What is a regular language? - syntax

Question

I'm trying to understand the concept of languages levels (regular, context free, context sensitive, etc.).
I can look this up easily, but all explanations I find are a load of symbols and talk about sets. I have two questions:
Can you describe in words what a regular language is, and how the languages differ?
Where do people learn to understand this stuff? As I understand it, it is formal mathematics? I had a couple of courses at uni which used it and barely anyone understood it as the tutors just assumed we knew it. Where can I learn it and why are people "expected" to know it in so many sources? It's like there's a gap in education.
Here's an example:
Any language belonging to this set is a regular language over the alphabet.
How can a language be "over" anything?

Answer 1

In the context of computer science, a word is the concatenation of symbols. The used symbols are called the alphabet. For example, some words formed out of the alphabet {0,1,2,3,4,5,6,7,8,9} would be 1, 2, 12, 543, 1000, and 002.
A language is then a subset of all possible words. For example, we might want to define a language that captures all elite MI6 agents. Those all start with double-0, so words in the language would be 007, 001, 005, and 0012, but not 07 or 15. For simplicity's sake, we say a language is "over an alphabet" instead of "a subset of words formed by concatenation of symbols in an alphabet".
In computer science, we now want to classify languages. We call a language regular if it can be decided if a word is in the language with an algorithm/a machine with constant (finite) memory by examining all symbols in the word one after another. The language consisting just of the word 42 is regular, as you can decide whether a word is in it without requiring arbitrary amounts of memory; you just check whether the first symbol is 4, whether the second is 2, and whether any more numbers follow.
All languages with a finite number of words are regular, because we can (in theory) just build a control flow tree of constant size (you can visualize it as a bunch of nested if-statements that examine one digit after the other). For example, we can test whether a word is in the "prime numbers between 10 and 99" language with the following construct, requiring no memory except the one to encode at which code line we're currently at:
if word[0] == 1:
if word[1] == 1: # 11
return true # "accept" word, i.e. it's in the language
if word[1] == 3: # 13
return true
...
return false
Note that all finite languages are regular, but not all regular languages are finite; our double-0 language contains an infinite number of words (007, 008, but also 004242 and 0012345), but can be tested with constant memory: To test whether a word belongs in it, check whether the first symbol is 0, and whether the second symbol is 0. If that's the case, accept it. If the word is shorter than three or does not start with 00, it's not an MI6 code name.
Formally, the construct of a finite-state machine or a regular grammar is used to prove that a language is regular. These are similar to the if-statements above, but allow for arbitrarily long words. If there's a finite-state machine, there is also a regular grammar, and vice versa, so it's sufficient to show either. For example, the finite state machine for our double-0 language is:
start state: if input = 0 then goto state 2
start state: if input = 1 then fail
start state: if input = 2 then fail
...
state 2: if input = 0 then accept
state 2: if input != 0 then fail
accept: for any input, accept
The equivalent regular grammar is:
start → 0 B
B → 0 accept
accept → 0 accept
accept → 1 accept
...
The equivalent regular expression is:
00[0-9]*
Some languages are not regular. For example, the language of any number of 1, followed by the same number of 2 (often written as 1n2n, for an arbitrary n) is not regular - you need more than a constant amount of memory (= a constant number of states) to store the number of 1s to decide whether a word is in the language.
This should usually be explained in the theoretical computer science course. Luckily, Wikipedia explains both formal and regular languages quite nicely.

Answer 2

Here are some of the equivalent definitions from Wikipedia:
[...] a regular language is a formal language (i.e., a possibly
infinite set of finite sequences of symbols from a finite alphabet)
that satisfies the following equivalent properties:
it can be accepted by a deterministic finite state machine.
it can be accepted by a nondeterministic finite state machine
it can be described by a formal regular expression.
Note that the "regular expression" features provided with many programming languages
are augmented with features that make them capable of recognizing
languages which are not regular, and are therefore not strictly
equivalent to formal regular expressions.
The first thing to note is that a regular language is a formal language, with some restrictions. A formal language is essentially a (possibly infinite) collection of strings. For example, the formal language Java is the collection of all possible Java files, which is a subset of the collection of all possible text files.
One of the most important characteristics is that unlike the context-free languages, a regular language does not support arbitrary nesting/recursion, but you do have arbitrary repetition.
A language always has an underlying alphabet which is the set of allowed symbols. For example, the alphabet of a programming language would usually either be ASCII or Unicode, but in formal language theory it's also fine to talk about languages over other alphabets, for example the binary alphabet where the only allowed characters are 0 and 1.
In my university, we were taught some formal language theory in the Compilers class, but this is probably different between different schools.