inputtostate

Inputtostate refers to the mechanism by which a system moves from one state to another in response to an input. In the context of finite state machines, this is usually described by a transition function that takes the current state and an input symbol and yields the next state. The concept is central to how automata process sequences of inputs and produce outputs or accept strings.

Formally, a deterministic finite automaton (DFA) consists of a set of states Q, an input alphabet Σ, a start state, and a transition function δ that maps each pair (q, a) with q in Q and a in Σ to a next state in Q. This can be written δ: Q × Σ → Q. In a nondeterministic finite automaton (NFA), the transition function maps to a set of possible next states, δ: Q × Σ → P(Q). Some models allow partial transition functions, which can be completed by introducing a dead state for undefined inputs.

In addition to the state-to-state transition, many models specify output behavior. In Mealy machines, the output is determined by the current state and input (transition function together with an output relation), whereas in Moore machines the output depends only on the current state. Representations include transition tables and state diagrams, which encode the input-to-state relationships graphically.

Applications of input-to-state mappings include lexical analysis, protocol handling, control logic in digital circuits, and various modeling tasks in computer science and engineering. The term should not be confused with input-to-state stability in control theory, which describes how a system’s state remains bounded in response to external inputs; that is a separate concept from the state transition mechanism in automata.