Theory of Computation Tutorial I: Closed Operations and Quotient in Automata Theory, Slides of Theory of Computation

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Theory of Computation

Tutorial I

Speaker: Yu-Han Lyu

September 26, 2006

Closed operations

• Union
• Concatenation
• Star
• Complement: L’= Σ*- L
  • Final state! non-final state
  • Non-final state! final state
• Difference
  • L-M = (L’∪M)’

Another Proof

• Let two DFAs D A

=(Q

A ,Σ,δ A ,q A

,F

A ) and D B

=(Q

B ,Σ,δ B ,q B

,F

B

), L( D

A

)=A, L( D

B

)=B

• Parallel run two machines, if both accept then accept, otherwise reject.
• Formally, we construct DFA D =(Q,Σ,δ,q,F)
  • Q=Q A ×Q B (two tuple)
  • F=F A ×F B
  • Start state=(q A ,q B )
  • δ((p,q),a)=(δ A (p,a),δ B (q,a))
• Finally, we should prove L( D ) = L(A∩B)

Example

p q 0 1 0, r s 0 1 0, pr ps qr qs 1 0 0 1 1 0 0,

Quotient

• A/B={w | wx∈A for some x∈B}
• Run A’s DFA
• Non-deterministically choose one state in

A and guess x

Assignment 1

• Due: 3:20 pm, October 13, 2006 (before

class)

• Late submission will not be marked
• No cheating
• Can exchange high-level idea
• Problems 1 ~ 3 are easy
• Problem 4
• Use closed operation property to prove this language is not regular.

Idea

• When reading a character a, we should

know

• This character is in odd or even position
• The current state in A and B
• Problem 6 is similar

Problem 7

• Answer is in the textbook
  • After understanding, write it down in you words, otherwise..
• x and y are distinguishable by L
  • Some string z exists whereby exactly one of the strings xz and yz is a member of L
• We say that X is pairwise distinguishable by L
  • Every two distinct strings in X are distinguishable by L.
• index of L
  • Maximum number of elements in any set of strings that is pairwise distinguishable by L

Myhill-Nerode Theorem

• L is regular if and only if it has finite index
• Application
  • Minimization DFA’s state in unique (NFA??)
  • Proof for non-regular
• Example: L={x | x is palindrome, x=x R }
  • X={a i | i≧0}
  • a and aa are distingushable, by choosing z= ba
  • aa and aaa are distingushable, by choosing z= baa
  • index is infinite, so this language is not regular

Problem 8

• A 1/

={ x | for some y, |x| = |y| and xy∈A}

• No hint
• Harder problem: A 3/

={ z | for some x, y,

|x| = |y| = |z| and xyz∈A}

• Can your method extend to this problem?