In automata theory, a permutation automaton, or pure-group automaton, is a deterministic finite automaton such that each input symbol permutes the set of states.[1]
Formally, a deterministic finite automaton may be defined by the tuple (Q, Σ, δ, q0, F),where Q is the set of states of the automaton, Σ is the set of input symbols, δ is the transition function that takes a state q and an input symbol x to a new state δ(q,x), q0 is the initial state of the automaton, and F is the set of accepting states (also: final states) of the automaton. is a permutation automaton if and only if, for every two distinct states and in Q and every input symbol in Σ, δ(qi,x) ≠ δ(qj,x).
A formal language is p-regular (also: a pure-group language) if it is accepted by a permutation automaton. For example, the set of strings of even length forms a p-regular language: it may be accepted by a permutation automaton with two states in which every transition replaces one state by the other.
The pure-group languages were the first interesting family of regular languages for which the star height problem was proved to be computable.[2]
Another mathematical problem on regular languages is the separating words problem, which asks for the size of a smallest deterministic finite automaton that distinguishes between two given words of length at most n - by accepting one word and rejecting the other. The known upper bound in the general case is
O(n2/5(logn)3/5)
O(n1/2)