F2007Assignment1_Review - COT 5310 Fall 2006 Review...

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COT 5310 Fall 2006 Review Assignment #1 Name:_ KEY __________________ 4 1. Present a Mealy Model finite state machine that reads an input x {0, 1}* and produces the binary number that represents the result of incrementing x by 1 (assumes all numbers are positive, including results). Assume that x is read starting with its least significant digit. Examples: 0010 0011; 1011 1100; 1111 0000 (wrong answer due to overflow) 6 2. Let L be defined as the language accepted by the finite state automaton M : Using the technique of collapsing states, replacing transition letters by regular expressions, I have included the diagrams , B , then C , in that order. You must use this approach of collapsing one state at a time, showing the resulting regular expressions. S F M : S C F M : S B C F M : S A B C F M : λ 0 1 0, 1 0 1 0 1
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COT 5310 – 2 – Review Assignment#1 – Hughes 3. Let L be defined as the language accepted by the finite state automaton M : 6 a.) Present the regular equations associated with each of M 's states, solving for the language recognized by M . b.) Assuming that we designate A as state 1 , B as state 2 and C as state 3 . Kleene’s Theorem allows us to associate regular expressions with M where is the set of strings formed by going from state i to State j without passing through states whose indices are higher than k . 4 Expressed in terms of these i , j k R , what is the language recognized by M ? L( A ) = 4 c.) Present a regular grammar that generates L , the language recognized by M . A C B M : 0 1 1 1 0 0 0
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COT 5310 – 3 – Review Assignment#1 – Hughes 4. Consider the regular grammar G : S 0 S | 1 A | 0 A A 1 S | 0 A | 0 B | λ B 0 S | 0 B 4 a.) Present an automaton M that accepts the language generated by G : b.) Regular grammars generate the class of regular languages. Regular expressions denote the class of regular sets. The equivalence of these is seen by a proof that every regular set is a regular language and vice versa. The first part of this, that every regular set is a regular language, can be done by first showing that the basis regular sets ( Ø , { λη } , { a | a Σ } ) are each generated by a regular grammar over the alphabet . 3 i.) Demonstrate a regular grammar for each of the basis regular sets. The empty set has been done for you. Ø G = { {S}, Σ , S, {S → S}} { λ } { a } Let L be generated by the regular grammar G = ( N , Σ , S , P ) .
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F2007Assignment1_Review - COT 5310 Fall 2006 Review...

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