hw1 - CSCI 2400 - Models of Computation Homework 1...

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CSCI 2400 - Models of Computation Homework 1 Solutions Problem 1 (25 points) Prove that if A and B are languages over the same alphabet Σ and A B , then A * B * . Solution: We will prove by mathematical induction that A k B k , for all k 0. Base case: A 0 B 0 ⇒ { λ } ⊆ { λ } Inductive hypothesis: Assume that A k B k for some k 0. Inductive step: Given that A k B k , demonstrate that A k +1 B k +1 : A k B k A k · A B k · A B k · B A k +1 B k +1 Thus, we can show that A k B k for any positive, Fnite k by building up from our base case, one step at a time. It follows that because A * = A 0 A 1 . . . and B * = B 0 B 1 . . . , then A * B * . Problem 2 (25 points) Prove that there does not exist a language L such that: L * = { a } * { b } * . Solution: Assume that there is such a language, and attempt to Fnd a contra- diction. { a
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This note was uploaded on 04/07/2008 for the course CSCI 2400 taught by Professor Carothers during the Spring '08 term at Rensselaer Polytechnic Institute.

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hw1 - CSCI 2400 - Models of Computation Homework 1...

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