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Nondeterministic Finite Automata (NFAs) Reminder: Deterministic Finite Automata (DFA) For every state q in Q and every character in , one and only one transition of the following form occurs: q > s b a p q’ b a b q r a a a b t b Expressiveness of DFA •DFA are a type of programs •Contains concepts of imperative languages: sequences, branching and loops. •Can implement a newspaper vendor machine or even control a character in a video game •Can implement pattern matching: find all text containing “Britney Spears” (or “Justin Timberlake”) •Can only implement programs that require a constant amount of memory Another Example a b A b > s a q a What is the language recognized by A? a B a > s q b b What is the language recognized by B? r b a r b Nondeterministic Finite Automata a b a q b r a > s a b a b q q’ b r r’ a b 1. Why is this automaton nondeterministic? 2. What is the language accepted by this automaton? Nondeterministic Finite Automata (NFA) For every state q in S and every character in , one of the following will happen: •No transition: q •One or more transitions: q q’ occurs … q’ p occurs Nondeterministic Finite Automaton (NFA) A nondeterministic finite automaton (NFA) is a 5-tuple (Q,,,s,F) where: •Q is a finite set of elements called states • is a finite input alphabet “the empty word” • Q × ( {e}) × Q •s Q called the start state •F Q called the favorable states The crucial point is that is a relation Book says: (Q,, ,s,F) where: is a transition function, Q × ( {e}) × (Q) Are and representing the same transitions? “power set” Nondeterministic Finite Automata (NFA) What else can occur in NFAs? q e q’ If next word to process in q is aaabbb and apply this transition, what is the word to be processed in q’? aaabbb “e-transitions” do not process any characters, they just allow to “jump” between states Formal Definition of Computation for NFAs • Given a nodeterministic finite automaton N= (Q,,,s,F), and let w = w1w2 …wn, where each wi is in M accepts w if we can write w as: w = y1y2 …ym , where each yi is in ( {e}) and there is a sequence of states r0 r1, r2 … rm in Q such that: 1. r0 is the start state of M 2. ri+1 (ri , wi+1) for i = 0, …, m-1 3. rm F • Language recognized by N: {w in * : N accepts w} Same Example a b a q b r a > s a b a b q q’ b r r’ a b Language accepted by this automaton is the set of all strings containing either the substring “aa” or the substring “bb” Example a >s b q b r b 1. Why is this automation nondeterministic? 2. What is the language accepted by this automaton? Why We Study Nondeterministic Computation? • Makes it easier to prove properties about Automata – In particular, makes it easier to prove certain properties: • If A is a regular language then AR is also regular • It also makes it possible to understand the boundaries of computation – P = NP? Oracle in NFAs •The oracle explanation of NFAs: given a choice between possible transitions, there is a device, called the oracle, that always chooses the transition that leads to a favorable state ... … ... Oracle chooses this transition “current state” because it leads ... to a favorable state •Is the oracle explanation compatible with our definition of acceptability in NFAs? •Sometimes it is useful to use the oracle when creating NFAs NFA vs DFA • Every DFA can be seen as an NFA • But not every NFA can be seen as a DFA • Are DFA’s more expressive? What does this means? It means that there an NFA can be constructed that accepts a language L for which no DFA can be constructed that accepts L • For example, is there an DFA that accepts the same language accepted by the following NFA? a >s b q b r b NFA vs DFA (2) • How about this one? a b a q b r a > s a b a b q q’ b r r’ a b • It turns out that for every NFA, a DFA can be constructed such they both accept the same language. • We will study a formal proof of this on Wednesday