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.

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.

Topic #1: Languages: Examples

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.

Topic #2: Grammar: Examples

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)

Topic #3: Automata: Examples

S2S0 S3S1

Topic #3: Automata: Examples

S2S0 S1

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

If you have any questions or suggestions

Please email me at:

Loren.Schwappach@peterson.af.mil

or

Lschwappach@yahoo.com