COM273: Automata, Computability, and Formal Languages
REVIEW NOTES

Connected graph A graph G is connected if given any vertices u and v in G, there is a path from u to v (or v to u). That is, in a connected graph, we can get from any vertex to any other vertex on a path. If there’s no path between some pair of vertices then the graph is called disconnected. The following shows a connected graph and a disconnected graph. Example: a b c b a c

e d connected graph

e d disconnected graph

Cycle A cycle (or a circuit) is a path of non-zero length from u to v with no repeated edges. A simple cycle is cycle from v to v, in which there are no repeated vertices (except for the initial and final vertices which are both equal to v) Example: a b d e c (a, b, d, c, b, e, c, a) is a cycle (but not simple) (a, d, c, b, e, a) is a simple cycle of length 5.

Notice that a simple path of length n contains (n+1) distinct vertices, which a simple cycle of length n contains n distinct vertices.

Eulerian graphs A cycle in a connected graph G that includes all the edges and all the vertices of G is called an Euler cycle (“Euler” is pronounced as “oiler”).

A connected graph containing an Euler cycle is called an Eulerian graph.

Finding an Euler cycle in a graph is the same as playing the following “game”. Draw the entire graph without lifting your pen which drawing all the edges exactly once, and visiting all the vertices, starting from and ending at the same vertex. Example: Try playing the “game” described above to show that graphs (1) – (5)is the following graphs are Eulerian. Graph (6) is not Eulerian because you can’t draw the entire graph starting and ending with the same vertex.

(1) (2)

(3)

(4)

(5)

(6)

Special Types of Graphs There are certain types of graphs to which we give special names. 1. 2. 3. 4. 5. Trivial graph – a graph of order 1 and size 0. Null graph – a graph of order n and size 0; denoted by Nn. Cycle graph – a graph of...

...Society is changing at a rapid pace and informality in the English language is on the rise. In places where formallanguage was once a must, informality has taken over, and questions are raised over whether formallanguage still has a place in modern English. However, despite changes, formallanguage is still in use and is still expected and considered the most appropriate method of address in many situations. In written and spoken communication, formallanguage has the ability to convey knowledge and provide a sense of occasion, while informality has the ability to minimise social distance and is easily understood. Situations in which formallanguage is the most appropriate form of address may have diminished, however it retains an important role in society.
From chatting on the phone to a friend to asking about a neighbour’s day, spoken informal language is the main form of language used in most people’s everyday lives, however it is now being used in more situations. For example, a shift in the way businesses run has been reflected in a less formal method of communicating. Whereas the old business model was focused on the boss being above their employees, hence, a formal manner of communication between the two groups. Businesses now run on the basis of...

...Languages, Grammars, and Automata Theory
Discrete Mathematics
Linda Chalk
Colorado Technical University
Professor Timothy Manzke
December 17, 2010
Languages, grammars and automata theory are all related to computer applications. Grammar is the rule for language structure regardless of the meaning. For computer programming languages context free grammar (CFG) is commonly used.Formallanguage as opposed to natural language must be used with computer applications because natural language is too vague for computer applications. Automata is a way to test an expression to determine if it is part of the language (Pfeifer, 2005).
A deterministic finite automata (DFA) is a machine that is the most simple to understand which will help with the more complicated machines because many important properties of the DFA are found in the more complicated machines. DFA’s are found in many things like vending machines and elevators. The DFA works starting in a ‘start state’ then by reading a string, then producing an output of true or false according to how the machine is setup. As long as there is data entering the machine the output will change according to what was read in and how the configuration of the machine states what the output reports. A deterministic algorithm will always run the...

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Deque Automata for all classes of Formallanguages
B. Asha latha1
Department of computers
SRKIT Engineering
Vijayawada Andhra Pradesh (India)
T.Vishnupriya2
Department of Electronics
SRKIT Vijayawada, Andhra Pradesh (India)
N.Himabindu3
Department of computers
KBN College of Vijayawada, Andhra Pradesh (India)
Abstract: The purpose of computation involves solving problems by communicating them to a computational model by means of a suitablelanguage .A number of languages have been developed for this purpose. To recognize these languages some computational models has been developed and they are finite state machine, push down automata, queue automata and turing machines. But these machines are restricted to only one specific formallanguages like regular, context free ,etc. In this paper we proposed a machine called a Dequeue automaton that is capable of recognizing different classes of automata. We also shown that the simulation results from the Deque automata.
Keywords: Formallanguages, Finite automata, PDA ,TM.
I. Introduction
A finite automaton was the first abstract model as well as the mathematical model of digital computers. It is very powerful model of computation. It can recognize and accept regular languages. But finite...

