Complexity and Computability Theory I

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Complexity and Computability Theory I

• Lecture 4
• Rina Zviel-Girshin
• Leah Epstein
• Winter 2002-2003

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Overview

• Nondeterminism
• Examples
• Equivalence of the nondeterministic model to the
  deterministic model

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Nondeterminism (briefly)

• Each state can be accepting or not.
• The automaton reads one input word symbol at each
  time unit.
• An NFA also can change it state without reading
  an input - ? transition
• If a state in which it stops reading an input
  (reading it till the end) is an accepting state-
  the automaton accepts (recognizes) the input.

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Nondeterminism (briefly)

• On each symbol
• it moves to a new state (defined by the
  transition function) as a function of
• the current state
• the symbol read or an ? transition

• or halts if no valid move defined

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Formal Definition

• A finite automaton is a 5-tuple (Q, ?, ?, q0, F),
  where
• Q - a finite set of states
• ? - alphabet
• ? - transition function - Q?(???)?2Q or
  P(Q)(given a state and an input symbol - what can
  be the next state)
• q0 - q0?Q is the start state
• F - F?Q the set of accept states

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DFA Example

• Consider the following language L (ab ? aba)
• The deterministic automaton

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NFA Example

• The nondeterministic automaton

• The basic idea of the construction
• An aba path
• An ab path

This simplifies the deterministic automaton.
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How does an NFA compute?

• A nondeterministic computation is a tree of
  possibilities.
• The root of the tree is a start state of the
  automaton.
• Each branching point is a computation at which
  automaton has multiple choices.
• An NFA accepts an input string if there exists at
  least one computation that leads to an accept
  state.

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NFA vs. DFA

• The difference between a deterministic finite
  automaton DFA and a non-deterministic one NFA is
  the number of paths in a computation tree.
• In DFAs we have only one path one chain.
• In NFAs we have a tree.

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The language of an NFA

• Informally
• The words that an NFA A accepts
  (recognizes), i.e. when reading them it stops in
  an accepting state.
• Formally
• L(A)w w??, ?( q0,w)?F??
•   means
• There exist a path from q0 to qi?F for
• the input string w.

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Example

• Construct a nondeterministic finite automaton for
  the following language
• L w1 w is over ?0,1
•   A

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What does A do on the input w110?
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What does A do on the input w101?
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Additional examples

• Construct a nondeterministic finite automaton
  recognizing the following language
• Lw w?(?0,1,2),
• w starts with 01 and ends with 1

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Additional examples

• Construct a nondeterministic finite automaton
  recognizing the following language

0i i mod 3 1
L0i1j i mod 3 1, j mod 2 1
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Additional examples
0i1j i mod 3 1 j mod 2 1
L0i1j i mod 3 1, j mod 2 1
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Equivalence of FA

•  Definition
• Two automata are equivalent if they
• recognize the same language.

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Example
A
B
L(A)L(B)
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Equivalency

• Theorem
• Every NFA has an equivalent DFA.
• Proof idea
• We will give a constructive proof - by giving an
  algorithm to build a DFA equivalent to a given
  NFA.

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How can we do it?

• Consider an example in which we can get from some
  state q on an input letter ? to more than one
  state.

• The next state - ?(q,?) - is one of qi, qj, qk.

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How can we do it?

• That means that the next state in an NFA is one
  state among several states.
• But in a DFA only one state can be the next
  state.
• Conclusion
• One state in a DFA A set of states in an NFA

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How can we do it?

• Consider an example. What states can we reach
  starting from some state q on two letters input
  ??1.

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How can we do it?

• Conclusion
• ?(q, ??1) ?(qi,qj,qk,?1)
• ?(qi,?1)? ?(qj,?1)? ?(qk,?1)
• Informally
• The next state in a DFA A set of all the
  states in an NFA to which you can get from the
  set of all the current states in an NFA.

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The ?- closure

• Formally
• The ?- closure of a state is a set of all the
  states reached from the current state using
  ?-transitions only.
• E(q) p?Q ?(q,?i) p, igt0
• Informally
• Use all the ? transitions you can. Add the
  states you reach to ?-closure. Try to reach as
  many states as you can.

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The ?- closure
E(q2) q2 E(q3) q3, q4 E(q4) q4
E(q0) q0, q1, q2 E(q1) q1, q2
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A DFA construction algorithm

• Let A be an NFA where A (QA, ?, ?A, q0A, FA)
  .
• We construct a DFA M equivalent to A,
• where M ( Q, ?, ?, q0, F).
• Q P(QA)
• ? ?
• For each R in Q and ? in ?, ?(R,?) is

• q0 E(q0A)
• F R?Q there exists r?R such that r?FA

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Explanation

• Create all the subsets of set of states of A.
  These subsets will become the states of M.
• The alphabet remains the same.
• Transition function
• for each state m in M and a letter ? find what
  are the states qi..qj in A included in m are

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Explanation

• for each qi?m find a set R of states which you
  can reach in NFA A using ?-closure, the letter ?
  and the ?-closure

?i ? ?j
?k ? ?l
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Explanation

• unite all the sets R you reach into one set S

?.. ? ? ?.. ?

• the next state of m on ? in M is S.

S
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Explanation

• The initial state of M is the set which includes
  only E(q0A)
• The final states of M are all the sets in which
  at least one state is a final (accepting) state
  of A.
• Eliminate all the unreachable states in M -
  states to which the is no path from the initial
  state of M.
• The automaton you have is a deterministic
  automaton equivalent to A.

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Example

• Convert a given nondeterministic finite
  automaton into a deterministic finite automaton.

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Example

• Construction of the DFA
• ?(q0,a)q2,q3
• ?(q0,b)q2
• ?(q1,a)q2,q3
• ?(q1,b)q2
• Therefore
• ?(q0,q1,a)q2,q3
• ?(q0,q1,b)q2

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Example

• ?(q2,a)q1
• ?(q2,b)q2

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Example

• ?(q1,a)q2,q3
• ?(q1,b)q2

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Example

• ?(q2,q3,a)q1,q2
• ?(q2,q3,b)q2

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Example

• ?(q1,q2,a)q1,q2,q3
• ?(q1,q2,b)q2

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?(q1,q2,q3,a)q1,q2,q3 ?(q1,q2,q3,b)q2
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Any Questions?