Abstract
A finitary automaton group is a group generated by an invertible, deterministic finite-state letter-to-letter transducer whose only cycles are self-loops at an identity state. We show that, for this presentation of finite groups, the uniform word problem is coNP-complete. Here, the input consists of a finitary automaton together with a finite state sequence and the question is whether the sequence acts trivially on all input words. Additionally, we also show that the respective compressed word problem, where the state sequence is given as a straight-line program, is PSpace-complete. In both cases, we give a direct reduction from the satisfiablity problem for (quantified) boolean formulae.
J. Ph. Wächter: The second author is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 492814705.
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Notes
- 1.
This makes sense as we will later on define a left action of the states on the words.
- 2.
Note that any complete finite automaton must contain a cycle and, therefore, every finitary \(\mathscr {G}\)-automaton has an identity state.
- 3.
The fact can be proved using a simple induction on the structure of the balanced iterated commutators, see [23, Fact 4].
- 4.
Such an element exists since there are two five-cycles in \(A_5\) whose commutator is again a five-cycle and since five-cycles are always conjugate (see [1, Lemma 1 and 3]).
- 5.
This is a well-known classical NP-complete problem, see e. g. [18, Problem 9.5.5].
- 6.
Note that the right-most letter here corresponds to the first variable \(x_1\). We could have done this the other way round as well but it turns out that this numbering has some technical advantages.
- 7.
A formula is closed if it does not have any free variables, i. e. if all appearing variables are bound by a quantifier.
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Acknowledgements
The authors would like to thank Armin Weiß for many discussions around the presented topic. The results are part of the first author’s Bachelor thesis, which was advised by the second author (while he was at FMI).
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Kotowsky, M., Wächter, J.P. (2023). The Word Problem for Finitary Automaton Groups. In: Bordihn, H., Tran, N., Vaszil, G. (eds) Descriptional Complexity of Formal Systems. DCFS 2023. Lecture Notes in Computer Science, vol 13918. Springer, Cham. https://doi.org/10.1007/978-3-031-34326-1_7
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