Describing periodicity in two-way deterministic finite automata using transformation semigroups

Michal Kunc and Alexander Okhotin

Abstract. A framework for the study of periodic behaviour of two-way deterministic finite automata (2DFA) is developed. Computations of 2DFAs are represented by a two-way analogue of transformation semigroups, every element of which describes the behaviour of a 2DFA on a certain string x. A subsemigroup generated by this element represents the behaviour on strings in x+. The main contribution of this paper is a description of all such monogenic subsemigroups up to isomorphism. This characterization is then used to show that transforming an n-state 2DFA over a one-letter alphabet to an equivalent sweeping 2DFA requires exactly n + 1 states, and transforming it to a one-way automaton requires exactly max0 ≤ n G(n) + + 1 states, where G(k) is the maximum order of a permutation of k elements.

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