Decision problems about regular languages

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Lecture 23

• Decision problems about regular languages
• Programs can be inputs to other programs
• FSAs, NFAs, regular expressions
• Basic problems are solvable
• halting, accepting, and emptiness problems
• Solvability of other problems
• many-one reductions to basic problems

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Problems with programs as inputs
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Decision problems with numbers as inputs

• Examples
• Primality
• Input integer n
• Yes/No question Is n prime?
• Zero
• Input integer n
• Yes/No question Is n 0?
• Equal
• Input integers m, n
• Yes/No question Is mn?

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Decision problems with programs as inputs

• Examples
• Lines of code
• Input program P, integer n
• Yes/No question Does P have less than n lines of
  code?
• Empty language
• Input program P
• Yes/No question Is L(P) ?
• Equal
• Input programs P, Q
• Yes/No question Is L(P) L(Q)?

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Language representation

• Empty language problem
• Input program P
• Yes/No question Is L(P) ?
• How might we represent input program P?
• As a string P over ASCII alphabet
• What is the language that corresponds to empty
  language decision problem?
• The set of strings representing programs which
  are yes instances to the empty language problem

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Programs

• In this unit, our programs are the following
  three types of objects
• FSAs
• NFAs
• regular expressions
• Previously, they were C programs
• Review those topics after mastering todays
  examples

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Basic Decision Problems(and algorithms for
solving them)
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Divisibility Problem

• Input
• integers m,n
• Question
• Is m a divisor of n?
• Algorithm DIV for solving divisibility problem
• Apply Euclids algorithm for finding the greatest
  common divisor to m and n
• If greatest common divisor is m THEN yes ELSE no

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Halting Problem

• Input
• FSA M
• Input string x to M
• Question
• Does M halt on x?
• Algorithm for solving halting problem
• Output yes
• Correct because an FSA always halts on all input
  strings

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Accepting Problem

• Input
• FSA M
• Input string x to M
• Question
• Is x in L(M)?
• Algorithm ACCEPT for solving accepting problem
• Run M on x
• If halting configuration is accepting THEN yes
  ELSE no
• Note this algorithm actually has to do something

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Primality Problem

• Input
• integer n
• Question
• Is n a prime number?
• Algorithm for solving primality problem
• for i 2 to n-1 Do
• apply algorithm DIV to inputs i,n
• If answer for any i is yes THEN no ELSE yes

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Empty Language Problem

• Input
• FSA M
• Question
• Is L(M)?
• How would you solve this problem?

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Algorithms for solving empty language problem

• Algorithm 1
• View FSA M as a directed graph (nodes, arcs)
• See if any accepting node is reachable from the
  start node
• Algorithm 2
• Let n be the number of states in FSA M
• Run ACCEPT(M,x) for all input strings of length lt
  n
• If any are accepted THEN no ELSE yes
• Why is algorithm 2 correct?
• Same underlying reason for why algorithm 1 works.
• If any string is accepted by M, some string x
  must be accepted where M never is in the same
  state twice while processing x
• This string x will have length at most n-1

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Solving Other Problems(using many-one reductions)
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Complement Empty Problem

• Input
• FSA M
• Question
• Is (L(M))c?
• Use many-one reductions to solve this problem
• We will show that the Complement Empty problem
  many-one reduces to the Empty Language problem
• How do we do this?
• Apply the construction which showed that LFSA is
  closed under set complement

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Algorithm Description

• Convert input FSA M into an FSA M such that
  L(M) (L(M))c
• We do this by applying the algorithm which we
  used to show that LFSA is closed under complement
• Feed FSA M into algorithm which solves the empty
  language problem
• If that algorithm returns yes THEN yes ELSE no

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Reduction Illustrated
Algorithm for solving empty language problem
FSA M
Yes/No
Algorithm for complement empty problem
The complement construction algorithm is the
many-one reduction function. If M is a yes input
instance of CE, then M is a yes input instance
of EL. If M is a no input instance of CE, then
M is a no input instance of EL.
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NFA Empty Problem

• Input
• NFA M
• Question
• Is L(M)?
• Use many-one reductions to solve this problem
• We will show that the NFA Empty problem many-one
  reduces to the Empty Language problem
• How do we do this?
• Apply the construction which showed that any NFA
  can be converted into an equivalent FSA

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Algorithm Description

• Convert input NFA M into an FSA M such that
  L(M) L(M)
• We do this by applying the algorithms which we
  used to show that LNFA is a subset of LFSA
• Feed FSA M into algorithm which solves the empty
  language problem
• If that algorithm returns yes THEN yes ELSE no

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Reduction Illustrated
Algorithm for solving empty language problem
NFA M
Yes/No
Algorithm for NFA empty problem
The subset construction algorithm is the many-one
reduction function. If M is a yes input instance
of NE, then M is a yes input instance of EL. If
M is a no input instance of NE, then M is a no
input instance of EL.
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Equal Problem

• Input
• FSAs M1 and M2
• Question
• Is L(M1) L(M2)?
• Use many-one reductions to solve this problem
• Try and reduce this problem to the empty language
  problem
• If L(M1) L(M2), then what combination of L(M1)
  and L(M2) must be empty?

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Algorithm Description

• Convert input FSAs M1 and M2 into an FSA M3 such
  that L(M3) (L(M1) - L(M2)) union (L(M2) -
  L(M1))
• We do this by applying the algorithm which we
  used to show that LFSA is closed under symmetric
  difference (similar to intersection)
• Feed FSA M3 into algorithm which solves the empty
  language problem
• If that algorithm returns yes THEN yes ELSE no

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Reduction Illustrated
Algorithm for solving empty language problem
FSA M1 FSA M2
Yes/No
Algorithm for Equal problem
The 2FSA-gt1FSA construction algorithm is the
many-one reduction function. If (M1,M2) is a
yes input instance of EQ, then M3 is a yes input
instance of EL. If (M1,M2) is a no input
instance of EQ, then M3 is a no input instance of
EL.
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Summary

• Decision problems with programs as inputs
• Basic problems
• You need to develop algorithms from scratch based
  on properties of FSAs
• Solving new problems
• You need to figure out how to combine the various
  algorithms we have seen in this unit to solve the
  given problem