Question 14.4.4: Show how to compute inductively the set State(S) of all ......
Show how to compute inductively the set State(S) of all situations coherent with a given string S.
(1) If a string S is coherent with a situation [K → U_V, t] and the first character in V is J, that is, V = JW, then SJ is coherent with the situation [K → UJ_W, t].
This rule determines completely which situations that do not start with underscore are coherent with SJ. It remains to find out for which nonterminals K and terminals t the string SJ belongs to Left(K, t). This is done according to the following two rules:
(2) If the situation [L → U_V, g] is coherent with SJ (according to (1)) and V starts with a nonterminal K, then SJ belongs to Left(K, s) for any terminal s that may appear as a first symbol in a string derivable from V \ K (the string V without the first symbol K), as well as for s = t, if the empty string is derivable from V \ K.
(3) If SJ is in Left(L, t) for some L and g, and L → V is a production rule, and V starts with a nonterminal K, then SJ belongs to Left(K, s) for any nonterminal s that may appear as a first symbol in a string derivable from V \ K, as well as for s = t, if the empty string is derivable from V \ K.