<sentence> ::= abc | a<thing>bc
     <thing>b   ::= b<thing>
     <thing>c   ::= <other>bcc 
     a<other>   ::= aa | aa<thing> 
     b<other>   ::= <other>b

  <sentence> ⇒ a<thing>bc
             ⇒ ab<thing>c
             ⇒ ab<other>bcc 
             ⇒ a<other>bbcc 
             ⇒ aa<thing>bbcc
             ⇒ aab<thing>bcc 
             ⇒ aabb<thing>cc
             ⇒ aabb<other>bccc 
             ⇒ aab<other>bbccc
             ⇒ aa<other>bbbccc 
             ⇒ aaabbbccc

     <string> ::= a <B> | b <A>
     <A> ::= a | a <string> | b <A> <A>
     <B> ::= b | b <string> | b <B> <B>

This grammar generates a language, which is too complicated to be described easily. However, the following grammar generates a language, in which all strings have equal numbers of a's and b's.

   <string> ::= a <B> | b <A>
   <A> ::= a | a <string> | b <A> <A>
   <B> ::= b | b <string> | a <B> <B>

     <stmt> ::= if <expr> then <stmt> | <matchedStmt>
     <matchedStmt> ::= if <expr> then <matchedStmt> else <stmt> | other 
     <expr> ::= 0 | 1

The solutions use the following grammar, which is slightly different from the above grammar.

    <statement>    ::= if ( <exp> ) <statement> | <matched-stmt>
    <matched-stmt> ::= if ( <exp> ) <matched-stmt> else <statement> | other
    <exp>          ::= 0 | 1

   <roman> ::= <tens> <units>
   <tens> ::= <low tens> | XL | L <low tens> | XC
   <low tens> ::= ε | X | XX | XXX
   <units> ::= <low units> | IV | V <low units> | IX
   <low units> ::= ε | I | II | III