## Artifact b7b18856ea69672e08faa94777221ec9ddaa5933:

D 2017-03-12T23:12:09.151 L rewriting\sprecedenced\srules N text/x-markdown P 2aa4c51f9851327a3d07ca9e67f9f23b4e9e5580 U aku W 3356 Up: [Notes](wiki?name=Notes) # Rewriting of precedenced rules to embed the precedence into the grammar ## Precedences * Precedences are integers <= 0. * The tighest precedence is 0. * The loosest precedence is some `K` <= 0 (dependent on the LHS) Rules: looser(x) = x-1 for x > K, x in [K, 0] = K for x = K tighter(x) = x+1 for x < 0, x in [K, 0] = 0 for x = 0 tighest = 0 loosest = K ## Rewriting In what follows, `E` will represent the LHS symbol of the precedenced rule, i.e. the operator. We will need to have a unique symbol for `E` at each level of precedence. We will represent the symbol for `E` at level `x` as `E[x]`. ### Top level The top level rule for the precedenced rule will be a rule whose LHS is the precedenced rule's LHS, and whose RHS is the symbol for `E` at the __lowest__ precedence (`K`). That is, the top level rule will be E ::= E[K] ### The "spine" In many expressions, there is no logic at a given precedence level, so that we need to simply "fall through" to the next-tighter level of precedence. Therefore, for every precedence `x`, such that `x` is less than `tightest`, we add the rule E[x] ::= E[tighter(x)], x < 0 For example, if the precedence ran from `loosest == -4` to `tightest == 0`, we would add the BNF rules E[-4] ::= E[-3] E[-3] ::= E[-2] E[-2] ::= E[-1] E[-1] ::= E[0] ### Left association We now can deal with the RHS alternatives in the work list. Let `curr` be the precedence level of the RHS alternative. If a RHS alternative has left association (the default) we rewrite the RHS, replacing all occurrences of `E`. We replace the __leftmost__ occurrence of `E` with `E[curr]`, and the others with `E[tighter(curr)]`. We then add a rule with the rewritten RHS, and `E[curr]` as the LHS. For example, if the RHS alternative is E + E we add the BNF rule E[curr] ::= E[curr] + E[tighter(curr)] Note that `E` may not occur on the RHS, in which case no RHS replacements will be necessary. ### Right association Right association is handled in a way that is symmetric with left association. Again, let `curr` be the precedence of the RHS alternative. If a RHS alternative has right association, we rewrite the RHS, replacing all occurrences of `E`. We replace the __rightmost__ occurrence of `E` with `E[curr]`, and the others with `E[tighter(curr)]`. We then add a rule with the rewritten RHS, and `E[curr]` as the LHS. For example, if the RHS alternative is E ** E we add the BNF rule E[curr] ::= E[tighter(curr)] ** E[curr] ### Group association The archetypal case for group association is the parenthesis operator. Intuitively, group association allows any expression to occur inside another "surrounding" expression. In the case of the parentheses, of course, they actually surround their contents lexically. Let `curr` be the precedence of the RHS alternative. If a RHS alternative has group association, we rewrite the RHS, replacing all occurrences of `E` with `E[K]`, We then add a rule with the rewritten RHS, and `E[curr]` as the LHS. For example, if the RHS alternative is ( E ) we add the BNF rule E[curr] ::= ( E[K] ) Z 8dbe83f25bfa3e830bdafd18596f11f3