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37 Slides--Finite Automata II

37 Slides--Finite Automata II - CS103 HO#37 Slides-Finite...

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CS103 HO#37 Slides--Finite Automata II May 3, 2010 1 CS103 Mathematical Foundations of Computing 5/3/10 Midterm Exam: Tuesday, May 4, 7 – 9 pm Coverage is through induction (no automata) Rooms by last name: A – F Educ 128 G – Ke Gates B12 Ki – Z 420-041 Please go to the correct room and sit in alternate seats. Note: It is important to do the reading in the Sipser text. You should try to understand it line by line. Come to office hours , write us with questions, or talk to your classmates! Let = {a, b, c, d} Let L Missing = {w | there is a symbol from not in w} Start state: all letters missing After one character, the state could be a read, b, c, d still missing b read, a, c, d still missing c read, a, b, d still missing d read, a, b, c still missing After two characters, the state could be any of the previous, or a, b read, c, d still missing a, c read, b, d still missing ... After three characters, ... Some Important Definitions Let M = (Q, , , q 0 , F) be a finite automaton and let w = w 1 , w 2 , ..., w n be a string where each w i   . Then m accepts w if there exists a sequence of states r 0 , r 1 , ..., r n in Q such that: 1. r 0 = q 0 , 2. (r i , w i+1 ) = r i+1 for i = 0, 1, ..., n – 1, and 3. r n F. We say that M recognizes language A if A = { w | M accepts w } A language is called a regular language if some finite automaton recognizes it. More Important Definitions Let A and B be languages. We define the regular operations union, concatenation, and star as follows: Union : A B = { x | x A or x B } Concatenation : A B = { xy | x A and y B } Star : A * = { x 1 , x 2 , ..., x k | k 0 and each x i A } A set is closed under an operation if applying that operation to members of the set returns an object still in the set. E.g: N = { 1, 2, 3, ... } is closed under addition but not under division.
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