Write a regular expression that captures the same language as the following context-free grammar:

  S → p R v
  S → q R v
  R → u R
  R → u

Give two six-symbol strings that are in the language defined by the following context-free grammar:

  S → a S d
  S → a T d
  T → b c
  T → b T c

Consider the following context-free grammar:

  A → t A e A
  A → t A
  A → x

1. Write two sentences consisting of five or more symbols that are in the language specified by the grammar.

2. Is the grammar ambiguous? If so, write a sentence and two different parse trees that illustrate that fact. If not, give an argument why not.

      def matchE(tokens):
           if len(tokens) == 0:
                raise ParseError('unxpected end of input')
           elif tokens[0] == "n":
                matchChar(tokens, 'n')
           else:
                matchChar(tokens, '(')
                matchE(tokens)
                matchChar(tokens, '+')
                matchE(tokens)
                matchChar(tokens, ')')

      def matchChar(tokens, c):
           if len(tokens) == 0:
                raise ParseError('unxpected end of input')
           elif tokens[0] == c:
                tokens.pop(0)   # eat character c from token stream
           else:
                raise ParseError('%c expected' % c)

1. Write a context-free grammar that corresponds to this parser.

2. Suppose we wrote a similar parser for this grammar:

     E → n
     E → E + E
   

   What problems might we encounter when we run our program?

Do one of the following:

1. Write a computer program in a language of your choosing that takes a context-free grammar for an input and returns true if and only if the grammar is unambiguous.

   or

2. Write a brief explanation of why you feel that the first part of this question is not a fair question to ask on this problem set.