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(k n )

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