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CCM 305 Lecture 04 Fall 98 Announcements See Web Page Homework for official homework assignments. U of M wins!!! 59-20 Lions lose 29 -6 Mcguire hits 65 (66 sort of) Chapter 5 Finite Automata accept string reject Definition A finite automation is a collection of three things; 1. A finite number of states, one of which is the start state, and some (or none) of which are designated as final(accept) states. 2. An alphabet , of possible input letters. 3. A finite set of rules or transitions that tell for each state and for each letter of the input alphabet which state to go to next. Picture (called a transition diagram) Start State - Any State Final State + Transition from state 1 to state 2 on reading an a 1 a 2 My problem : Find a FA that accepts all strings that contain abb. a b a b a b + a,b Find a FA to accept all strings that end with abb a b a b b + a b a Read abb Read ab nothing Read a Build a FA that accepts only the language of all words with b as the second letter. - a,b b a + a,b a,b 3. Build an FA that accepts only the words baa, ab, and abb. a + a b b b a,b a,b - a,b a a a b b + a a b b a, b a, b a a a b a, b b + 4. Build a Finite Automata with 3 states that accepts all strings. -+ a,b + a,b a,b + 7. Build a Finite Automata that accepts only those words that begin or end with a double letter. First we will build a FA to accept words that start with aa or bb. Then add words that end with aa or bb. b accepts First we build a,b only b the FA that those words that do begin + with ba. a a a a,b b Now we will build a FA to accept words that start with aa or bb. Then add words that end with aa or bb. - b b + a a b a a b + a b Now put them together a + a a a a b b - + b + a b b b b a + a,b a,b 17. Describe in english the languages accepted by the following FAs: b a,b a + a,b Try listing some to get an idea. a, aaa, aba, baa, bba, aaaaa, aaaba, aabaa, aabba, abaaa, ababa, abbaa, abbba, baaaa, … The answer is odd length ends with a. Chapter 6 Transition Graphs Relaxing the inputs Allow words on an edge or Allow 0 or 1 edges from a state Allow duplicated edges Allow more than 1 start state We know have non-determinism Now the machine can crash We will accept a word if there is some path that leads to +. Examples: - abb a a - b a + This accepts, ba, aa, abb,aba More examples of TGs and what they accept a - b b abba Test a reject b reject babbaa accept a a b + b a babbaaba accept ba accept aaa accept Do pg 88, 2 TG1 Generalized TG allows regular expressions on the edges. Pg 90 # 10 Proof that TG are closed under concatenation. Assume that L1 and L2 are 2 languages accepted by the Transitions graphs TG1 and TG2 respectively. We are going to build TG3 that accepts L1L2. We create TG3 by adding new transitions from the + states of TG1 going to all the – states of TG2, and then removing the – ‘s from TG2. We then remove the +’s from the states that were part of TG1.