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RAJALAKSHMI ENGINEERING COLLEGE
THEORY OF COMPUTATION 11/19/2018
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UNIT- I AUTOMATA 11/19/2018
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Finite Automata 11/19/2018
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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 11/19/2018
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UNIT - II REGULAR EXPRESSIONS AND LANGUAGES 11/19/2018
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Questions about Regular Languages
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Question contd… 11/19/2018
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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 11/19/2018
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Context Free Grammars 11/19/2018
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Cont… 11/19/2018
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Cont… 11/19/2018
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Defining CFG 11/19/2018
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Notational conventions For CFGs
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OTHER CFG EXAMPLES 11/19/2018
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Languages of CFG 11/19/2018
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Languages of CFG 11/19/2018
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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 11/19/2018
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Formal Construction of Gu
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Derivations 11/19/2018
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Sentential Form 11/19/2018
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Left Most and Right Most Derivation
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Right-Linear Grammars
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Ambiguous Grammars 11/19/2018
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Ambiguous Grammars 11/19/2018
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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? 11/19/2018
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Cont… 11/19/2018
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Cont… 11/19/2018
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Example PDA for{0n1n|n≥0}
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Formal Definition 11/19/2018
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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… 11/19/2018
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Deterministic PDA 11/19/2018
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CONTEXT-FREE LANGUAGES
UNIT - IV PROPERTIES OF CONTEXT-FREE LANGUAGES 11/19/2018
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Chomsky Normal Form 11/19/2018
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Defining CNF 11/19/2018
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What is the Big Deal about CNF?
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Cont… 11/19/2018
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Converting CFGs into CNF
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Eliminating ε-Productions
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Cont… 11/19/2018
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Cont… 11/19/2018
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Nullability 11/19/2018
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Generating an ε-Production-free CFGs
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Converting CFGs into CNF
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Eliminating Unit Productions
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Cont… 11/19/2018
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U(G,A) and New Productions
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Example 11/19/2018
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Cont… 11/19/2018
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Formal Construction 11/19/2018
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Eliminating Terminal Productions
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Example 11/19/2018
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Formal Construction 11/19/2018
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Pumping Lemma For CFLs 11/19/2018
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Cont… 11/19/2018
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Derivation Trees 11/19/2018
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Defining Derivation Tree
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Example 11/19/2018
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Derivation Trees and CNF
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Pumping Lemma for CFL 11/19/2018
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Proving Languages Non Context Free using Pumping Lemma
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Prove that L= {aNbNcN|N ≥0} is Not a CFL
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Proof Cont… 11/19/2018
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Turing Machines 11/19/2018
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A Finite Automaton 11/19/2018
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A Pushdown Automaton 11/19/2018
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A Turing Machine 11/19/2018
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Cont.. 11/19/2018
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Differences 11/19/2018
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Example 11/19/2018
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Formal Definition 11/19/2018
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Cont… 11/19/2018
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Configurations 11/19/2018
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Cont… 11/19/2018
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More Configuration 11/19/2018
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Accepting a Language 11/19/2018
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Enumerable Languages 11/19/2018
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UNIT – V UNDECIDABILITY 11/19/2018
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Enumerable Languages 11/19/2018
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Enumerable Languages 11/19/2018
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Example 11/19/2018
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Example 11/19/2018
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Example 11/19/2018
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Element Distinctness 11/19/2018
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Element Distinctness 11/19/2018
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Element Distinctness 11/19/2018
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Element Distinctness 11/19/2018
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Element Distinctness 11/19/2018
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Variants 11/19/2018
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Multitape Turing Machine
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Equivalence 11/19/2018
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Simulation 11/19/2018
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Simulation 11/19/2018
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Non Deterministic Turing Machine
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Decidability 11/19/2018
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Example 11/19/2018
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Theorem 11/19/2018
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Theorem 11/19/2018
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Theorem 11/19/2018
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Theorem 11/19/2018
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Halting Problem 11/19/2018
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Cont.. 11/19/2018
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Cont.. 11/19/2018
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Turing Machines are countable
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Cont.. 11/19/2018
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Post Correspondence Problem
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NP-complete Problem 11/19/2018
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To keep things simple, we will mainly concern ourselves with
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? 11/19/2018
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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. 11/19/2018
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The class NP (meaning non-deterministic polynomial time) is the
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''. 11/19/2018
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Technically speaking:
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? 11/19/2018
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Another way of thinking of NP is it is the set of problems that can
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? 11/19/2018
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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! 11/19/2018
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As we will see, some problems are at least as hard to solve as any
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? 11/19/2018
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All evidence indicates that these conjectures are true.
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) PNP (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-complete: 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 P=NP. 11/19/2018
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NP-Complete Problems 11/19/2018
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