... pages
I. Essay Writing _________________________________ 2
II Essay Bibliography _________________________________ 9
III Book Summary (Chapter 17) __________________________ 10
IV Book Summary (Chapter 10) __________________________ 13
V Book Summary _______________________________ 16
ESSAY
The environment necessary for the development of language can be enhanced to structure the child from birth to six years. How is language encouraged in the Montessori nursery class?
A Montessori nursery classroom is a prepared environment for children from the ages of 2 1/2 to 6 years of age. The classroom contains only materials that respond to the developmental stages of a child from 2 1/2 to 6 years of age. There are no toys as the Montessori materials fulfill the various needs of the children at all stages. Lessons/presentations are given only when the child is ready. Generally lessons are given individually, however some lessons are given in a small group setting. Before a lesson is given the child must have successfully completed any preparatory lessons. If the concepts...

...Alphabets, Strings and Languages
Example : Consider the string 011 over the binary alphabet. All the prefixes, suffixes and substrings of this string are listed below.
Prefixes: e, 0, 01, 011.
Suffixes: e, 1, 11, 011.
Substrings: e, 0, 1, 01, 11, 011.
Note that x is a prefix (suffix or substring) to x, for any string x and e is a prefix (suffix or substring) to any string.
A string x is a proper prefix (suffix) of string y if x is a prefix (suffix) of y and x y.
In the above example, all prefixes except 011 are proper prefixes.
Powers of Strings : For any string x and integer , we use to denote the string formed by sequentially concatenating n copies of x. We can also give an inductive definition of as follows:
= e, if n = 0 ; otherwise
Example : If x = 011, then = 011011011, = 011 and
Powers of Alphabets :
We write (for some integer k) to denote the set of strings of length k with symbols from . In other words,
= { w | w is a string over and | w | = k}. Hence, for any alphabet, denotes the set of all strings of length zero. That is, = { e }. For the binary alphabet { 0, 1 } we have the following.
The set of all strings over an alphabet is denoted by . That is,
The set contains all the strings that can be generated by iteratively concatenating symbols from any number of times.
Example : If = { a, b }, then = { e, a, b, aa, ab, ba, bb, aaa, aab, aba, abb, baa, …}.
Please note that if , then that is . It may...

...Language and Identity
First of all, Identity is the belief as “who we are and how we are” which we all have as individuals and it is based on many fundamental factors such as Ethnic group,
Racial, National, Gendered, Social Class, Language, Sexual and Religious. As it is clear that identity has many essentials and it is not made of just one fact but has something to do with many, I will focus on the relationship especially betweenlanguage and identity in this paper.
The concept of “identity” has talked a lot in many societies these days. It is mainly because world has been becoming more civilized and English is becoming an international language through entire societies dramatically. English has spread around the world and the fact that it is spoken and used to communicate by many non-native
English speakers is raising some controversial questions such as “Do you have to become a native English speaker in your mentality in order to speak a perfect English?”, “Are you a different person when you speak a different language?”, “Can you become a near native speaker without losing who you are when you migrate in other countries?” These questions are interesting to many people who speak more than one language other than their mother tongue because this identity crisis usually happens in second language learning. What I mean by “identity crisis” is that...

...Importance of language
The importance of language is essential to every aspect and interaction in our everyday lives. We use language to inform the people around us of what we feel, what we desire, and question/understand the world around us. We communicate effectively with our words, gestures, and tone of voice in a multitude of situation. Would you talk to a small child with the same words you would in a business meeting. Being able to communicate with each other, form bonds, teamwork, and it’s what separates humans from other animal species. Communication drives our lives and better ourselves.
Origins of why their are so many different languages as plagued scholars and linguistics for centuries and will continue to puzzle them far beyond our lifetimes to come. In most cultures have myths that there was a common language spoke among the people with a deity getting angry and confusing the people or separating them from each other/segmenting the people to create their own language. Prime examples of stories like this is the “Tower of Babel”, Hindu with the story of the “Knowledge Tree”, and even Native Americans believing in a “Great Deluge(Flood)” separating people and speech.
The importance of communication can be often overlooked. Even with the ability to communicate with each other. Misunderstandings happen. Remember, communication is a two way street that should be embraced and not...

...PROBLEMS OF RELIGIOUS LANGUAGE
HUME’S FORK David Hume divides knowledge into two classes: ‘relations of ideas’ (i.e. tautologies) and ‘matters of fact’ (i.e. empirical statements). His book concludes (on p.165) with the following paragraph: “When we run over libraries, persuaded of these principles, what havoc must we make? If we take in our hand any volume; of divinity or school metaphysics, for instance; let us ask, Does it contain any abstract reasoning concerning quantity of number? No. Does it contain any experimental reasoning concerning matter of fact or existence? No. Commit it then to the flames: for it can contain nothing but sophistry and illusion.” LOGICAL POSITIVISM Hume’s Fork was updated by modern logical positivists (such as A.J.Ayer, Antony Flew and Gilbert Ryle) who proposed the Verification Principle. This claims that sentences are only meaningful if they are tautologies (which are true because of the definitions of the terms involved, e.g. a square has four sides, six is bigger than four), or if they are in some way empirically verifiable (i.e. connected with actual experience, e.g. Harold lost at Hastings, electrons are both particles and waves). Any other statements will be meaningless, because their truth is not decided by either definitions or evidence. According to Ayer, this makes discussion about religion and morality meaningless. Religious statements like God is love are not false, they are incapable of being either true or false....