Basic Concepts and Terminologies

Some terminologies used in this course include the follws. For more, please see the textbook from page $14$ to page $29$.

Alphabet:

It is finite set of symbols, often denoted by $\Sigma$. For example, the alphabet of English is $a, b, c, \ldots, x, y, z$, and $A, B, C, \ldots, X, Y, Z$, punctuation, space and number digits $0, 1, \ldots, 8, 9$. For the sake of convenience, we often use $a-z$ to denote the alphabets $a, b, c, \ldots, x, y, z$. Similarly to use $A-Z$ to denote all capital English alphabets, use $0-9$ to denote all decimal digits. For computer, the alphabets are often binary $0, 1$.

String:

It is finite sequence of symbols taken from alphabet $\Sigma$. For example, in English: Hello world. A sequence of binary bits $011001$ is a string. Notice that the set of all strings is denoted by $\Sigma^*$. For example, $\{ 0,1 \}^* = \{ \epsilon,0,1,00,01,10,11,000,001,...\}$. It is countably infinite.

Language:

It is set of strings, i.e., subset of $\Sigma^*$. distinguishing strings into those in/out of the language

Machine:

It is formal abstraction of a computer based on states and transitions between states. It computes some output from input. Typically, there are three uses of machine, which can be related to each other.

acceptance:

input = string, output = yes/no membership in language

enumeration:

input = nothing, output = all strings in language

function:

input = string, output = string

Grammar:

The rules for deriving and parsing strings, akin to high school grammar rules for natural languages.

Expression:

A meta-string that "denotes" a language.