Abstract
We study the reachability problems in various nondeterministic polynomial maps in \(\mathbb {Z}^n\). We prove that the reachability problem for very simple three-dimensional affine maps (with independent variables) is undecidable and is \(\texttt {PSPACE}\)-hard for two-dimensional quadratic maps. Then we show that the complexity of the reachability problem for maps without functions of the form \(\pm x+b\) is lower. In this case the reachability problem is \(\texttt {PSPACE}\)-complete in general, and \(\mathtt{NP}\)-hard for any fixed dimension. Finally we extend the model by considering maps as language acceptors and prove that the universality problem is undecidable for two-dimensional affine maps.
This work was supported by EPSRC grant “Reachability problems for words, matrices and maps” (EP/M00077X/1).
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Notes
- 1.
In [6], additive polynomials were called counter-like as they are similar to updates in counter machines and VASSs.
- 2.
Ackermann is a complexity class containing decision problems solvable in time bounded by Ackermann function, which is computable but not primitive-recursive.
- 3.
The automata of [9] were defined without final states but the weighted automaton with undecidable universality problem was constructed in such a way that a non-empty word can be accepted only in a single state.
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Ko, SK., Niskanen, R., Potapov, I. (2018). Reachability Problems in Nondeterministic Polynomial Maps on the Integers. In: Hoshi, M., Seki, S. (eds) Developments in Language Theory. DLT 2018. Lecture Notes in Computer Science(), vol 11088. Springer, Cham. https://doi.org/10.1007/978-3-319-98654-8_38
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