Summary Summary Complexity Classification of problems into those

Summary

Summary Complexity Classification of problems into those can solved fast, and other slow. • Class P e. g. Given a map of JR routes in Osaka, determine if one can go from Sugimotocho to Morinomiya. • Class NP e. g. Given a list of cities and the distances between each pair of cities, find the shortest route that visits each city once and returns to the origin city (travelling salesman problem).

Summary Complexity • • • Time complexity Big-O Algorithm time analysis using big-O Class P and examples Class NP and examples • • NP-complete (k-clique problem) Space complexity

Summary Computability Classification of problems into those are solvable (decidable), and those are unsolvable (undecidable). • Decidable e. g. Travelling salesman problem • Undecidable e. g. Given a program, determine if it always halts.

Summary Computability • • Definition of algorithm High-level description of Turing machines Decidability Examples of decidable problems Undecidability Examples of undecidable problems Unrecognizability and example

Summary Computation models Definitions and properties of mathematical models of computation and the corresponding languages. • Finite automaton (regular language) Finite memory • Push-down automaton (context-free language) Infinite memory, restricted access • Turing machine (Turing-decidable/recognizable language) Infinite memory, unrestricted access

Summary Computation models • • • Alphabet, strings, languages Deterministic finite automata and operations Nondeterministic finite automata and conversion to DFA Regular expressions and conversion to NFA Regular languages and pumping lemma Context-free grammars Push-down automata Context-free languages and pumping lemma Turing machines and Turing-recognizable languages Deciders and Turing-decidable languages Multi-tape TM and Nondeterministic TM

Summary 1. Computation models 2. Computability 3. Complexity What are the fundamental capabilities and limitations of computers?