# Exercises 01

## Exercise 1

## Exercise 2

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 3

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 denoting that expression optional
- Limiting Kleene repetition with a maximum and minimum bound

Design your own operator for regular expressions to make it expressible for describing nested comments.

## Exercise 4

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 5

Let be the language of strings on defined by that is, .

- Construct a DFA that accepts .
- 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*)

## Exercise 7

Homomorphism is function from letters to strings such that

- Prove if language is regular then is regular.
- Using Pumping Lemma prove that the language is not regular.
- Prove that equally many digits before and after decimal point is not regular.