Nextstate

Nextstate, in the context of automata theory and state machines, refers to the state that results from applying the transition function to the current state and an input symbol. It is a fundamental concept used to model how a system changes over time in response to events or inputs.

In a deterministic finite automaton (DFA), the next-state function, often denoted as δ, maps a pair consisting of a current state and an input symbol to a single next state: δ: Q × Σ → Q. For a nondeterministic finite automaton (NFA), δ maps to a set of possible next states: δ: Q × Σ → P(Q). Some machines also pair transitions with outputs, such as Mealy machines, where the next state and output are determined by the current state and input.

The next-state function underpins how state machines are specified and analyzed. It enables the description of control flow in software, hardware design, protocol verification, and lexical analysis in compilers. A complete specification of a state machine includes the finite set of states, the input alphabet, the transition function, the start state, and the set of accepting or final states (if recognition is the goal).

A simple example: a DFA with two states, A and B, and an alphabet {0, 1}. Let the start state be A and the accepting state be B. Define transitions δ(A,0) = A, δ(A,1) = B, δ(B,0) = A, δ(B,1) = B. Reading a sequence of inputs determines a unique path through states, ending in the last state defined by the next-state function. This mechanism extends to more complex systems, including synchronous circuits, protocol state machines, and event-driven software workflows.

See also automata theory, finite automaton, transition function, Mealy machine, Moore machine.