Finite-state machines are crucial in modeling computation with limited memory. They use states and transitions to process input, making them perfect for tasks like vending machines, traffic lights, and pattern matching in text editors.

There are two main types: deterministic (DFA) and nondeterministic (NFA) finite automata. DFAs have predictable behavior, while NFAs can be in multiple states at once. Both can be represented using state diagrams or transition tables.

Finite-State Automata Fundamentals

Components and Structure of FSAs

• Finite-state automaton (FSA) models computation with a finite number of states and transitions
• States represent distinct conditions or situations within the system
• Transitions define rules for moving between states based on input
• State diagram visually represents FSA using circles for states and arrows for transitions
• Transition table provides alternative representation showing states, inputs, and resulting states

Practical Applications and Examples

• FSAs model vending machines tracking inserted coins and selected items
• Traffic light systems use FSAs to manage light sequences and timing
• Text editors employ FSAs for find-and-replace functionality
• Regular expression engines utilize FSAs for pattern matching in strings

Types of Finite-State Automata

Deterministic Finite Automaton (DFA)

• DFA has exactly one transition for each input symbol from every state
• Deterministic nature ensures predictable behavior for given input
• DFA can be in only one state at a time during computation
• Formal definition includes set of states, alphabet, transition function, start state, and accept states
• DFA for recognizing binary strings ending in "01" consists of three states and transitions for "0" and "1" inputs

Nondeterministic Finite Automaton (NFA)

• NFA allows multiple transitions for the same input symbol from a state
• Epsilon transitions permit state changes without consuming input
• NFA can be in multiple states simultaneously during computation
• NFAs offer more compact representations for certain languages compared to DFAs
• NFA for recognizing strings containing "ab" uses epsilon transitions to simplify the state diagram

Regular Expressions and Language Recognition

Regular Expressions and Their Components

• Regular expression provides concise way to describe patterns in strings
• Basic building blocks include individual characters, concatenation, alternation, and repetition
• Kleene star () represents zero or more occurrences of preceding element
• Plus (+) denotes one or more occurrences of preceding element
• Question mark (?) indicates zero or one occurrence of preceding element
• Regular expression (a|b)abb matches strings of a's and b's ending with "abb"

Language Recognition and Automata Equivalence

• Language recognition determines whether a given string belongs to a specified language
• FSAs and regular expressions have equivalent expressive power for defining regular languages
• Every regular expression can be converted to an equivalent NFA
• NFAs can be transformed into equivalent DFAs through subset construction algorithm
• Minimization algorithms reduce DFAs to their smallest equivalent form
• Pumping lemma provides method for proving certain languages are not regular