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- 17. 12/28/2022 17 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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- 41. 12/28/2022 41 UNIT - II REGULAR EXPRESSIONS AND LANGUAGES
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- 80. 12/28/2022 80 Questions about Regular Languages
- 82. 12/28/2022 82 Singleton Languages are Regular
- 83. 12/28/2022 83 Finite Languages are regular
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- 89. 12/28/2022 89 Closure Properties for Regular Languages
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- 94. 12/28/2022 94 UNIT III CONTEXT-FREE GRAMMAR AND LANGUAGES
- 95. 12/28/2022 95 Context Free Grammars
- 99. 12/28/2022 99 Notational conventions For CFGs
- 100. 12/28/2022 100 OTHER CFG EXAMPLES
- 101. 12/28/2022 101 Languages of CFG
- 102. 12/28/2022 102 Languages of CFG
- 103. 12/28/2022 103 Regular Languages and CFL
- 104. 12/28/2022 104 Translating FAs into CFG
- 105. 12/28/2022 105 Formalizing the Translation
- 106. 12/28/2022 106 Closure Properties of CFL
- 107. 12/28/2022 107 Proving CFLs closed under Union
- 108. 12/28/2022 108 IDEA
- 109. 12/28/2022 109 Formal Construction of Gu
- 112. 12/28/2022 112 Left Most and Right Most Derivation
- 116. 12/28/2022 116 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?
- 117. 12/28/2022 117 Cont…
- 118. 12/28/2022 118 Cont…
- 119. 12/28/2022 119 Example PDA for{0n1n|n≥0}
- 122. 12/28/2022 122 Language accepted by PDA
- 123. 12/28/2022 123 Language accepted by PDA
- 124. 12/28/2022 124 Language accepted by PDA
- 125. 12/28/2022 125 Equivalence of CFGs and PDAs
- 126. 12/28/2022 126 Cont…
- 128. 12/28/2022 128 UNIT - IV PROPERTIES OF CONTEXT-FREE LANGUAGES
- 129. 12/28/2022 129 Chomsky Normal Form
- 131. 12/28/2022 131 What is the Big Deal about CNF?
- 132. 12/28/2022 132 Cont…
- 133. 12/28/2022 133 Converting CFGs into CNF
- 135. 12/28/2022 135 Cont…
- 136. 12/28/2022 136 Cont…
- 138. 12/28/2022 138 Generating an ε-Production-free CFGs
- 139. 12/28/2022 139 Converting CFGs into CNF
- 140. 12/28/2022 140 Eliminating Unit Productions
- 141. 12/28/2022 141 Cont…
- 142. 12/28/2022 142 U(G,A) and New Productions
- 144. 12/28/2022 144 Cont…
- 146. 12/28/2022 146 Eliminating Terminal Productions
- 149. 12/28/2022 149 Pumping Lemma For CFLs
- 150. 12/28/2022 150 Cont…
- 152. 12/28/2022 152 Defining Derivation Tree
- 154. 12/28/2022 154 Derivation Trees and CNF
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- 156. 12/28/2022 156 Pumping Lemma for CFL
- 157. 12/28/2022 157 Proving Languages Non Context Free using Pumping Lemma
- 158. 12/28/2022 158 Prove that L= {aNbNcN|N ≥0} is Not a CFL
- 161. 12/28/2022 161 A Finite Automaton
- 162. 12/28/2022 162 A Pushdown Automaton
- 163. 12/28/2022 163 A Turing Machine
- 167. 12/28/2022 167 Cont…
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- 175. 12/28/2022 175 Cont…
- 177. 12/28/2022 177 Accepting a Language
- 179. 12/28/2022 179 UNIT – V UNDECIDABILITY
- 191. 12/28/2022 191 Multitape Turing Machine
- 195. 12/28/2022 195 Non Deterministic Turing Machine
- 205. 12/28/2022 205 Turing Machines are countable
- 207. 12/28/2022 207 Post Correspondence Problem
- 208. 12/28/2022 208 Cont…
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- 217. 12/28/2022 217 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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- 219. 12/28/2022 219 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.
- 220. 12/28/2022 220 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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- 222. 12/28/2022 222 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?
- 223. 12/28/2022 223 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?
- 224. 12/28/2022 224 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!
- 225. 12/28/2022 225 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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- 230. 12/28/2022 230 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. P NP co-NP
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