Complexity and Computability
ICSG-704/ICSS-481
Spring 2001

Instructor

Stanislaw Radziszowski

bldg. 10, room 0A182,
(716) 475-5193, spr@cs.rit.edu
http://www.cs.rit.edu/~spr
office hours TR 8-10pm

Lectures

Tuesday/Thursday, 6-8pm, 12-3105

Books and Other Reading

• M. Sipser, Introduction to the Theory of Computation, PWS Publ, 1997 (required textbook).
• C. Papadimitriou, Computational Complexity, Addison-Wesley 1994 (complementary reading).
• M. Garey and D. Johnson, Computers and intractability : a guide to the theory of NP-completeness, W. H. Freeman 1979 (suggested reference).
• D. Hofstadter, Gödel, Escher, Bach : an eternal golden braid, New York Basic Books 1979 (suggested reading).
• E. Nagel and J.R. Newman, Gödel's Proof, New York University Press 1958 (suggested reading).
• Journal articles.

Prerequisites

Evaluation

• 30% Homeworks
• 30% Presentation / Research paper
• 15% Class participation
• 25% Final exam

Contents

Introduction to the complexity and computability theories. Overview of basic complexity classes, with special focus on NP-theory related problems. Study of problems complete in NP and PSPACE. Church-Turing thesis. Undecidability of chosen classical problems. Some advanced topics in computability, like degrees of unsolvability, recursion theorem or Gödel's incompleteness theorem, are included.

Topics

1. Turing machines based computability
   • Church-Turing Thesis
   • Hilbert's problems
   • Diagonalization method
   • (Non)-decidable-solvable-computable-recursive problems
2. Undecidable problems
   • Halting problem
   • Undecidability by reductions
   • Program equivalence
   • Printable = recursively enumerable (r.e.)
   • Problems in formal languages
3. Selected advanced topics
   • Beyond r.e.
   • Recursion theorem
   • Creative sets
   • Arithmetical hierarchy
   • Complexity of proving theorems
   • Gödel's incompleteness theorem
4. Complexity Fundamentals
   • Models of computation
   • Computational problems, algorithms, complexity
   • P versus NP
   • Reducibility and completeness
5. NP-complete problems
   • Satisfiability problem
   • Chosen graph problems
   • Traveling salesman
   • Knapsack
   • Scheduling
   • Garey-Johnson catalog
6. Complexity classes
   • NP, coNP, primality testing
   • PSPACE, pebbling and games
   • Logarithmic space, path problem
   • Interactive proof systems
7. Hierarchies
   • Time and space hierarchy
   • Polynomial hierarchy
   • Beyond PSPACE
   • Other miscellaneous classes