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][2]
Formally, a deterministic finite automaton A 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. A is a permutation automaton if and only if, for every two distinct states qi and qj in Q and every input symbol x 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.
^McNaughton, Robert (August 1967), "The loop complexity of pure-group events", Information and Control, 11 (1–2): 167–176, doi:10.1016/S0019-9958(67)90481-0
^Thierrin, Gabriel (March 1968). "Permutation automata". Theory of Computing Systems. 2 (1): 83–90. doi:10.1007/BF01691347.
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