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# MELJUN CORTES Automata Theory

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MELJUN CORTES Automata Theory

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### MELJUN CORTES Automata Theory

1. 1. MELJUN CORTES
2. 2.  Automata are abstract mathematical models of machines that perform computations on an input by moving through a series of states or configurations.
3. 3.  Automaton  An automaton is represented formally by the 5- tuple of elements {Q,Σ,δ,q0,A}, where: ▪ Q is a finite set of states. ▪ Σ is a finite set of symbols.
4. 4.  Automaton  An automaton is represented formally by the 5- tuple of elements {Q,Σ,δ,q0,A}, where: ▪ δ is the transition function. ▪ q0 is the start state, where q0∈ Q. ▪ A is a set of states of Q (i.e. A⊆Q) called accept states.
5. 5.  Input word  An automaton reads a finite string of symbols a1,a2,...., an , where ai ∈ Σ, which is called a input word.  Accepting word  A word w ∈ Σ* is accepted by the automaton if qn ∈ A.
6. 6.  Run  A run of the automaton on an input word w = a1,a2,...., an ∈ Σ*, is a sequence of states q0,q1,q2,...., qn, where qi ∈ Q such that q0 is the start state and qi = δ(qi-1,ai) for 0 < i ≤ n.
7. 7.  A finite automaton (FA) is a simple idealized machine used to recognize patterns within input taken from some character set (or alphabet) C.
8. 8.  A finite automaton consists of:     a finite set S of N states a special start state a set of final (or accepting) states a set of transitions T from one state to another, labeled with chars in C  As noted above, we can represent a FA graphically, with nodes for states, and arcs for transitions.
9. 9.  We execute our FA on an input sequence as follows:  Begin in the start state  If the next input char matches the label on a transition from the current state to a new state, go to that new state  Continue making transitions on each input char ▪ If no move is possible, then stop ▪ If in accepting state, then accept
10. 10.  It is a finite state machine which accepts the word "nice". In this FSM the only accepting state is number 7.
11. 11.  For each input symbol in Σ, there is exactly one transition of each state (possibly back to the state itself).  It do not accept empty strings.
12. 12.  A deterministic finite automaton (DFA) consists of  A finite set of states (often denoted Q)  A finite set Σ of symbols (alphabet)  A transition function that takes as argument a state and a symbol and returns a state (often denoted δ)
13. 13.  A start state often denoted q0  A set of ﬁnal or accepting states (often denoted F)  We have q0 ∈ Q and F ⊆ Q
14. 14.  So a DFA is mathematically represented as a 5-uple  (Q, Σ, δ, q0, F )  The transition function δ is a function in  Q×Σ→Q  Q × Σ is the set of 2-tuples (q, a) with q ∈ Q and a ∈ Σ
15. 15.  How to present a DFA? With a transition table
16. 16.  How to present a DFA? With a transition table  Inputs 0 1
17. 17.  How to present a DFA? With a transition table  Inputs  States 0 q0 q1 q2 1
18. 18.  How to present a DFA? With a transition table  The → indicates the start state 0  q0 q1 q2 1
19. 19.  How to present a DFA? With a transition table  The → indicates the start state  The ∗ indicates the ﬁnal state(s) 0  q0 q1 * q2 1
20. 20.  How to present a DFA? With a transition table  δ (q0,0) = q1  δ (q0,0) = q0 0  q0 q1 * q2 1 q1 q0
21. 21.  How to present a DFA? With a transition table  δ (q0,0) = q1  δ (q0,0) = q0  δ (q1,0) = q2  δ (q1,0) = q1 0 1  q0 q1 q0 q1 q2 q1 * q2
22. 22.  How to present a 0 1  q0 q1 q0 q1 q2 q1 * q2 q2 q2 DFA? With a transition table  δ (q0,0) = q1  δ (q0,0) = q0  δ (q1,0) = q2  δ (q1,0) = q1  δ (q2,0) = q2  δ (q2,0) = q2
23. 23.  Construct a DFA that accepts the language  L = {010, 1}  ( Σ = {0, 1} )
24. 24.  Construct a DFA that accepts the language  L = {010, 1}  ( Σ = {0, 1} )
25. 25.  Set of all strings over {0,1} where 1 is always even.
26. 26.  Set of all strings over {0,1} where 1 is always even.
27. 27.  Set of all strings over {0,1} where 1 is always odd.
28. 28.  Set of all strings over {0,1} where 1 is always odd.