 不相等
scalarltsel 小于等于
scalargtsel 大于等于
areasel
positionsel
contsel
iclikesel
icnlikesel
regexeqsel
likesel
icregexeqsel
regexnesel
nlikesel
icregexnesel
rangesel
networksel
tsmatchsel
arraycontsel
(20 rows)
当然，用户如果自定义数据类型的话，也可以自定义选择性函数，或者使用以上标准的选择性函数，只是可能需要实现一下类型转换。
源码中的介绍：
src/backend/utils/adt/selfuncs.c
/*----------
* Operator selectivity estimation functions are called to estimate the
* selectivity of WHERE clauses whose top-level operator is their operator.
* We divide the problem into two cases:
* Restriction clause estimation: the clause involves vars of just
* one relation. 一种是符合WHERE条件的选择性（百分比）。
* Join clause estimation: the clause involves vars of multiple rels.
* Join selectivity estimation is far more difficult and usually less accurate
* than restriction estimation. -- JOIN的选择性评估通常没有WHERE条件的选择性准确。
*
* When dealing with the inner scan of a nestloop join, we consider the
* join's joinclauses as restriction clauses for the inner relation, and
* treat vars of the outer relation as parameters (a/k/a constants of unknown
* values). So, restriction estimators need to be able to accept an argument
* telling which relation is to be treated as the variable.
在使用nestloop JOIN时，一个表的字段将作为变量，另一个表的字段（及其统计信息）与操作符作为JOIN评估子句。
*
* The call convention for a restriction estimator (oprrest function) is
*
* Selectivity oprrest (PlannerInfo *root,
* Oid operator,
* List *args,
* int varRelid);
* 评估选择性需要4个参数:
* root: general information about the query (rtable and RelOptInfo lists
* are particularly important for the estimator). plannerinfo信息。
* operator: OID of the specific operator in question. 操作符的OID
* args: argument list from the operator clause. 操作符子句中的参数列表
* varRelid: if not zero, the relid (rtable index) of the relation to
* be treated as the variable relation. May be zero if the args list
* is known to contain vars of only one relation. 表示where条件所包含的参数来自哪些relation。
*
* This is represented at the SQL level (in pg_proc) as
*
* float8 oprrest (internal, oid, internal, int4); 在pg_proc数据字典中表示为oprrest指定的函数。
*
* The result is a selectivity, that is, a fraction (0 to 1) of the rows
* of the relation that are expected to produce a TRUE result for the
* given operator. 选择性函数的评估结果就是一个百分比。乘以pg_class.reltuples就可以得到记录数。
*
* The call convention for a join estimator (oprjoin function) is similar
* except that varRelid is not needed, and instead join information is
* supplied:
* JOIN选择性的计算函数与WHERE选择性的计算函数参数有轻微差别，么有varRelid, 增加了join信息的参数。
* Selectivity oprjoin (PlannerInfo *root,
* Oid operator,
* List *args,
* JoinType jointype,
* SpecialJoinInfo *sjinfo);
*
* float8 oprjoin (internal, oid, internal, int2, internal);
*
* (Before Postgres 8.4, join estimators had only the first four of these
* parameters. That signature is still allowed, but deprecated.) The
* relationship between jointype and sjinfo is explained in the comments for
* clause_selectivity() --- the short version is that jointype is usually
* best ignored in favor of examining sjinfo.
*
* Join selectivity for regular inner and outer joins is defined as the
* fraction (0 to 1) of the cross product of the relations that is expected
* to produce a TRUE result for the given operator. For both semi and anti (半连接与预连接)
* joins, however, the selectivity is defined as the fraction of the left-hand
* side relation's rows that are expected to have a match (ie, at least one
* row with a TRUE result) in the right-hand side.
*
* For both oprrest and oprjoin functions, the operator's input collation OID
* (if any) is passed using the standard fmgr mechanism, so that the estimator
* function can fetch it with PG_GET_COLLATION(). Note, however, that all
* statistics in pg_statistic are currently built using the database's default
* collation. Thus, in most cases where we are looking at statistics, we
* should ignore the actual operator collation and use DEFAULT_COLLATION_OID.
* We expect that the error induced by doing this is usually not large enough
* to justify complicating matters.
*----------
4. join，是joinsel即join的选择性计算函数。
对应pg_operator.oprjoin
postgres=# select distinct oprjoin from pg_operator order by 1;
oprjoin
------------------
-
eqjoinsel
neqjoinsel
scalarltjoinsel
scalargtjoinsel
areajoinsel
positionjoinsel
contjoinsel
iclikejoinsel
icnlikejoinsel
regexeqjoinsel
likejoinsel
icregexeqjoinsel
regexnejoinsel
nlikejoinsel
icregexnejoinsel
networkjoinsel
tsmatchjoinsel
arraycontjoinsel
(19 rows)
5. hashes
6. merges
hashes和merges表示该操作符是否允许hash join和merge join, 只有返回布尔逻辑值的二元操作符满足这个要求。
我们在pg_operator这个catalog中也可以查看到对应的介绍：
| Name | Type | References | Description |
|---|---|---|---|
| oid | oid | Row identifier (hidden attribute; must be explicitly selected) | |
| oprname | name | Name of the operator | |
| oprnamespace | oid | pg_namespace.oid | The OID of the namespace that contains this operator |
| oprowner | oid | pg_authid.oid | Owner of the operator |
| oprkind | char | b = infix ("between"), l = prefix ("left"), r = postfix ("right") 指定操作符在什么位置，例如中间，左侧，右侧 |
|
| oprcanmerge | bool | This operator supports merge joins 此操作符是否支持merge join |
|
| oprcanhash | bool | This operator supports hash joins 此操作符是否支持hash join |
|
| oprleft | oid | pg_type.oid | Type of the left operand 操作符左侧的数据类型 |
| oprright | oid | pg_type.oid | Type of the right operand 操作符右侧的数据类型 |
| oprresult | oid | pg_type.oid | Type of the result 返回结果的数据类型 |
| oprcom | oid | pg_operator.oid | Commutator of this operator, if any |
| oprnegate | oid | pg_operator.oid | Negator of this operator, if any |
| oprcode | regproc | pg_proc.oid | Function that implements this operator |
| oprrest | regproc | pg_proc.oid | Restriction selectivity estimation function for this operator |
| oprjoin | regproc | pg_proc.oid | Join selectivity estimation function for this operator |
