Bottom-up Parsing

Bottom-up Parsing

1. Recall that in a bottom-up parser, we repeatedly simplify sentential forms (starting with the input) by replacing the right hand side of a production by the left hand side.
   • Again consider the process of parsing the string 'bbaababa' relative to the grammar:

     < S > a < R >  |  b < S > b < R >
     < R > b < R >  |  a

     Head of sentential form	Remaining input
     	bbaababa
     b	baababa
     bb	aababa
     bba	ababa
     bbaa	baba
     bbaR	baba
     bbS	baba
     bbSb	aba
     bbSba	ba
     bbSbR	ba
     bS	ba
     bSb	a
     bSba
     bSbR
     S

   • Note that the parser depends on a stack to remember details of decisions it has already made (somewhat as in a top-down parser).
     • In a bottom up parser, the stack holds a prefix of a sentential form that the parser hopes the input will complete.
     • In a top down parser, the stack holds a suffix of a sentential form which the remaining input must match.

   • At each step a three part decision must be made:
     1. Whether it is time to quit (i.e. either the parse is complete or an error has been encountered).
     2. If we are to continue, should we shift more input symbols onto the stack or are we ready to replace the top of the stack with the left-hand side of some production.
     3. If we are to reduce, which production should be used.

   • As a result, bottom-up parsers are also known as shift-reduce parsers.

   • Note that one can extract a right-most derivation by reversing the steps taken during a bottom-up parse.

   • To be "determinisitic", a bottom-up parser must make these decisions based on examining at most k symbols of the remaining input.
2. One cannot always parse bottom-up by simply reducing as soon as one can match the right hands side of a production.
   • Consider

     If we begin parsing `bbaababa' by reducing the first `a' to obtain 'bbRababa' we will get stuck because the string we obtain is not a sentential form of the language.

   • Consider the effects of eps productions.

3. Assuming that the string being considered is a sentential form, the correct string to reduce is the handle as defined by:

   simple phrase

   Given a grammar G and a string w = such that

   1. w, , , (V_n U V_t)^* ,

   2. for some U V_n, U P and U is a sentential form of G

   we say that is a simple phrase of the sentential form w.

   handle

   The leftmost simple phrase of a sentential form is called the handle.

4. The problem one must solve to construct a bottom-up parser is to find rules for determining when one is looking at the handle.

   • These rules can only involve the current contents of the stack and a bounded number of input characters.

5. The class of grammars for which one can parse bottom-up deterministically (i.e. in without backup) looking ahead at most k characters at each step is called the class of LR(k) grammars.

Bottom-up Parsing