Nondeterministic Finite Automata (NFAs)

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Nondeterministic Finite Automata (NFAs)
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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
q
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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

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Another Example
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What is the language recognized by A?
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What is the language recognized by B?
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Nondeterministic Finite Automata
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1. Why is this automaton nondeterministic?
2. What is the language accepted by this automaton?

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Nondeterministic Finite Automata (NFA)
For every state q in S and every character ? in
?, one of the following will happen
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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
• s ? Q called the start state
• F ? Q called the favorable states

• ? ? Q (? ? e) Q

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?
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Nondeterministic Finite Automata (NFA)
What else can occur in NFAs?
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
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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
• r0 is the start state of M
• ri1 ? ?(ri , wi1) for i 0, , m-1
• rm ? F
• Language recognized by N w in ? N accepts w

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Same Example
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Language accepted by this automaton is the set of
all strings containing either the substring aa
or the substring bb

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Example
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1. Why is this automation nondeterministic?
2. What is the language accepted by this automaton?

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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?

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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 because it leads
to a favorable state
current state
...

• Is the oracle explanation compatible with our
  definition of acceptability in NFAs?
• Sometimes it is useful to use the oracle when
  creating NFAs

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NFA vs DFA

• Every DFA can be seen as an NFA
• But not every NFA can be seen as a DFA
• Are DFAs more expressive? What does this means?
• For example, is there an DFA that accepts the
  same language accepted by the following NFA?

It means that there an NFA can be constructed
that accepts a language L for which no DFA can be
constructed that accepts L
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NFA vs DFA (2)

• How about this one?

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• 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