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Final Exam Review

Final Exam Review. Cummulative Chapters 0, 1, 2, 3, 4, 5 and 7. Chapter 0: Discrete Math Review. Sets, Sequences Venn Diagrams Boolean Logic Equivalence Relations Concept of an Equivalence Class Symbols, Alphabets, Strings & Languages. Chapter 0: Discrete Math Review. Proof by Induction

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Final Exam Review

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  1. Final Exam Review Cummulative Chapters 0, 1, 2, 3, 4, 5 and 7

  2. Chapter 0: Discrete Math Review • Sets, Sequences • Venn Diagrams • Boolean Logic • Equivalence Relations • Concept of an Equivalence Class • Symbols, Alphabets, Strings & Languages

  3. Chapter 0: Discrete Math Review • Proof by Induction • Proof by Contradiction (Pumping Lemma) • Proof by Construction (Machine Construction and formal definitions)

  4. Chapter 1: Regular Languages • Describing FSA’s with set/sequence descriptions • Language Description (words and sets)  FSA • FSA  Language description • Language Description  Regular Expression • Regular Expression  Language description • How are the 3 regular operations implemented with FSA’s

  5. Chapter 1: Regular Languages • FSA  Regular Expression • Non-determinism: 3 forms • NFA  DFA • Pumping Lemma for proving that a language is not Regular

  6. Chapter 2: Context Free Languages • Context Free Grammars (CFG) • Push-down Automata (PDA) • Language descriptions  CFG • CFG  Language Description • Language description  PDA • PDA  Language Description • Non-determinism in PDA’s

  7. Chapter 2: Context Free Languages • CFG  PDA • PDA  CFG • Pumping Lemma to prove that a language is NOT Context-free (and also NOT Regular) • Implications of adding 2 or more stacks

  8. Chapter 3: Turing Machines • Turing Machines = Algorithms • Turing Machines = Recursively Enumerable Languages • Turing Machine Tape can be implemented with 2 stacks

  9. Chapter 3: Turing Machines • Language Description  Turing Machine • Turing Machine  Language Description • Computability • 2 or more tapes adds no power • Complexity • 2 or more tapes can add efficiency

  10. Chapter 4 & 5 • Turing Decidable vs. Turing Recognizable • Un-decidable Problems • ATM • The Halting Problem • Review Theorems in these Chapters

  11. Chapter 7: P vs. NP • Examples of Polynomial Problems • PATH Problem • Examples of NP Problems • Hamiltonian PATH • Decider vs. Verifier • Understanding NP • Polynomial Verification with a Non-deterministic Turing Machine  Exponential Computation O(kn)

  12. Chapter 7: P vs. NP • NP-Complete Problems • New Problem <==>Satisfiability Problem • X <==> Y Polynomial-time reduction • The input of one problem X can be transformed into the input of another problem Y such that solving problem Y also yields a solution for problem X

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