Understanding Strings, Languages, and Automata in Computing

Posted on Jul 10, 2025 in Commerce

Understanding Character Strings

A character string, word, or phrase (often simply called a string) is an ordered sequence of elements of arbitrary, though finite, length, belonging to a certain alphabet. Generally, a character string is a sequence of letters, numbers, or other symbols. In usual mathematics, the letters w, x, y, z are often used to refer to strings. From a programming standpoint, a string can consist of any finite combination of characters from the available character set.

Defining Alphabets

An alphabet is an ordered set of symbols used in a language. It is the group of graphemes, arranged in a particular order, used to represent a language that serves as a communication system. In mathematics, an alphabet is defined as a finite, ordered set of symbols.

Formal Languages and Their Specification

A formal language is a set of words (strings) of finite length, formed from a finite alphabet (the set of characters). The term ‘language’ is justified because these structures are formed according to specific rules (syntax) and have a semantic interpretation (meaning). Formal languages can be specified in a variety of ways, including:

  • Strings produced by a formal grammar
  • Strings produced by a regular expression
  • Strings accepted by an automaton, such as a Turing machine

What is a Regular Expression?

A regular expression, also called a pattern, is a way of representing regular languages. It is constructed using characters from an alphabet, specifically defining the language. Regular expressions are built using operations like Kleene closure.

Understanding Concatenation

Concatenation (or linking) is the operation by which two characters come together to form a string, or by which two strings, or a character and a string, are joined to form a larger string.

Kleene Closure (Star Closure)

Kleene Closure (or Kleene Star), denoted as A*, for a language A over an alphabet, is defined as follows: A* = ⋃n=0 An. This represents the set of all possible strings that can be formed by concatenating zero or more strings from A, including the empty string (ε).

Positive Closure

Positive Closure, denoted as A+, is defined as follows: A+ = ⋃n=1 An. This indicates that the character or string must appear at least once.

Finite State Machines (Automata)

A finite state machine, or finite automaton, is a mathematical model of a system with discrete inputs and outputs. The system can exist in any of a finite number of configurations or states. The current state of the system summarizes information regarding previous inputs and is crucial for determining the system’s behavior for subsequent inputs. A regular language is said to be accepted by a finite automaton.

Deterministic Finite Automaton (DFA)

A Deterministic Finite Automaton (DFA) consists of a set of states and a set of transitions from state to state that occur based on input symbols taken from an alphabet. For each input symbol, there is only one possible transition from each state. A DFA has an initial state and one or more acceptance (or final) states. A DFA is typically associated with a directed graph known as a transition diagram.

Formal Definition of a DFA:

AFD = (Q, Σ, δ, q0, F)

Where:

  • Q: A finite set of states
  • Σ: A finite input alphabet
  • δ: The transition function (or transition table)
  • q0: The initial state (an element of Q)
  • F: A set of final (or accepting) states (a subset of Q)

Applications of Finite Automata

Finite automata are positioned at the lowest level of the Chomsky hierarchy of machines and languages. One significant application of finite automata is in the construction of compilers. Compilers must recognize various layers, such as strings and symbols, within the source program, treating them as representations of individual objects. For example, they identify variable names, numeric constants, and keywords. This crucial pattern recognition task is handled by the compiler’s lexer (or lexical analyzer).