övergångsfunktionen

Övergångsfunktionen, often translated as "transition function" or "state transition function" in English, is a fundamental concept in the study of automata theory and discrete dynamical systems. It describes how a system moves from one state to another. In the context of a finite automaton, the transition function dictates the next state based on the current state and the input symbol received.

Formally, for a deterministic finite automaton (DFA), the transition function is typically represented as a function δ. This function takes two arguments: the current state and the input symbol. The output of the function is the next state the automaton will enter. Mathematically, it can be expressed as δ: Q × Σ → Q, where Q is the set of states and Σ is the input alphabet.

For a non-deterministic finite automaton (NFA), the transition function is similar but allows for multiple possible next states, or no next state at all, for a given state and input symbol. In this case, the function's output is a set of states, represented as δ: Q × Σ → P(Q), where P(Q) is the power set of Q. The concept of ε-transitions, which allow a state change without consuming an input symbol, also modifies the NFA transition function.

Beyond automata, the term övergångsfunktion can also refer to the way a system's state evolves over time in other discrete-time models. It's a key component for defining the behavior and predictability of computational and abstract systems.