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Complexity Classes

Complexity Classes. Chapter 8 (Abbreviated). Overview. Problems are classified according to the resources they use on computing machines (serial or parallel) Serial models are RAM and Turing machines Parallel models are the circuit and PRAM Terms of discussion are first defined

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Complexity Classes

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  1. Complexity Classes Chapter 8 (Abbreviated)

  2. Overview • Problems are classified according to the resources they use on computing machines (serial or parallel) • Serial models are RAM and Turing machines • Parallel models are the circuit and PRAM • Terms of discussion are first defined • Tasks that are well-defined • Resource measures • Machine models • Serial complexity classes are discussed • P-complete problems • NP-complete problems • PSPACE-complete problems

  3. Languages and Problems

  4. Resource Bounds • “Feasible” problems are defined currently as those problems which can be solved with a deterministic TM in polynomial time  known as the serial computation thesis • Problems can be classified according to their use and requirement of resources, r(n) • Resource bounds expressed in terms of these functions:

  5. Serial Computational Models • Random-Access Machine: this flavor allows for words that have potentially unbounded length

  6. A non-deterministic, single-tape Turing machine Serial Computational Models (cont.) • Turing Machine: will use deterministic (DTM) and non-deterministic (NDTM) versions, as well as multi-tape flavors of each

  7. Classification ofDecision Problems • Problems in P are currently defined as “feasible” problems

  8. Classification ofDecision Problems (cont.) • Is this last result evidence that PNP?

  9. Reductions • Generalize the notion of reductions to include bounds on resources needed and preservation of complexity class • Define new notation and terminology to include these notions:

  10. Hard and Complete Problems • Define the notions of “hard” and “complete” problems for different complexity classes • The following result follows naturally from our definitions:

  11. P-Complete Problems • These problems are P-Complete (there are hundreds of others):

  12. NP-Complete Problems • These problems are NP-Complete (there are literally thousands of others):

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