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On Affine Reachability Problems

Authors Stefan Jaax, Stefan Kiefer

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Author Details

Stefan Jaax
  • Technische Universität München, Germany
Stefan Kiefer
  • University of Oxford, UK

Funding

  • Jaax, Stefan: Supported by an ERC Advanced Grant (787367: PaVeS). This research was also funded in part by TUM IGSSE Grant 10.06 (PARSEC).
  • Kiefer, Stefan: Supported by a Royal Society University Research Fellowship.

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Stefan Jaax and Stefan Kiefer. On Affine Reachability Problems. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 48:1-48:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.MFCS.2020.48

Abstract

We analyze affine reachability problems in dimensions 1 and 2. We show that the reachability problem for 1-register machines over the integers with affine updates is PSPACE-hard, hence PSPACE-complete, strengthening a result by Finkel et al. that required polynomial updates. Building on recent results on two-dimensional integer matrices, we prove NP-completeness of the mortality problem for 2-dimensional integer matrices with determinants +1 and 0. Motivated by tight connections with 1-dimensional affine reachability problems without control states, we also study the complexity of a number of reachability problems in finitely generated semigroups of 2-dimensional upper-triangular integer matrices.

Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • Counter Machines
  • Matrix Semigroups
  • Reachability

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