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Introduction to Deterministic Finite Automata: Understanding Languages Accepted by Machines

Explore the basic concepts of Deterministic Finite Automata (DFAs) with readings from Sipser's book. Understand state diagrams, accepted languages, regular languages, designing DFAs, and different examples. Dive into automata computation and the acceptance conditions of DFAs.

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Introduction to Deterministic Finite Automata: Understanding Languages Accepted by Machines

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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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