​​ Exercise 2.2.1 Consider the context-free grammar S  S S + | S S * | a a. Show how the string aa + a* can be generated by this grammar. b. Construct a parse tree for this string. c. What language does this grammar generate? Justify your answer. Exercise 2.2.2 What language is generated by the following grammars? In each case justify your answer. Also describe each language informally using a sentence or two in English. a. S  0 S 1 | 0 1 b. S  + S S | - S S | a c. S  S ( S ) S | ε d. S  a S b S | b S a S | ' e. S  a | S + S | S S | S * | ( S ) Exercise 2.2.3 Which of the grammars in Exercise 2.2.2 are ambiguous? Exercise 2.3.1 Construct a syntax-directed translation scheme that translates arithmetic expressions from infix notation into prefix notation in which an operator appears before its operands; e.g., -xy is the prefix notation for x – y. Give annotated parse trees for the inputs 9-5+2 and 9-5*2.