LL parser

An LL parser is called an LL(k) parser if it uses k tokens of look-ahead when parsing a sentence. If such a parser exists for a certain grammar and it can parse sentences of this grammar without backtracking then it is called an LL(k) grammar. Of these grammars LL(1) grammars, although fairly restrictive, are very popular because the corresponding LL parsers only need to look at the next token to make their parsing decisions.

Architecture of an LL parser

A table-based top-down parser can be schematically presented as in Figure 1.

         +---+---+---+---+---+---+
  Input: | ( | 1 | + | 1 | ) | $ |
         +---+---+---+---+---+---+
                   ^
                   |
  Stack:           |
              +----------+
  +---+       |          |
  | + |<------|  Parser  |----->  Output
  +---+       |          |
  | S |       +----------+
  +---+            ^                  
  | ) |            |             
  +---+            |                   
  | $ |       +----------+
  +---+       | Parsing  | 
              |  table   |
              +----------+

(1) S -> F

(2) S -> ( S + F )

(3) F -> 1

The parsing table for this grammar looks as follows:

Note that there is also a column for the special terminal $ that is used to indicate the end of the input stream.

When the parser starts it always starts on its stack with

 [ S, $ ]

where $ is a special terminal to indicate the bottom of the stack and the end of the input stream, and S is the start symbol of the grammar. The parser will attempt to rewrite the contents of this stack to what it sees on the input stream. However, it only keeps on the stack what still needs to be rewritten. For example, let's assume that the input is "( 1 + 1 )". When the parser reads the first "(" it knows that it has to rewrite S to "( S + F )" and writes the number of this rule to the output. The stack then becomes:

 [ (, S, +, F, ), $ ]

In the next step it removes the '(' from its input stream and from its stack:

 [ S, +, F, ), $ ]

Now the parser sees an '1' on its input stream so it knows that it has to apply rule (1) and then rule (3) from the grammar and write their number to the output stream. This results in the following stacks:

 [ F, +, F, ), $ ]
 [ 1, +, F, ), $ ]

In the next two steps the parser reads the '1' and '+' from the input stream and also removes them from the stack, resulting in:

 [ F, ), $ ]

In the next three steps the 'F' will be replaced on the stack with '1', the number 3 will be written to the output stream and then the '1' and ')' will be removed from the stack and the input stream. So the parser ends with both '$' on its stack and on its input steam. In this case it will report that it has accepted the input string and on the output stream it has written the list of numbers [ 2, 1, 3, 3 ] which is indeed a rightmost derivation if the input string in reverse.

As can be seen from the example the parser performs three types of steps depending on whether the top of the stack is a nonterminal, a terminal or the special symbol $:

• If the top is a nonterminal then it looks up in the parsing table on the basis of this nonterminal and the symbol on the input stream which rule of the grammar it should use to replace it with on the stack. The number of the rule is written to the output stream. If the parsing table indicates that there is no such rule then it reports an error and stops.
• If the top is a terminal then it compares it to the symbol on the input stream and if they are equal they are both removed. If they are not equal the the parser reports an error and stops.
• If the top is $ and on the input stream there is also a $ then the parser reports that it has successfully parse the input, otherwise it reports an error. In both cases the parser will stop.

Constructing an LL(1) parsing table

In order to fill the parsing table we have to establish what grammar rule the parser should choose if it sees a nonterminal A on the top of its stack and symbol a on its input stream. It is easy to see that such a rule should be of the from A -> w and that the language corresponding with w should have at least one string starting with a. For this purpose we define the First-set of w, written here as Fi(w), as the terminals with which the strings that belong to w start plus ε if the empty strings also belongs to w. Given a grammar with the rules A₁ -> w₁, ..., A_n -> w_n we can compute the Fi(w_i) and Fi(A_i) for every rule as follows:

1. initialize every Fi(w_i) and Fi(A_i) with the empty set
2. add Fi(w_i) to Fi(w_i) for every rule A_i -> w_i where Fi is defined as follows:
   • Fi(a w' ) = { a } for every terminal a
   • Fi(A w' ) = Fi(A) for every nonterminal A with ε not in Fi(A)
   • Fi(A w' ) = Fi(A) \ { ε } ∪ Fi(w' ) for every nonterminal A with ε in Fi(A)
   • Fi(ε) = { ε }
3. add Fi(w_i) to Fi(Ai) for every rule A_i -> w_i
4. repeat the steps 2 and 3 until all Fi sets stay the same.

Unfortunately the First-sets are not sufficient to compute the parsing table. This is because a right-hand side w of a rule might ultimately be rewritten to the empty string. So the parser should also use the a rule A -> w if ε is in Fi(w) and it sees on the input stream a symbol that could follow A. Therefore we also need the Follow-set of A, written as Fo(A) here, which is defined as the set of terminals a such that there is a string of symbols αAaβ that can be derived from the start symbol. Computing the Follow-sets for the nonterminals in a grammar can be done as follows:

1. initialize every Fo(A_i) with the empty set
2. if there is a rule of the form A_j -> wA_iw' then
   • if the terminal a is in Fi(w' ) then add a to Fo(A_i)
   • if ε is in Fi(w' ) then add Fo(A_j) to Fo(A_i)
3. repeat step 2 until all Fo sets stay the same.

Now we can define exactly which rules will be contained where in the parsing table. If T[A, a] denotes the entry in the table for nonterminal A and terminal a then

T[A,a] contains the rule A -> w iff

a is in Fi(w) or

ε is in Fi(w) and a is in Fo(A).

If the table will contain at most one rule in every one of its cells then the parser will always know which rule it has to use and can therefore parse strings without backtracking. It is precisely in this case that the grammar is called an LL(1) grammar.

Constructing an LL(k) parsing table

... yet to be written ...