Introduction to Automata

Introduction to Automata

• Automata is the plural of automaton, meaning "something that works automatically".
• Automata is used in the context of computing to describe a system that acts automatically.

Formal and Informal Languages

• Formal languages (also called syntactic languages) are defined with precise rules, like a programming language.
• Informal languages (also called semantic languages) rely on meaning and interpretation, like natural language.

Alphabets and Strings

• An alphabet (denoted by Σ) is a finite, non-empty set of symbols.
• A string is a finite sequence of symbols from an alphabet.
• The empty string, denoted by Λ, has a length of 0.

Valid & Invalid Alphabets

• An alphabet is valid if it is unambiguous and each letter can be uniquely tokenized (identified).
• This means that within any given alphabet, while it is permissible for a single letter to be composed of two or more individual symbols, it is crucial that no letter overlaps or begins with the same sequence of any other letter. This ensures clarity and prevents confusion during parsing or interpretation.

Length of Strings

• The length of a string (denoted by |s|) is the number of symbols in the string, including repetitions.

Reverse of a String

• The reverse of a string (denoted by Rev(s) or sr) is obtained by writing the letters in reverse order.

Defining Languages

• Languages can be defined in different ways, including:
  • Descriptive definition: Clearly describing the properties of the language's words.
  • Recursive definition: Using rules to build words from existing ones.
  • Regular expressions: Using pattern matching for defining the language.
  • Finite Automaton: Using a machine to recognize the language.

Examples of Languages

• EQUAL: Strings containing equal numbers of 'a's and 'b's (e.g., "ab", "aabb", "abab").
• EVEN-EVEN: Strings with even numbers of 'a's and 'b's (e.g., "aa", "bb", "aabb").
• INTEGER: Strings representing integers (e.g., "-2", "-1", "0", "1", "2", ...).
• EVEN: Strings representing even integers (e.g., "-4", "-2", "0", "2", "4", ...).
• {an bn}: Strings where the number of 'a's is equal to the number of 'b's (e.g., "ab", "aabb", "aaabbb").
• {an bn an}: Strings where 'a's appear at the start and end, with 'b's in the middle (e.g., "aba", "aabbaa", "aaabbbaaa").

Factorial, Double Factorial, and Square Languages

• factorial: Strings representing the factorial values (e.g., "1", "2", "6", "24", "120",...).
• FACTORIAL: Strings containing 'a' repeated n! times (e.g., "a", "aa", "aaaaaa", ...).
• DOUBLEFACTORIAL: Strings containing 'a' repeated n! times followed by 'b' repeated n! times (e.g., "ab", "aabb", "aaaaabbbbbb",...).
• SQUARE: Strings containing 'a' repeated n2 times (e.g., "a", "aaaa", "aaaaaaaa", ...).
• DOUBLESQUARE: Strings containing 'a' repeated n2 times followd by 'b' repeated n2 times (e.g., "ab", "aaaabbbb", "aaaaaaaabbbbbbbbb", ...).

PRIME Language

• PRIME: Strings containing 'a' repeated p times, where p is a prime number (e.g., "aa", "aaa", "aaaaa", "aaaaaaa", ...).

PALINDROME Language

• PALINDROME: Strings that are the same forward and backwards (e.g., "a", "aa", "aba", "bab", "bbb", ...).
• Remark: There are an equal number of palindromes with lengths 2n and 2n-1.

Counting Strings

• The total number of possible strings of length 'm' created from an alphabet of 'n' symbols is nm.
  • For example, there are 22 = 4 strings of length 2 from the alphabet {a, b}

Dive into the foundational concepts of automata, including the differences between formal and informal languages. Explore the components of alphabets and strings, and learn what constitutes valid alphabets in computing. This quiz will help solidify your understanding of automata in relation to computer science.