# hw1 - CS 181 Winter 2008 Formal Languages and Automata...

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CS 181 — Winter 2008 Problem Set #1 Formal Languages and Automata Theory Due Jan 16, 2008 Problem 1.1. a. Show that Σ 1 Σ 2 implies Σ * 1 Σ * 2 . Is the converse true? b. Let ( Q, Σ , δ, q 0 , F ) be an FA. Define ˆ δ : Q × Σ * Q by induction as follows: ˆ δ ( q, ) = q ˆ δ ( q, w.a ) = δ ( ˆ δ ( q, w ) , a ) where w Σ * and a Σ. Argue (using induction) that a word w Σ * is accepted iff ˆ δ ( q 0 , w ) F . Problem 1.2. Give state-transition diagrams for DFA s that accept the following two languages. a. A 1 = { x ∈ { 0 , 1 } * | x has at least 3 letters, and x begins and ends with the same letter } . b. A 2 = { x ∈ { 0 , 1 } * | x contains an even number of occurrences of the substring 11 } . For example, the word 111 contains 2 (overlapping) occurrences of 11 . For this exercise, consider 0 to be an even number. Use the following four states: a . . . “have encountered an even number of 11 s, and the last input letter was 0 b . . . “have encountered an even number of 11 s, and the last input letter was 1 c . . . “have encountered an odd number of 11 s, and the last input letter was 0

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• Winter '08
• Rupak
• Formal language, Regular expression, Automata theory, Formal Languages and Automata Theory, Σ∗, input letter

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