Implement the current set of rules for precedence partial ordering

executable_semantics/syntax/README.md

@@ -15,3 +15,134 @@ techniques can be applied to other kinds of AST nodes as needed. See the
 handling of the `UNIMPL_EXAMPLE` token for an example of how this is done, and
 see [`unimplemented_example_test.cpp`](unimplemented_example_test.cpp) for an
 example of how to test it.
+
+## Precedence and associativity
+
+The [Bison expression grammar](parser.ypp) uses the
+[precedence climbing method](https://en.wikipedia.org/wiki/Operator-precedence_parser#Precedence_climbing_method)
+to model precedence and associativity, suitably modified to handle Carbon's
+partial precedence order without grammar ambiguities.
+
+Consider this example
+[precedence diagram](/docs/design/expressions/README.md#precedence):
+
+```mermaid
+graph BT
+    %%{init: {'themeVariables': {'fontFamily': 'monospace'}}}%%
+    minus["minus<br>-x"]
+    mul>"mul<br>x * y"]
+    add>"add<br>x + y"]
+    mod["mod<br>x % y"]
+    eq["eq<br>x = y"]
+
+    eq --> add & mod
+    add --> mul
+    mul & mod --> minus
+```
+
+For each precedence level, we have up to three grammar productions:
+
+-   `foo_expression` represents an expression at that precedence level or
+    higher, and includes as productions all of the expression kinds that are
+    immediately higher in the precedence graph:
+    ```bison
+    add_expression:
+      mul_expression | add_lhs '+' add_operand ;
+    ```
+-   `foo_operand` represents an operand of a `foo_expression` that is not itself
+    a `foo_expression`.
+    ```bison
+    eq_operand:
+      add_expression | mod_expression ;
+    ```
+-   For left-associative operators, `foo_lhs` represents either a `foo_operand`
+    or a `foo_expression`.
+    ```
+    add_lhs:
+      add_operand | add_expression ;
+    ```
+
+The above approach leads to (benign) reduce-reduce conflicts. In our example
+precedence diagram, the expression `-x == y` has two different parses:
+
+-   _eq_expression_
+    -   _eq_operand_
+        -   _add_expression_
+            -   _mul_expression_
+                -   _minus_expression_
+                    -   `-`
+                    -   `x`
+    -   `==`
+    -   _eq_operand_
+        -   ...
+            -   `y`
+
+and
+
+-   _eq_expression_
+    -   _eq_operand_
+        -   _mod_expression_
+            -   _minus_expression_
+                -   `-`
+                -   `x`
+    -   `==`
+    -   _eq_operand_
+        -   ...
+            -   `y`
+
+These would invoke the same parsing actions, so the states can be combined, but
+Bison isn't smart enough to see that.
+
+In order to eliminate these conflicts, if there are multiple paths through the
+precedence graph between a higher-precedence level `foo` and some lower
+precedence level `bar` -- that is, if there's a diamond in the precedence graph
+with `foo` at the top and `bar` at the bottom -- `foo_expression`s are excluded
+from all intermediate `_expression` productions on the diamond between `foo` and
+`bar`, and are added back in the downstream `_operand` productions in the
+diamond instead:
+
+```bison
+minus_expression:
+  identifier | '-' identifier ;
+
+// In the real grammar, trivial productions like this are inlined.
+mul_operand:
+  minus_expression ;
+mul_lhs:
+  mul_operand | mul_expression ;
+// A minus_expression is not a mul_expression, even though it's a
+// higher-precedence expression, because there are multiple paths from
+// eq_expression to minus_expression, and this production is on such a path.
+mul_expression:
+  mul_lhs '*' mul_operand
+
+// minus_expression is listed here because it is excluded from mul_expression.
+add_operand:
+  minus_expression | mul_expression ;
+// This is notionally
+//   add_operand | add_expression
+// but that introduces another kind of reduce-reduce conflict, because there
+// would be two ways to interpret a mul_expression as an add_lhs.
+add_lhs:
+  minus_expression | add_expression ;
+// A mul_expression is an add_expression, because multiplication is
+// higher-precedence, and mul is not at the top of a diamond in the precedence
+// graph. minus_expression is excluded because we are within a diamond with it
+// at the top.
+add_expression:
+  mul_expression | add_lhs '+' add_operand ;
+
+mod_operand:
+  minus_expression ;
+mod_expression:
+  mod_operand '%' mod_operand ;
+
+// We add back minus_expression here because it was excluded from add_expression
+// and mod_expression.
+eq_operand:
+  minus_expression | add_expression | mod_expression ;
+// We also include minus_expression here because this is the bottom of the
+// precedence diamond.
+eq_expression:
+  minus_expression | add_expression | mod_expression | eq_operand '=' eq_operand ;
+```