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Unformatted text preview: NP CookLevin Theorem CS 473: Algorithms Chandra Chekuri [email protected] 3228 Siebel Center University of Illinois, UrbanaChampaign Fall 2009 Chekuri CS473ug NP CookLevin Theorem P and NP and Turing Machines P : set of decision problems that have polynomial time algorithms NP: set of decision problems that have polynomial time nondeterministic algorithms Chekuri CS473ug NP CookLevin Theorem P and NP and Turing Machines P : set of decision problems that have polynomial time algorithms NP: set of decision problems that have polynomial time nondeterministic algorithms Question: What is an algorithm? Chekuri CS473ug NP CookLevin Theorem P and NP and Turing Machines P : set of decision problems that have polynomial time algorithms NP: set of decision problems that have polynomial time nondeterministic algorithms Question: What is an algorithm? Depends on the model of computation! Chekuri CS473ug NP CookLevin Theorem P and NP and Turing Machines P : set of decision problems that have polynomial time algorithms NP: set of decision problems that have polynomial time nondeterministic algorithms Question: What is an algorithm? Depends on the model of computation! What is our model of computation? Chekuri CS473ug NP CookLevin Theorem P and NP and Turing Machines P : set of decision problems that have polynomial time algorithms NP: set of decision problems that have polynomial time nondeterministic algorithms Question: What is an algorithm? Depends on the model of computation! What is our model of computation? Formally speaking our model of computation is Turing Machines. Chekuri CS473ug NP CookLevin Theorem Turing Machines: Recap Turing Machines X 1 X 2 · · · X n finitestate control tape head Unrestricted memory: an infinite tape A finite state machine that reads/writes symbols on the tape Can read/write anywhere on the tape Tape is infinite in one direction only (other variants possible) Infinite tape Finite state control Input at beginning of tape Special tape letter “blank” t Head can move only one cell to left or right Chekuri CS473ug NP CookLevin Theorem Turing Machines: Formally A TM M = ( Q , Σ , Γ ,δ, q , q accept , q reject ): Q is set of states in finite control q start state, q accept is accept state, q reject is reject state Σ is input alphabet, Γ is tape alphabet (includes t ) δ : Q × Γ → { L , R } × Γ × Q is transition function δ ( q , a ) = ( q , b , L ) means that M in state q and head seeing a on tape will move to state q while replacing a on tape with b and head moves left. L ( M ): language accepted by M is set of all input strings s on which M when started in q on tape cell 1 and s on tape halts in q accept ....
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 Fall '08
 Chekuri,C
 Algorithms, Computational complexity theory, NPcomplete, CookLevin Theorem, Circuit Satisfaction

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