Add some notes about the basic mathematical laws that the system presumes

tglsfdc · tglsfdc · commit d83235415bc2 · 2007-01-12T17:04:54.000Z
hold true for operators in a btree operator family.  This is mostly to
clarify my own thinking about what the planner can assume for optimization
purposes.  (blowing dust off an old abstract-algebra textbook...)
diff --git a/src/backend/access/nbtree/README b/src/backend/access/nbtree/README
@@ -1,4 +1,4 @@
-$PostgreSQL: pgsql/src/backend/access/nbtree/README,v 1.16 2007/01/09 02:14:10 tgl Exp $
+$PostgreSQL: pgsql/src/backend/access/nbtree/README,v 1.17 2007/01/12 17:04:54 tgl Exp $
 
 This directory contains a correct implementation of Lehman and Yao's
 high-concurrency B-tree management algorithm (P. Lehman and S. Yao,
@@ -485,4 +485,47 @@ datatypes to supply us with a comparison procedure via pg_amproc.
 This procedure must take two nonnull values A and B and return an int32 < 0,
 0, or > 0 if A < B, A = B, or A > B, respectively.  The procedure must
 not return INT_MIN for "A < B", since the value may be negated before
-being tested for sign.  See nbtcompare.c for examples.
+being tested for sign.  A null result is disallowed, too.  See nbtcompare.c
+for examples.
+
+There are some basic assumptions that a btree operator family must satisfy:
+
+An = operator must be an equivalence relation; that is, for all non-null
+values A,B,C of the datatype:
+
+	A = A is true						reflexive law
+	if A = B, then B = A					symmetric law
+	if A = B and B = C, then A = C				transitive law
+
+A < operator must be a strong ordering relation; that is, for all non-null
+values A,B,C:
+
+	A < A is false						irreflexive law
+	if A < B and B < C, then A < C				transitive law
+
+Furthermore, the ordering is total; that is, for all non-null values A,B:
+
+	exactly one of A < B, A = B, and B < A is true		trichotomy law
+
+(The trichotomy law justifies the definition of the comparison support
+procedure, of course.)
+
+The other three operators are defined in terms of these two in the obvious way,
+and must act consistently with them.
+
+For an operator family supporting multiple datatypes, the above laws must hold
+when A,B,C are taken from any datatypes in the family.  The transitive laws
+are the trickiest to ensure, as in cross-type situations they represent
+statements that the behaviors of two or three different operators are
+consistent.  As an example, it would not work to put float8 and numeric into
+an opfamily, at least not with the current semantics that numerics are
+converted to float8 for comparison to a float8.  Because of the limited
+accuracy of float8, this means there are distinct numeric values that will
+compare equal to the same float8 value, and thus the transitive law fails.
+
+It should be fairly clear why a btree index requires these laws to hold within
+a single datatype: without them there is no ordering to arrange the keys with.
+Also, index searches using a key of a different datatype require comparisons
+to behave sanely across two datatypes.  The extensions to three or more
+datatypes within a family are not strictly required by the btree index
+mechanism itself, but the planner relies on them for optimization purposes.
