lecture20 - NP Cook-Levin Theorem CS 473 Algorithms Chandra...

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Unformatted text preview: NP Cook-Levin Theorem CS 473: Algorithms Chandra Chekuri [email protected] 3228 Siebel Center University of Illinois, Urbana-Champaign Fall 2009 Chekuri CS473ug NP Cook-Levin 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 non-deterministic algorithms Chekuri CS473ug NP Cook-Levin 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 non-deterministic algorithms Question: What is an algorithm? Chekuri CS473ug NP Cook-Levin 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 non-deterministic algorithms Question: What is an algorithm? Depends on the model of computation! Chekuri CS473ug NP Cook-Levin 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 non-deterministic algorithms Question: What is an algorithm? Depends on the model of computation! What is our model of computation? Chekuri CS473ug NP Cook-Levin 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 non-deterministic 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 Cook-Levin Theorem Turing Machines: Recap Turing Machines X 1 X 2 · · · X n finite-state 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 Cook-Levin 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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lecture20 - NP Cook-Levin Theorem CS 473 Algorithms Chandra...

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