CSCI 6610		  Automata and Formal Languages		     Spring 2004

		            Two algorithms for a CFG


These algorithms are not discussed in Sipser, but were covered in CSCI 2670.
The "text" referred to is Hopcroft, Motwani and Ullman.


Def: X is a generating symbol of G iff X is in V and X =*=> w for some w in T*.

Def: b is a reachable symbol of G iff S =*=> w for some w containing b (so b may
       be in V or T).

Def: b is a useful symbol of G iff b belongs to some string in a derivation of
     S =*=> w for some w in T*.
      

Let G = (V,T,P,S) be an arbitrary CFG.  The algorithms given on p. 258 of the
text for finding the generating symbols of G and the reachable symbols of G
are inductive, but the exact form of induction is not specified.  We reformulate
those algorithms to make the induction explicit.  Problems NT.1 and NT.2(a) of
HW set 8 ask for answers in this format.

For the generating symbols, let PI_0 = T, and for each i >= 0 let 
PI_{i+1} = (PI_i union { A in V : there is A --> w in P for some w in PI_i* }).
When PI_{n+1} = PI_n then PI_n is the set of all generating symbols of G.

For the reachable symbols, let R_0 = {S}, and for each i >= 0 let 
R_{i+1} = (R_i union {symbols of w : there is A --> w in P for some A in R_i }).
When R_{n+1} = R_n then R_n is the set of all reachable symbols of G.


For example 7.1 of the text ( S --> AB | a, A --> b ) we have

	PI_0 = {a,b},
	PI_1 = {a,b,S,A},
	PI_2 = PI_1 so those are the generating symbols;

	R_0 = {S},
	R_1 = {S,A,B,a},
	R_2 = {S,A,B,a,b},
	R_3 = R_2 so those are the reachable symbols.

Note that here the reachable symbols algorithm was applied to the original CFG.
As discussed in the text, to eliminate useless symbols one should first find all
generating symbols, eliminate any non-generating symbols to give a modified
CFG G', and then find all reachable symbols in G'.  Removing any non-reachable
symbols from G' produces a CFG G'' which is equivalent to G and contains no
useless symbols (provided that L(G) is nonempty, so that at least S remains
as a useful symbol  --  Thm. 7.2 of the text).