Introduction to the Theory of Computation

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Introduction to the Theory of Computation

• John Paxton
• Montana State University
• Summer 2003

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Textbook

• Introduction to the Theory of Computation
• Michael Sipser
• PWS Publishing Company
• 1997

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Humor

• One night, Bill Clinton was awakened by George
  Washington's ghost in the White House. Clinton
  saw him and asked, "George, what is the best
  thing I could do to help the country?"       
        "Set an honest and honorable example, just
  as I did," advised George.              The
  next night, the ghost of Thomas Jefferson moved
  through the dark bedroom. "Tom, what is the best
  thing I could do to help the country?" Clinton
  asked.              "Cut taxes and reduce the
  size of government," advised Tom.       
        Clinton didn't sleep well the next night,
  and saw another figure moving in the shadows. It
  was Abraham Lincoln's ghost. "Abe, what is the
  best thing I could do to help the country?"
  Clinton asked.              "Go to the
  theatre."

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Chapter 1 Regular Languages

• Very simple model of computation

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1.1 Finite Automaton

• A finite state automaton (FSA) is a very simple
  model of computation.
• Studying an FSA allows us to better understand
  more complex models of computation.
• An FSA can be used in a compiler to tokenize the
  input.

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Example 1
0
1
1
q0
q1
0
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Example 1 Questions

• What is the start state?
• What is the set of accept states?
• What sequence of states are gone through on input
  0111001?
• Is the empty string input (e) accepted?

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Formal Definition

• A finite automaton is a 5-tuple (Q, S, d, q0, F)
  where
• Q is a finite set called the states
• S is a finite set called the alphabet
• d Q x S ? Q is the transition function
• q0 Î Q is the start state
• F Í Q is the set of accept states

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Example 1 Formal Definition

• Q q0, q1
• S 0, 1
• F q1
• d

0 1
q0 q0 q1
q1 q0 q1
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Language of Machine M

• L(M) A where A is the set of all strings that
  machine M accepts.
• L(Example 1) w w ends in a 1
• M recognizes A

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Acceptance

• Let M (Q, S, d, q0, F) be a finite automaton
  and w w1w2 wn be a string over the alphabet S.
  Then M accepts w if a sequence of states
  r0r1rn exists in Q with the following three
  conditions
• r0 q0
• d(ri, wi1) ri1 for 0 lt i lt n 1
• rn Î F

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Regular Language

• A language is called a regular language if some
  finite automaton recognizes it.

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Exercises

• Design a finite automaton that recognizes the
  regular language of all strings that contain the
  string 001 as a substring. The alphabet is 0,
  1, 2.
• Design a finite automaton that recognizes all
  binary numbers that are multiples of 3.
• Design a finite automaton that recognizes all
  binary strings with a 1 in the kth position where
  k gt 0.

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Regular Operations

• Union A U B x x Î A or x Î B
• Concatenation A B xy x Î A
  and y Î B
• Star A x1x2xk k gt 0 and xi Î A

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Theorem

• The class of regular languages is closed under
  the union operation.
• In other words, if A1 and A2 are regular
  languages, so is A1 U A2.

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Proof

• M1 (Q1, S, d1, q1, F1)
• M2 (Q2, S, d2, q2, F2)
• Construct M (Q, S, d, q0, F)

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Proof (by construction)

• Q (r1, r2) r1 Î Q1 and r2 Î Q2
• q0 (q1, q2)
• F (r1, r2) r1 Î F1 or r2 Î F2
• d((r1,r2), a) (d1 (r1, a), d2 (r2, a) )

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Exercise

• Use the preceding construction to construct a FSA
  that accepts the union of L(A1) w w ends in
  1 and L(A2) w w contains the substring
  00

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More Theorems

• The class of regular languages is closed under
  the concatenation operation.
• The class of regular languages is closed under
  the star operation.
• Proofs easier with the concept of nondeterminism!