Constructing the LR(0) Machine for a Grammar

Constructing the LR(0) Machine for a Grammar

1. To make the ideas we talked about last time precise (and eventually prove that we can use them to construct a correct parser) we need to refresh your knowledge of finite automata a bit.
2. First, recall the structure of a deterministic finite state machine.
   1. A finite set of states, .
   2. An input alphabet, .
   3. A transition function : x -> .
   4. A subset F of called the set of final states.
   5. An element ₀ of called the initial state.
3. While you are at it, recall (or at least note) that we can explain the behavior of a deterministic finite state machine by defining a function that extends to strings over the input alphabet. In particular, we can define : x ^* -> recursively as
   • ( , ) =
   • ( , x ) = ( ( , ), x )
   and then state that the language accepted by the machine is

   { ^*  | ( ₀, ) F }

4. Now, we can give a general definition of the LR(0) machine for an arbitrary grammar G.

   Of course, we need a few more definitions:

   goto

   Given a set of LR(0) items for a grammar G, we define

   goto( , x ) = { [ N ₁ x . ₂ ]  | [ N ₁ . x ₂ ] }

   closure

   Given a set of LR(0) items for a grammar G with productions P, we define closure() to be the smallest set of LR(0) items such that:

   1. closure()
   2. if [N_{1 1} . N_{2 2}] closure() and N_{2 3} P then [N₂ . ₃] closure()

5. The closure of a set of LR(0) items can be computed using a simple (but important) little algorithm
   • An algorithm to compute closure( )
     1. set ' equal to .
     2. while there is some [ N ₁ . M ₂ ] ' such that M ₃ P and [ M . ₃ ] ' add [M . ₃ ] to '.
6. With these definitions and the assumption that the start symbol S of G is replaced by a new start symbol S' and that the rule S' S $ is added to the set of productions (The $ just stands for end-of-input). The definition of the LR(0) machine is:
   • Let the set of states of the machine be the set of all sets of LR(0) items for G.
   • Let the set of final states be all states except the state corresponding to the empty set of LR(0) item.
   • Let the initial state be the state corresponding to the set

     closure ( {[ S' . S $]} )

   • Let the transition function : x (V_n U V_t) -> be defined by:

     (,x) = closure(goto(,x))

7. A somewhat interesting example.

   < S` >	< S > $
   < S >	< S > a < S > b  |  c

   

Constructing the LR(0) Machine for a Grammar