Exercises 1

Exercise 1

Assume the following extensions to the regular expressions. In each case describe why the modification does not actually change the expressibility.

  • The intersection of two regular expressions.
  • The optional expression $(r)?$ denoting that expression $r$ optional
  • Limiting Kleene repetition with a maximum and minimum bound

Design your own operator that extends regular expressions to make it possible to express nested comments.

Exercise 2

Convert the following NFAs to deterministic finite automata.

a) b)

Exercise 3

Integer literals are in three forms in Scala: decimal, hexadecimal and octal. The compiler discriminates different classes from their beginning. Decimal integers are started with a non-zero digit. Hexadecimal numbers begin with 0x or 0X and may contain the digits from 0 through 9 as well as upper or lowercase digits A to F afterwards. If the integer number starts with zero, it is in octal representation so it can contain only digits 0 through 7. There can be an l or L at the end of the literal to show the number is Long.

  • Draw a single DFA that accepts all the allowable integer literals.
  • Write the corresponding regular expression.

Exercise 4

Design a DFA which accepts all the binary numbers divisible by 6. For example your automaton should accept the words 0, 110 (6 decimal) and 10010 (18 decimal).

Exercise 5

Let $L$ be the language of strings on $\Sigma = \{<,=\}$ defined by $L = \{<,=,<====^{*}\}$ that is, $\{ <, = \} \cup \{ <=^{n} \mid n \ge 3 \}$.

  • Construct a DFA that accepts $L$.
  • Describe how the lexical analyzer will tokenize the following inputs.
    • <=====
    • ==<==<==<==<==
    • <=====<

Exercise 6

For each of the following languages find the first set. Determine if the language is nullable.

(a|b)*(b|d)((c|a|d)* | a*)