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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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