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Intro to DFAs

Intro to DFAs. Readings: Sipser 1.1 (pages 31-44) With basic background from Sipser 0. Intuition: finite automata. Don’t hit anyone!. Front pad. Back pad. State diagram. What is accepted? 001? 000? 010? Can we come up with a description of the language accepted by this machine?.

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Intro to DFAs

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  1. Intro to DFAs Readings: Sipser 1.1 (pages 31-44) With basic background from Sipser 0

  2. Intuition: finite automata • Don’t hit anyone! Front pad Back pad

  3. State diagram • What is accepted? • 001? 000? 010? • Can we come up with a description of the language accepted by this machine?

  4. More formally… • A (deterministic) finite automaton (DFA) is a 5-tuple (Q, Σ, δ,q0, F), where • Q is a finite set called the states • Σ is a finite set called the alphabet • δ: Q × Σ → Qis the transition function • q0∈Q is the start state • F⊆Q is a set of accept states • In-class exercise:

  5. Languages • The set of all strings accepted by a DFA M is called the language of M and is denoted L(M) • We say that “M recognizes the language L(M)”

  6. What language is accepted here?

  7. Automata computation • More formally: • Let M=(Q, Σ, δ, q0, F)be a DFA and let w=w1w2w3…wn be a string over the alphabet Σ • Then Macceptsw if a sequence of states s0,s1,s2,…,sn exists in Q with the following conditions: • s0=q0 • δ(si,wi+1) = si+1fori = 0,…,n-1 • sn∈F 100

  8. Regular languages • A language is a regular language if some DFA recognizes it • Examples: • L(M1)={w | w contains at least one 1 and an even number of 0s follow the last 1} • L(M2)={w | w is a string over {a,b} that starts and ends with the same symbol}

  9. Designing your own • Is {w | wis a string of 0s and 1s containing an even number of 1s} a regular language? • Is {w | w is a string of as and bs containing the substring aba} a regular language? • How could we prove a “yes” answer?

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