Consider the following grammar fragment for if-then-else statements in
a programming language (where teminals are in lower case and
non-terminals are upper case:

    STATEMENT --> if BOOLEAN-EXPRSSION then STATEMENT else STATEMENT
    STATEMENT --> if BOOLEAN-EXPRSSION then STATEMENT 

1) Show that a statement of the form

       if BOOLEAN-EXPRSSION then 
       if BOOLEAN-EXPRSSION then STATEMENT 
       else STATEMENT
    
   is ambiguous according to the grammar fragment.


2) Rewrite the grammar fragment so that it is not ambiguous.  In particular,
   ambiguities should be eliminated in such a way that in the example of 1), 
   the "else" part is associated with the nearest the second "if".
   That is, the statement will read like a), not b).

   a)  (if BOOLEAN-EXPRSSION then 
           (if BOOLEAN-EXPRSSION then STATEMENT else STATEMENT))

   b)  (if BOOLEAN-EXPRSSION 
       then (if BOOLEAN-EXPRSSION then STATEMENT)
       else STATEMENT)
   

3) As the grammar stands, there is no way for a programmer to specify
   a statement with meaning b).  One way to enable this, would be to
   incorporate parentheses (or begin-end pairs) into the language so
   that a programmer could write a) or b) explicitly.  Modify your
   grammar to facilitate this.