# Automata construction

In automata theory, automata construction is an important mathematical technique used to demonstrate the existence of an automaton with a certain desired property. Very often, it is presented as an algorithm that takes a desired property as input and produces as output an automaton with the property.

Many hard problems in automata theory involve finding the right construction of an automaton such that the problem can be answered. For example, the famous construction in McNaughton's Theorem answered the question if non-deterministic Büchi automaton can always be translated into a deterministic Muller automaton.

## Example

Powerset construction is an algorithm to construct a deterministic finite automaton from a given nondeterministic finite automaton.

## Optimality of a construction

An automata construction is called optimal if there is an input to the construction such that there exist no automaton that satisfy the desired property with smaller size complexity than output of the construction.