COMS W3261
Computer Science Theory
Lecture 1: September 9, 2009

1. Teaching Staff

Instructor

Professor Alfred V. Aho

http://www1.cs.columbia.edu/~aho

aho@cs.columbia.edu

513 Computer Science Building

Office hours: Mondays and Wednesdays 2:45-3:45pm

Course webpage: http://www1.cs.columbia.edu/~aho/cs3621

Course bulletin board: https://courseworks.columbia.edu

TAs

Sumeet Jindal

sj2405@columbia.edu

Office hours: Mondays and Fridays 10:00-11:00am

TA Room: 122 Mudd

Injung Kim

ik2249@columbia.edu

Office hours: Wednesdays 3:00-4:00pm and Thursdays 1:00-2:00pm

TA Room: 122 Mudd

Liang Zhang

lz2232@columbia.edu

Office hours: Mondays and Wednesdays 9:45-10:45am

TA Room: 122 Mudd

2. Schedule

• Lectures: Mondays and Wednesdays, 1:10-2:25pm, 702 Hamilton Hall

3. Course Objectives

• Understanding the fundamental models of computation that underlie modern computer hardware, software, and programming languages.
• Learning that there are problems no computer can solve.
• Learning that there are limits as to how fast a computer can solve a problem.
• Learning the foundations of automata theory, computability theory, and complexity theory.
• Learning about applications of theory to other areas of computer science such as algorithms, programming languages, compilers, natural language translation, operating systems, and software verification.

4. Course Syllabus

• Languages and proofs
• Finite automata
• Regular expressions
• Properties of regular languages
• Context-free grammars
• Pushdown automata
• Properties of context-free languages
• Turing machines
• Lambda calculus
• Undecidability
• Complexity theory
• Other models of computation

5. Textbooks and References

• The course text is
John E. Hopcroft, Rajeev Motwani, and Jeffrey D. Ullman
Introduction to Automata Theory, Languages, and Computation, Third Edition
Pearson/Addison-Wesley, 2007, ISBN 0-321-45536-3



• Other good references are
• Michael Sipser
Introduction to the Theory of Computation, Second Edition
Thomson, 2006


• Alfred V. Aho and Jeffrey D. Ullman
Foundations of Computer Science, C Edition
W. H. Freeman, 1995
An online version of this book is available here.

6. Course Requirements

• First midterm (30% of final grade)
• Second midterm (30% of final grade)
• Final (40% of final grade)

7. Languages

• An alphabet Σ is a finite, nonempty set of symbols.
• Examples: {0,1}, ASCII, Unicode
• A string is a finite sequence of symbols chosen from some alphabet.
• Examples: ε (the empty string), 0, 01, 011
• Common operations on strings: concatenation, reversal
• Terms associated with strings: prefix, suffix, substring, subsequence
• A language over Σ is a set of strings whose symbols are chosen from Σ. Examples:
• the empty set, ∅
• {0,1}
• P = {10, 11, 101, 111, 1011, 1101, ... } (the binary representations of the prime numbers)
• Operations on languages: union, concatenation, Kleene closure
• Example: {0,1}* = {ε, 0, 1, 00, 01, 10, 11, 000, ... } (all strings of 0s and 1s)
Question: How many strings are there in {0,1}*?
• A problem is the question of deciding whether a given string is a member of some particular language.
• Example: Given a binary number x, is x in P?
• Question: How many languages over {0,1} are there?

8. Proofs

• What is a theorem?
• A theorem is a statement that has been proven true by a convincing logical argument.
• What is a proof?
• A (deductive) proof is a sequence of statements each one of which is a given fact or follows by a logical rule from some previous statements in the proof.
• Types of proof
• By deduction
• By construction
• By induction
• By structural induction
• By contradiction
• Example: the square root of two is not a rational number.
• Many others

9. Practice Problems

1. How many (a) prefixes, (b) suffixes, (c) substrings, (d) subsequences are there in a string of length n?
2. What does it mean for a string to have balanced parentheses?
3. Prove that the following recursive definition generates all and only all strings of balanced parentheses.
• Rule 1 (basis): The empty string is a balanced string.
• Rule 2 (induction): If x and y are balanced strings, then (x)y is a balanced string.
4. Research problem: If x is a string of length m and y is a string of length n, then what is the maximum possible number of longest common subsequences between x and y as a function of m and n?

10. Reading Assignment

• HMU: Ch. 1