4.3 cont

Left Factoring

A technique useful for producing a better grammar for parsing
Helps simplify the choice between two productions.
Consider the following grammar (again)

stmt → if expr then stmt else stmt

| if expr then stmt
It is not possible to decide which production to use when we see this if.
So let's factor the grammar to fix that.

stmt → if expr then stmt ifend
ifend → else stmt

| ε

In general, we transform

A → αΒ₁|αΒ₂

into 
A → αA'
A' → Β₁ | Β₂

Algorithm 4.21

Input G, a grammar

 For each nonterminal A
    find the longest possible prefix α common to two or more of its alternatives
    if α ≠ ε
         replace all alternatives of the form A → αΒ_n with  A → αA'
         Other productions (A → γ) remain.
         Add a rule A' → α Β₁ | αΒ₂ | ... | αΒ_n
         If A→ α was an alternative, add A' → ε
         GOTO step 2

Non-Context Free Language Constructs.
- We are provided with two things you might want to do with a grammar, but no context free grammar can.
- Recognize declaring a variable then using it. (But we have this already)
  - L = { wcw | w ∈ (a|b)^*}
  - "It is beyond the scope of this book ..."
  - So we get back a <id> and use the symbol table.
- Checking the length of arguments against formal parameter lists.
  - L = {aⁿb^mcⁿd^m | n ≥ 1 and m ≥ 1}
  - a and b represent declarations, c and d represent invocations.
  - Parameters are done with
    
    stmt → id (expr_list)
    
    expr_list → expr_list, expr
    
    | expr

stmt	→	if expr then stmt else stmt
	\|	if expr then stmt

stmt	→	if expr then stmt ifend
ifend	→	else stmt
	\|	ε

stmt	→	id (expr_list)
expr_list	→	expr_list, expr
	\|	expr