Understanding Automata Theory and Formal Languages

Nondeterministic Finite-State Machine

It is a set of states and sets of transitions from state to state that occur on input symbols taken from an alphabet.

Language

A language is a set of words (strings) of finite length formed from a finite alphabet.

Regular Expression

A way of representing regular languages using alphabet characters on which the language is defined.

Networking

This seems to be a misunderstanding or mislabeling in the original text. Based on the examples, it describes Concatenation.

Concatenation is the combination of characters or strings. For example:

• 'z' concatenated with '132' = "Z132"
• '456' concatenated with 'AB' = "456AB"
• 'z' concatenated with 'A' = "ZA"

Alphabet

An alphabet is an ordered set of symbols for a language. It is the group of graphics used to represent the language, serving as a communication system in an ordered, finite mathematical symbol set.

String

A string is a sequence of characters, a word, or a phrase. It is an ordered sequence of finite length, composed of elements belonging to a specific alphabet.

Kleene Star

If A is a language over an alphabet, the Kleene star is defined as A* = ∪_n=0^∞ Aⁿ. This means a character can appear in a string 0, 1, or infinitely many times.

Positive Closure

Defined as A⁺ = ∪_n=1^∞ Aⁿ. This indicates that the character must appear at least once. For example: "ho+la" could result in "hola", "hoola", "hooola", etc.

Finite Automata

A finite automaton is a mathematical model of a system with discrete inputs and outputs.

Formal Grammar Example (G)

L(G) = {x | x is (w and z)ⁿ for n from 1 to infinity}

• Transitions (→):
• X → Are
• Sequence → w y z
• Sequence → w y z sequence

G = (V, N, V₀, →)

• V = S ∪ N
• S = {w, z}
• N = {V₀, sequences}
• V₀ = V₀

Language Example (LG)

LG = {950, .85, 830.20, 0, .79, 1.5, ...}

Grammar definition for decimal numbers:

G = (V, N, V₀, →)

• V = S ∪ N
• S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
• N = {V₀, decimal, decimal fraction, unsigned integer, digit}
• V₀ = V₀
• Transitions (→):
• V₀ → decimal number
• Decimal number → unsigned decimal | unsigned decimal fraction
• Unsigned decimal → unsigned decimal fraction
• Unsigned decimal fraction → unsigned fraction . unsigned
• Unsigned fraction → unsigned
• Unsigned → digit unsigned
• Unsigned → digit
• Digit → {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}