lect4-fa3

lect4-fa3 - 1 Nondeterministic Finite Automata...

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Unformatted text preview: 1 Nondeterministic Finite Automata Nondeterminism Subset Construction 2 Nondeterminism A nondeterministic finite automaton has the ability to be in several states at once. Transitions from a state on an input symbol can be to any set of states. 3 Nondeterminism (2) Start in one start state. Accept if any sequence of choices leads to a final state. Intuitively : the NFA always guesses right. 4 Example : Moves on a Chessboard States = squares. Inputs = r (move to an adjacent red square) and b (move to an adjacent black square). Start state, final state are in opposite corners. 5 Example : Chessboard (2) 1 2 5 7 9 3 4 8 6 1 r b b 4 2 1 5 3 7 5 1 3 9 7 r b 1 2,4 5 2 4,6 1,3,5 3 2,6 5 4 2,8 1,5,7 5 2,4,6,8 1,3,7,9 6 2,8 3,5,9 7 4,8 5 8 4,6 5,7,9 9 6,8 5 * Accept, since final state reached 6 Formal NFA A finite set of states, typically Q. An input alphabet, typically . A transition function, typically . A start state in Q, typically q . A set of final states F Q. 7 Transition Function of an NFA (q, a) is a set of states. Extend to strings as follows: Basis : (q, ) = {q} Induction : (q, wa) = the union over all states p in (q, w) of (p, a) 8 Language of an NFA A string w is accepted by an NFA if (q , w) contains at least one final state. The language of the NFA is the set of strings it accepts. 9 Example : Language of an NFA For our chessboard NFA we saw that rbb is accepted. If the input consists of only bs, the set of accessible states alternates between {5} and {1,3,7,9}, so only even-length, nonempty strings of bs are accepted. What about strings with at least one r? 1 2 5 7 9 3 4 8 6 10 Equivalence of DFAs, NFAs A DFA can be turned into an NFA that accepts the same language. If D (q, a) = p, let the NFA have N (q, a) = {p}. Then the NFA is always in a set containing exactly one state the state the DFA is in after reading the same input. 11 Equivalence (2) Surprisingly, for any NFA there is a DFA that accepts the same language....
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This note was uploaded on 03/30/2012 for the course CS 154 taught by Professor Motwani,r during the Spring '08 term at Stanford.

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lect4-fa3 - 1 Nondeterministic Finite Automata...

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