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CS310

CS310. Finite Automata Sections: Sep 3, 2008. Quick Review. Alphabet: ∑ = {a,b} ∑*: Closure: String: any finite sequence of symbols from a given alphabet. |w| = length Concatenation/Prefix/Suffix/Reverse Language L over ∑ is a subset of ∑* L= { x | rule about x}

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CS310

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  1. CS310 Finite Automata Sections: Sep 3, 2008 CS 310 – Fall 2008 Pacific University

  2. Quick Review • Alphabet: ∑ = {a,b} ∑*: Closure: • String: any finite sequence of symbols from a given alphabet. |w| = length Concatenation/Prefix/Suffix/Reverse • Language L over ∑ is a subset of ∑* L= { x | rule about x} Concatenation/Union/Kleene Star Recursive Definition CS 310 – Fall 2008 Pacific University

  3. Finite Automata • How can we reason about computation? • Simple model of computation • Finite Automata • extremely limited amount of memory • represent states of computation CS 310 – Fall 2008 Pacific University

  4. Front pad Rear pad Door Door State: Open, Closed Inputs: Front, Rear, Both, Neither State Transition Table CS 310 – Fall 2008 Pacific University

  5. Closed Open State Diagram CS 310 – Fall 2008 Pacific University

  6. More uses… • Recognize patterns in data • Build an automata that can classify a string as part of a language or not Language: L = { x є {0,1}* | x contains at least one 1 and the last 1 is followed by even number of 0s} CS 310 – Fall 2008 Pacific University

  7. 1 0 1 0 q1 q2 q3 1, 0 Inputs: Accept or Reject? 1101 1110 0 1100 1 11 Set of all strings (A) accepted by a machine (M) is the Language of the Machine M recognizes A or M accepts A CS 310 – Fall 2008 Pacific University

  8. Formal Definition • Deterministic Finite Automata: 5-tuple (Q, ∑ , δ,q0, F) Q: finite set of states ∑: alphabet (finite set) δ : transition function (δ: Qx∑->Q) q0: start state F : accepting states (subset of Q) CS 310 – Fall 2008 Pacific University

  9. 1 0 1 0 q1 q2 q3 1, 0 Q: finite set of states ∑: alphabet δ : transition function q0: start state F : accepting states CS 310 – Fall 2008 Pacific University

  10. 1 0 1 Q: What strings get ∑: accepted? δ : q0: L(M) = { } F : q1 q2 0 CS 310 – Fall 2008 Pacific University

  11. Designing a DFA • Identify small pieces • alphabet, each state needs a transition for each symbol • finite memory, what crucial data does the machine look for? • can things get hopeless? do we need a trap? • where should the empty string be? • what is the transition into the accept state? • can you transition out of the accept state? • Practice! CS 310 – Fall 2008 Pacific University

  12. L(M) = { w | w = ε or w ends in 1} ∑ = { 0,1} Q: δ : q0: F : CS 310 – Fall 2008 Pacific University

  13. ∑ = {0,1}, L(M)={w | odd # of 1s} CS 310 – Fall 2008 Pacific University

  14. Build a DFA to do math! L(M) = Accept sums that are multiples of 3 ∑ = { 0,1,2, <Reset>} Keep a running total of input, modulo 3 q0 CS 310 – Fall 2008 Pacific University

  15. ∑ = {0,1}, L(M)={w | begins with 1, ends with 0} CS 310 – Fall 2008 Pacific University

  16. ∑ = {0,1}, L(M)={w | contains 110} CS 310 – Fall 2008 Pacific University

  17. ∑ = {0,1}, L(M)={w | does not contain 110} CS 310 – Fall 2008 Pacific University

  18. ∑ = {0,1}, L(M)={w | (01)* } CS 310 – Fall 2008 Pacific University

  19. ∑ = {0,1}, L(M)={w | w even #0s, odd #1s } CS 310 – Fall 2008 Pacific University

  20. ∑ = {0,1}, L(M)={w | w any string except 11 and 111 } CS 310 – Fall 2008 Pacific University

  21. Formal Definition of Computing • Given a machine M= (Q, ∑ , δ,q0, F) and a string w=w1w2…wn over ∑, then M accepts w if there exists a sequence of states r0,r1…rn in Q such that: • r0 = q0 : r0 is the start state • δ (ri, wi+1) = ri+1 ,i=0,…,n-1 : legal transitions • rnє F : stop in an accept state • M recognizes A if A={w | M accepts w} • Language A is regular if there exists a Finite Automata that recognizes A. CS 310 – Fall 2008 Pacific University

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