Announcement RIT is looking for a few good Deterministic Finite Automata programmers! ACM Programming Contest 3 people teams Tryouts Tues, Sept 23 rd at 5pm (ICL3) FREE FOOD! Contact Paul Tyman (ptt@cs.rit.edu) by Sept 22 nd . Scholarship Announcement Computer Science Community Women in Computing Scholarship CS Networking and Social Group Attend the 2008 Grace Hopper Celebration of Women in Computing (Keystone, CO) First Meeting http://gracehopper.org/2008/ Wednesday, Sept 10 th 6-8pm Contact instructor for more info if CS Student Center interested. Puzzles, pizza, soda, and prizes. Logistics First things Homework #1 E-mail From textbook Should have received mail from me. Page 47 -- Exercise 2b, d, e ( use JFLAP ) If not: Page 48 -- Exercise 9b, c, e ( use JFLAP ) Page 48 -- Exercise 19 Check RIT forwarding Page 55 -- Exercise 7 ( use JFLAP ) Page 62 -- Exercise 3 Due 9/16 1
Finite Automata Visualization Submitting JFLAP files JFLAP Via mycourses dropbox Java Formal Language and Automata Package Use obvious file names By Susan Rodger at Duke University Eg. 2b.jff, 2c.jff, etc. http://www.jflap.org Need jff extension! Available on mycourses (Content). UNIX: tar file Running JFLAP Windows: zip file UNIX java –jar JFLAP.jar Mac: sit/zip file java –jar ~jmg/JFLAP.jar Include a README with name(s) Windows/Mac: double click on JFLAP.jar Questions Languages Any questions before we start? Recall. What is a language? String Recognition machine Language Classes Recall, we will be looking at classes of Given a string and a definition of a languages: language (set of strings), is the string a Each class will have its own means for member of the language? describing the language YES, string is Each class will have its own machine model in Language for string recognition Language Input string recognition Languages and machines get more machine complex as we move forward in the NO, string is course. not in Language 2
Languages Regular Languages A language is a set of strings. Today we start looking at our first class of languages: Regular languages Machine for accepting: Finite Automata A class of languages is nothing more Means of defining: Regular Expressions than a set of languages Grammar: Regular Grammars We’ll be studying different aspects of regular languages for the next 3-4 weeks Deterministic Finite Automata Deterministic Finite Automata A deterministic finite automata (DFA) consists of Input tape A read tape A machine that can be in one or more “states” 1 2 Start state – The state the machine is in at the 3 beginning of execution 5 4 Accepting states – The state(s) the machine has to be in after execution in order for a string to be “accepted” State machine Deterministic Finite Automata Deterministic Finite Automata How the automaton works Testing a string for membership Read a character on the tape Place the string to be tested on the read tape Based on the character read and the current “state” of the machine, put your Place the machine into the start “state” machine into another “state” Let the machine run to completion Move the read head to the right If, upon completion, the machine is in an accepting “state”, the string is accepted, Repeat the above until all characters have been read. otherwise it is not. 3
Deterministic Finite Automata Deterministic Finite Automata Transition function L = {x ∈ {0,1} * | x contains an odd number of 0s and an even number of 1s} Defines what state the machine will move into given: 1 The current machine state 0 # 0s #1s 1 The character read off the tape 1 0 even even 0 0 2 1 odd even Sometimes illustrated as directed graph 1 0 where nodes represent states and edges 2 odd odd 1 represent transitions. 3 0 3 even odd Deterministic Finite Automata Deterministic Finite Automata This transition can also be given by a Another example character table. L = {x ∈ {0,1} * | x contains 01 as a 0 1 substring} → 0 1 3 s 1 0 0,1 *1 0 2 t a 0 1 2 t 2 3 1 0 1 e 3 2 0 Deterministic Finite Automata Deterministic Finite Automata Consists of Another example A set of states L = {x ∈ {0,1} * | x does not contains 01 A start state as a substring} A set of accepting states Read symbols Transition function Let’s see in JFLAP Let’s define an automata formally 4
Deterministic Finite Automata Transition Function A deterministic finite automaton (finite-state The transition function machine) is a 5-tuple ( Q , Σ , δ , q o , F ) where δ is a function from Q x Σ to Q Q is a finite set (of states) δ ( q, a ) = q’ where Σ is a finite alphabet of symbols q, q’ ∈ Q q o ∈ Q is the start state a ∈ Σ F ⊆ Q is the set of final states δ defines, given a current state q and δ is a function from Q x Σ to Q (transition reading character a , to which state the function) DFA will move. Transition Function on Strings Transition Function on Strings Applying the transition function to a string. Recursive definition of Let A = ( Q , Σ , δ , q o , F ) be an FA. δ * is a function from Q x Σ * to Q Define δ * : Q x Σ * → Q δ * ( q, x ) = q’ where For any q ∈ Q, δ * ( q , λ ) = q q, q’ ∈ Q 1. x ∈ Σ * For any y ∈ Σ * , a ∈ Σ , and q ∈ Q 2. δ * defines, given a current state q and reading a δ * ( q, ya ) = δ ( δ * ( q, y ) , a ) string x , to which state the DFA will move once all characters of x are read. Transition Function on Strings Language accepted by a DFA Example δ * ( q , λ ) = q Let A = ( Q , Σ , δ , q o , F ) be a DFA. δ * ( q, ya ) = δ ( δ * ( q, y ) , a ) c a b q q 1 q 2 q 3 A string x ∈ Σ * is accepted by A if (q, abc) = δ ( δ * (q,ab), c) δ * (q 0 , x) ∈ F = δ ( δ ( δ * (q, a), b), c) In other words, = δ ( δ ( δ * (q, λ a), b), c) Starting in your start state q 0 = δ ( δ ( δ ( δ * (q, λ ), a), b), c) Run the machine with input x = δ ( δ ( δ (q, a), b), c) The machine ends up in a final state = δ ( δ (q 1 , b), c) If a string x is not accepted by A it is said = δ (q 2 , c) to be rejected by A. = q 3 5
Language accepted by a DFA Regular Languages The language accepted or recognized Definition: by A is: A Language L is regular if and only if there L(A) = { x ∈ Σ * | x is accepted by A } exists a finite automata M such that L = L (M) If L is a language over Σ , L is accepted by A iff L = L(A). All regular languages are accepted by For all x ∈ L, x is accepted by A. some finite automata For all x ∉ L, x is rejected by A. Reality Check Reality Check What we’ve learned so far Questions? An DFA can be expressed by ( Q , Σ , δ , q o , F ) Transition function: δ : Q x Σ → Q Let’s take a break. Transition function on strings δ * : Q x Σ * → Q Defined recursively DFA accepting a string The language accepted by an DFA All languages accepted by a DFA are regular. 6
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