This document contains lecture notes for a discrete mathematics course taught through Colorado Technical University. The notes cover three topics: [1] languages, discussing formal languages and language set notation; [2] grammar, defining formal grammars and introducing Backus-Naur form; and [3] automata theory, defining deterministic and nondeterministic finite automata. Examples are provided for each topic to illustrate the concepts. The document was created on June 20, 2008 by TSgt Loren K. R. Schwappach, an electrical engineering major.

1. Colorado Technical University
MAT200, OLA: Discrete Mathematics
Professor: Ricky Streight
Created: 20 June 2008
Presented by: TSgt Loren K. R. Schwappach
Major: Electrical Engineering
NOTE: Please Use MS Power Point version 2007 for full compatibility with slides.

2. Topic #1: Languages
What are † Languages?:
 Formal Language: Wikipedia defines a formal language as an
organized set of symbols that can be precisely defined in terms of the
shapes and locations of the symbols. Colorado Technical University
explains a formal language is like a set of strings produced from a finite
alphabet, and uses the English language as an example. The set of
stings would include all of the words making up the language.
 For example the string l-i-f-e is the word life and is a part of the English
language while the string r-x-x-r-q-q-z has no meaning in the English
language.
 Language set notation: Formal languages are defined by rules the
language must obey. Thus set notation provides a powerful method for
logically approaching formal languages.

3. Topic #1: Languages: Examples

4. Topic #2: Grammar
What are † Formal Grammars?:
 Defined: Wikipedia says a formal grammar is a precise description
of a formal language. Grammar describes which of the possible
sequences of symbols (strings) in a language constitute valid words or
statements in a language, but does not describe their semantics.
Wikipedia, 2008)
 In other words grammar is the glue that makes the strings (words) in a
formal language work together.
 Backus-Naur form (BNF): One of the most common methods used
to describe a formal language. Wikipedia classifies BNF as a meta-
syntax used to express context-free grammars.
 BNF describes how a sentence is produced using a form known as
productions. Using BNF language strings can be made into sentences
 The best way to understand BNF is with a quick example.

5. Topic #2: Grammar: Examples

6. Topic #3: Automata
What is † Automata Theory?:
 Defined: The study of abstract machines and the problems they are
able to solve. (Wikipedia, 2008)
 Deterministic finite automata: An automata that has a transition for every
letter of the alphabet and where a state cannot have multiple outgoing
edges with the same label.
 Nondeterministic finite automata: An automata that may or may not
have a transition for every letter in the alphabet, or multiple transitions and
where a state can have multiple outgoing edges all with the same label.
 Automaton: A mathematical model of a finite-state machine † in
which the set of output symbols is {0,1} and where the current state
determines the last output. (Johnsonbaugh, 2009)
 A finite-state machine (FSM) is a machine that, given an input of
symbols “jumps” through a series of states according to a transition function
(which can be expressed as a table). (Wikipedia, 2008)

7. Topic #3: Automata: Examples
S0 S1 S2 S3

8. Topic #3: Automata: Examples
S0 S1 S2

9. CTU Online, (2008). Multimedia Material, Retrieved Jun, 19, 2008, from Colorado Technical University
Web site: https://campus.ctuonline.edu
Johnsonbaugh, R. (2009). Discrete Mathematics (7th ed.). Upper Saddle River, New Jersey: Pearson
Prentice Hall.
Rizzi, R. (2002). Complexity of Context-free Grammars with Exceptions and the inadequacy of
grammars as models for XML and SGML. Massachusetts Institute of Technology.
Wikipedia. (2008). Automata theory. Retrieved Jun, 9, 2008, from Wikipedia the free encyclopedia
Web site: http://en.wikipedia.org/wiki/Automata_theory
Wikipedia. (2008). Automaton. Retrieved Jun, 9, 2008, from Wikipedia the free encyclopedia Web site:
http://en.wikipedia.org/wiki/Automaton
Wikipedia. (2008). Backus-Naur Form. Retrieved Jun, 9, 2008, from Wikipedia the free encyclopedia
Web site: http://en.wikipedia.org/wiki/Backus-Naur_form
Wikipedia. (2008). Formal_grammar. Retrieved Jun, 9, 2008, from Wikipedia the free encyclopedia
Web site: http://en.wikipedia.org/wiki/Formal_grammar

10. If you have any questions or suggestions
Please email me at:
Loren.Schwappach@peterson.af.mil
or
Lschwappach@yahoo.com