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MIT6_045JS11_lec12

# MIT6_045JS11_lec12 - 6.045 Automata Computability and...

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6.045: Automata, Computability, and Complexity Or, GITCS Class 12 Nancy Lynch

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Today: Complexity Theory First part of the course: Basic models of computation Circuits, decision trees DFAs, NFAs: Restricted notion of computation: no auxiliary memory, just one pass over input. Yields restricted class of languages: regular languages. Second part: Computability Very general notion of computation. Machine models like Turing machines, or programs in general (idealized) programming languages. Unlimited storage, multiple passes over input, compute arbitrarily long, possibly never halt. Yields large language classes: Turing-recognizable = enumerable, and Turing-decidable. Third part: Complexity theory
Complexity Theory First part of the course: Basic models of computation Second part: Computability Third part: Complexity theory A middle ground. Restrict the general TM model by limiting its use of resources: Computing time (number of steps). Space = storage (number of tape squares used). Leads to interesting subclasses of the Turing-decidable languages, based on specific bounds on amounts of resources used. – Compare: Computability theory answers the question “What languages are computable (at all)?” Complexity theory answers “What languages are computable with particular restrictions on amount of resources?”

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Complexity Theory • Topics – Examples of time complexity analysis (informal). – Asymptotic function notation: O, o, Ω , Θ – Time complexity classes – P, polynomial time – Languages not in P – Hierarchy theorems • Reading: – Sipser, Sections 7.1, 7.2, and a bit from 9.1. • Next: – Midterm, then Section 7.3 (after the break).
Examples of time complexity analysis

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