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 \(\max_{0 \leqslant \ell \leqslant n} G(n-\ell)+\ell+1\) states, where G(k) is the maximum order of a permutation of k elements.
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Kunc, M., Okhotin, A. (2011). Describing Periodicity in Two-Way Deterministic Finite Automata Using Transformation Semigroups. In: Mauri, G., Leporati, A. (eds) Developments in Language Theory. DLT 2011. Lecture Notes in Computer Science, vol 6795. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22321-1_28
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DOI: https://doi.org/10.1007/978-3-642-22321-1_28
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