Homework 1

Due Wednesday, 5th October, 10:15am. Please hand in to Eva or Giuliano before the labs.

Problem 1

Find a regular expression that generates all alternating sequences of 0 and 1 with arbitrary length (including lengths zero, one, two, …). For example, the alternating sequences of length one are 0 and 1, length two are 01 and 10, length three are 010 and 101. Note that no two adjacent character can be the same in an alternating sequence.

Problem 2

Construct a DFA for the language of well-nested parenthesis with a maximal nesting depth of 3. For example, ε, ()(), (()(())) and (()())()(), but not (((()))) nor (()(()(()))), nor ())) .

By well-nested parentheses we mean those that are correctly nested, and are thus given by the grammar such as the one seen in the class:

  S ::= \epsilon \mid (S)S

Problem 3

Find two equivalent states in the automaton, and merge them to produce a smaller automaton that recognizes the same language. Repeat until there are no longer equivalent states.

Recall that the general algorithm for minimizing finite automata works in reverse. First, find all pairs of inequivalent states. States X, Y are inequivalent if X is final and Y is not, or (by iteration) if $X \xrightarrow{a} X'$ and $Y \xrightarrow{a} Y'$ and X' and Y' are inequivalent. After this iteration ceases to find new pairs of inequivalent states, then X, Y are equivalent, if they are not inequivalent.

Problem 4

Let $D$ be any deterministic finite automaton. Assume that $D$ contains exactly $n$ states. Show that if it accepts at least one string of length $n$ or greater then the accepted language is infinite.

Problem 5

Let rtail be a function that returns all the symbols of a string except the last one. For example, rtail(Lexer) = Lexe. rtail is undefined for an empty string. If $R$ is a regular expression, then $\textit{rtail}(R)$ applies the function to all the elements. For example, $\textit{rtail}(1 2 0) = 1 2$.

Prove that $\textit{rtail}(R)$ is regular if $R$ is not nullable.

Bonus part: Let $|v|$ denote the length of the string $v$. Let $R$ be a regular language. Is the language $\{ u \mid uv \in R, |v|=7,\ \ u,v \in \Sigma^* \}$ always regular?