RAJALAKSHMI ENGINEERING COLLEGE

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RAJALAKSHMI ENGINEERING COLLEGE
THEORY OF COMPUTATION
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UNIT- I

• AUTOMATA

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Finite Automata
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• States
• Open
• Closed
• Sensors
• Front Someone on the Front pad
• Rear Someone on the Rear pad
• Both Someone on both the pads
• Neither No one on either pad

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UNIT - II

• REGULAR EXPRESSIONS
• AND
• LANGUAGES

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Questions about Regular Languages
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Question contd
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Singleton Languages are Regular
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Finite Languages are regular
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Closure Properties for Regular Languages
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Myhill-Nerode Theorem
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UNIT III

• CONTEXT-FREE GRAMMAR
• AND
• LANGUAGES

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Context Free Grammars
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Cont
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Cont
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Defining CFG
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Notational conventions For CFGs
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OTHER CFG EXAMPLES
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Languages of CFG
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Languages of CFG
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Regular Languages and CFL
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Translating FAs into CFG
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Formalizing the Translation
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Closure Properties of CFL
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Proving CFLs closed under Union
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IDEA
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Formal Construction of Gu
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Derivations
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Sentential Form
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Left Most and Right Most Derivation
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Right-Linear Grammars
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Ambiguous Grammars
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Ambiguous Grammars
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Pushdown Automata

• Recall our study of regular languages.
• They were defined in terms of regular expressions
  (syntax).
• We then showed that FAs provide the computational
  power
• needed to process them.
• We would like to mimic this line of development
  for CFLs.
• We have a syntactic definition of CFLs in terms
  of CFGs.
• What kind of computing power is needed to
  process (i.e.
• recognize) CFLs?
• Do FAs suffice?

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Cont
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Cont
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Example PDA for0n1nn0
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Formal Definition
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Instantaneous Description
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Language accepted by PDA
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Language accepted by PDA
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Language accepted by PDA
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Equivalence of CFGs and PDAs
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Cont
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Deterministic PDA
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UNIT - IV

• PROPERTIES
• OF
• CONTEXT-FREE LANGUAGES

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Chomsky Normal Form
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Defining CNF
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What is the Big Deal about CNF?
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Cont
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Converting CFGs into CNF
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Eliminating e-Productions
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Cont
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Cont
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Nullability
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Generating an e-Production-free CFGs
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Converting CFGs into CNF
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Eliminating Unit Productions
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Cont
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U(G,A) and New Productions
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Example
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Cont
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Formal Construction
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Eliminating Terminal Productions
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Example
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Formal Construction
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Pumping Lemma For CFLs
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Cont
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Derivation Trees
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Defining Derivation Tree
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Example
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Derivation Trees and CNF
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Pumping Lemma for CFL
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Proving Languages Non Context Free using Pumping
Lemma
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Prove that L aNbNcNN 0 is Not a CFL
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Proof Cont
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Turing Machines
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A Finite Automaton
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A Pushdown Automaton
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A Turing Machine
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Cont..
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Differences
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Example
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Cont
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Cont
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Cont
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Formal Definition
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Cont..
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Cont
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Cont
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Configurations
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Cont
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More Configuration
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Accepting a Language
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Enumerable Languages
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UNIT V

• UNDECIDABILITY

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Enumerable Languages
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Enumerable Languages
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Example
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Example
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Example
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Element Distinctness
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Element Distinctness
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Element Distinctness
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Element Distinctness
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Element Distinctness
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Variants
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Multitape Turing Machine
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Equivalence
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Simulation
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Simulation
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Non Deterministic Turing Machine
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Decidability
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Example
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Theorem
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Theorem
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Theorem
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Theorem
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Halting Problem
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Cont..
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Cont..
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Turing Machines are countable
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Cont..
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Post Correspondence Problem
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Cont
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Cont...
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Cont
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Cont
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Cont
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NP-complete Problem
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Decision Problems To keep things simple, we
will mainly concern ourselves with decision
problems. These problems only require a single
bit output yes'' and no''. How would you
solve the following decision problems? Is
this directed graph acyclic? Is there a
spanning tree of this undirected graph with
total weight less than w? Does this
bipartite graph have a perfect (all nodes
matched) matching? Does the pattern p
appear as a substring in text t?
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P P is the set of decision problems that can be
solved in worst-case polynomial time If
the input is of size n, the running time must be
O(nk). Note that k can depend on the
problem class, but not the particular instance.
All the decision problems mentioned above are
in P.
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Nice Puzzle The class NP (meaning
non-deterministic polynomial time) is the set of
problems that might appear in a puzzle magazine
Nice puzzle.'' What makes these problems
special is that they might be hard to solve, but
a short answer can always be printed in the back,
and it is easy to see that the answer is correct
once you see it. Example... Does matrix A have
an LU decomposition? No guarantee if answer is
no''.
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NP Technically speaking A problem is in
NP if it has a short accepting certificate.
An accepting certificate is something that we can
use to quickly show that the answer is yes''
(if it is yes). Quickly means in polynomial
time. Short means polynomial size. This
means that all problems in P are in NP (since we
don't even need a certificate to quickly show the
answer is yes''). But other problems in NP
may not be in P. Given an integer x, is it
composite? How do we know this is in NP?
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Good Guessing Another way of thinking of NP is
it is the set of problems that can solved
efficiently by a really good guesser. The
guesser essentially picks the accepting
certificate out of the air (Non-deterministic
Polynomial time). It can then convince itself
that it is correct using a polynomial time
algorithm. (Like a right-brain, left-brain sort
of thing.) Clearly this isn't a practically
useful characterization how could we build such
a machine?
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Exponential Upperbound Another useful property
of the class NP is that all NP problems can be
solved in exponential time (EXP). This is
because we can always list out all short
certificates in exponential time and check all
O(2nk) of them. Thus, P is in NP, and NP is in
EXP. Although we know that P is not equal to EXP,
it is possible that NP P, or EXP, or neither.
Frustrating!
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NP-hardness As we will see, some problems are
at least as hard to solve as any problem in NP.
We call such problems NP-hard. How might we
argue that problem X is at least as hard (to
within a polynomial factor) as problem Y? If X
is at least as hard as Y, how would we expect an
algorithm that is able to solve X to behave?
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co-NP
NP
P
One of the central (and widely and intensively
studied 30 years) problems of (theoretical)
computer science is to prove that (a) P?NP
(b) NP ? co-NP. ?All evidence indicates that
these conjectures are true. ? Disproving any of
these two conjectures would not only be
considered truly spectacular, but would also come
as a tremendous surprise (with a variety of
far-reaching counterintuitive consequences). NP-c
omplete Collection Z of problems is NP-complete
if (a) it is NP and (b) if polynomial-time
algorithm existed for solving problems in Z, then
PNP.
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NP-Complete Problems
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