Transcript Module 25

Module 25
• Decision problems about regular languages
– Basic problems are solvable
• halting, accepting, and emptiness problems
– Solvability of other problems
• answer-preserving input transformations to basic
problems
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Programs
• In this unit, our programs are the following
three types of objects
– FSA’s
– NFA’s
– regular expressions
• Previously, they were C++ programs
– Review those topics after mastering today’s
examples
2
Basic Decision Problems
(and algorithms for solving them)
3
Halting Problem
• Input
– FSA M
– Input string x to M
• Question
– Does M halt on x?
• Give an algorithm for solving the FSA halting problem.
4
Accepting Problem
• Input
– FSA M
– Input string x to M
• Question
– Is x in L(M)?
• Give an algorithm ACCEPT for solving the
accepting problem.
5
Empty Language Problem
• Input
– FSA M
• Question
– Is L(M)={}?
• Give an algorithm for solving the empty language problem.
– Don’t look ahead to the next slide.
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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 < n
– If any are accepted THEN no ELSE yes
• Why is algorithm 2 correct?
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Solving Other Problems
(using answer-preserving input
transformations)
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Complement Empty Problem
• Input
– FSA M
• Question
– Is (L(M))c = {}?
• Show how to use an answer-preserving input
transformation to help solve this problem
– Show that the Complement Empty problem transforms
to the Empty Language problem
– Don’t look at next two slides
9
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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Input Transformation Illustrated
FSA M
Complement FSA M’
Construction
Algorithm for
solving empty
language problem
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.
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NFA Empty Problem
• Input
– NFA M
• Question
– Is L(M)={}?
• Show how to use answer-preserving input
transformations to help solve this problem
– Show that the NFA Empty problem transforms to the
Empty Language problem
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Input Transformation
Yes/No
Algorithm for NFA empty problem
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Equal Problem
• Input
– FSA’s M1 and M2
• Question
– Is L(M1) = L(M2)?
• Show how to 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?
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Input Transformation Illustrated
Yes/No
Algorithm for Equal problem
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Summary
• Decision problems with programs as inputs
• Basic problems
– You need to develop algorithms from scratch
based on properties of FSA’s
• 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
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