Overview

Description

CALENDAR DESCRIPTION:

This course introduces students to formal models of computations such as Turing machines and RAMs. Notions of tractability and intractability are discusses both with respect to computability and resource requirements. The relationship of these concepts to logic is also covered.

COURSE DETAILS:

This course focuses on the inherent "complexity" of solving problems using a computer. The goal is to understand why some seemingly simple problems cannot be solved on computers and others have no efficient (ie fast) solution. In the course, we will see the formal notions of computers, computability and complexity. At the successful completion of this course students will understand why, for example, computer viruses are so pervasive and why no one will ever write a perfect virus checker. We will see how these concepts are related to logic, in particular, the famous Incompleteness Theorem of Godel. Finally, we will see a few surprising results from modern complexity, in particular, the results making use of randomness in computation.

Topics

• Preliminaries (if needed) - sets, functions, relations, alphabets, strings, asymptotics.
• Turing Machines as a formalization of the intuitive notion of an algorithm.
• Computability (Does a program exist?): basic computability (checking if a program is in an infinite loop), reducibilities and oracles, the Recursion Theorem (existence of computer viruses).
• Review of Logic and Godel's Incompleteness Theorem.
• Complexity Theory: Non-determinism, the class NP, reductions.
• Randomness in Computation: Interactive Proofs.
• Approximation algorithms and hardness of approximation: Probabilistically Checkable Proofs and the PCP Theorem

Grading

Materials

MATERIALS + SUPPLIES:

Reference Books

• Introduction to Automata Theory, Languages and Computation - 3rd Edition, J.E. Hopcroft , Rajeev Motwani, J.D. Ullman, , Addison Wesley, 2006, 9780321455369

REQUIRED READING: