1. COSC 3340: Introduction to Theory of Computation
Rakesh Verma
Computer Science Department
University of Houston
URL:
UofH - COSC Dr. Verma

2. What is this course about?
Course is about models of computation, their power, and relationships.
Hierarchy of models of increasing power: DFA < PDA < TM.
For each model, we study:
What can be computed. (easier)
What cannot be computed. (harder)
UofH - COSC Dr. Verma

3. Basic Definitions (Reading Assignment: Chapter 0 of text)
1. Alphabet - a finite set of symbols.
Notation:  .
Examples: Binary alphabet {0,1}, English alphabet {a,...,z,!,?,...}
2. String over an alphabet  - a finite sequence of symbols from .
Notation: (a) Letters u, v, w, x, y, and z denote strings.
(b) Convention: concatenate the symbols. No parentheses or commas used.
Examples: 0000 is a string over the binary alphabet.
a!? is a string over the English alphabet.
UofH - COSC Dr. Verma

4. Definitions (contd.)
3. Empty string: e or  denotes the empty sequence of symbols.
4. Language over alphabet  - a set of strings over .
Notation: L.
Examples:
{0, 00, 000, ...} is an "infinite" language over the binary alphabet.
{a, b, c} is a "finite" language over the English alphabet.
UofH - COSC Dr. Verma

5. Definitions (contd.)
5. Empty language - empty set of strings. Notation: .
6. Binary operation on strings: Concatenation of two strings u.v - concatenate the symbols of u and v.
Notation: uv
Examples:
00.11 = 0011.
.u = u. = u for every u. (identity for concatenation)
UofH - COSC Dr. Verma

6. Binary relations on strings
Prefix - u is a prefix of v if there is a w such that v = uw.
Examples:
 is a prefix of 0 since 0 = 0
apple is a prefix of appleton since appleton = apple.ton
Suffix - u is a suffix of v if there is a w such that v = wu.
0 is a suffix of 0 since 0 = ?
ton is a suffix of appleton since ?
UofH - COSC Dr. Verma

7. Binary relations (contd.)
Substring - u is a substring of v if there are x and y such that v = xuy.
Examples:
let is a substring of appleton since appleton = app.let.on
0 is a substring of 0 since 0 = epsilon.0.epsilon
Observe that prefix and suffix are special cases
of substring.
UofH - COSC Dr. Verma

8. Relevance of strings and languages for this course.
Consider decision problems. Decision problems are problems with Yes/No answers.
Examples: (a) Is an array sorted?
(b) Is x in that array A?
Consider instances, I, of a decision problem D.
Example: For problem (a) an instance is just an array.
UofH - COSC Dr. Verma

9. Relevance (contd.) We can encode instances of any problem by strings.
Associate language with decision problem D by taking all the strings that encode its yes instances.
UofH - COSC Dr. Verma

10. Applications of the Concepts
Are vast, and include:
Text processing (editors, etc.)
Compiler design
Verification through model checking
Real world examples are all around you and not just on the desk!
UofH - COSC Dr. Verma

11. New Developments DNA Computing Quantum Turing Machines
New Applications are being discovered as we speak!
UofH - COSC Dr. Verma

12. History of the Field Check out:
URL:
UofH - COSC Dr. Verma