Final Exam Review

CummulativeChapters 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• Proof by Contradiction (Pumping Lemma)• Proof by Construction

(Machine Construction and formal definitions)

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

Chapter 1: Regular Languages

• FSA Regular Expression• Non-determinism: 3 forms• NFA DFA• Pumping Lemma for proving that a language is

not Regular

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

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

Chapter 3: Turing Machines

• Turing Machines = Algorithms• Turing Machines = Recursively Enumerable

Languages• Turing Machine Tape can be implemented

with 2 stacks•

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

Chapter 4 & 5

• Turing Decidable vs. Turing Recognizable• Un-decidable Problems– ATM

• The Halting Problem• Review Theorems in these Chapters

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)

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