Decision problems about regular languages

1
Lecture 26

• Decision problems about regular languages
• Basic problems are solvable
• halting, accepting, and emptiness problems
• Solvability of other problems
• many-one reductions to basic problems

2
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

3
Basic Decision Problems(and algorithms for
solving them)
4
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

5
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

6
Empty Language Problem

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

7
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

8
Solving Other Problems(using answer-preserving
input transformations)
9
Complement Empty Problem

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

10
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

11
Input Transformation Illustrated
Algorithm for solving empty language problem
FSA M
Yes/No
Algorithm for complement empty problem
The complement construction algorithm is the
answer-pres. input transformation. 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.
12
NFA Empty Problem

• Input
• NFA M
• Question
• Is L(M)?
• Use answer-preserving input transformations to
  solve this problem
• We will show that the NFA Empty problem
  transforms 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

13
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

14
Input Transformation Illustrated
Algorithm for solving empty language problem
NFA M
Yes/No
Algorithm for NFA empty problem
The subset construction algorithm is the
ans.-pres. input transformation. 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.
15
Equal Problem

• Input
• FSAs M1 and M2
• Question
• Is L(M1) L(M2)?
• Use answer-preserving input transformations to
  solve this problem
• Try and transform this problem to the empty
  language problem
• If L(M1) L(M2), then what combination of L(M1)
  and L(M2) must be empty?

16
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

17
Input Transformation Illustrated
Algorithm for solving empty language problem
FSA M1 FSA M2
Yes/No
Algorithm for Equal problem
The 2FSA-gt1FSA construction algorithm is the
ans.-pres. input transformation. 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.
18
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