Lower Bounds for Graph-Walking Automata

Authors Olga Martynova , Alexander Okhotin

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Olga Martynova
  • Department of Mathematics and Computer Science, St. Petersburg State University, Russia
Alexander Okhotin
  • Department of Mathematics and Computer Science, St. Petersburg State University, Russia

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Olga Martynova and Alexander Okhotin. Lower Bounds for Graph-Walking Automata. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 52:1-52:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Graph-walking automata (GWA) traverse graphs by moving between the nodes following the edges, using a finite-state control to decide where to go next. It is known that every GWA can be transformed to a GWA that halts on every input, to a GWA returning to the initial node in order to accept, as well as to a reversible GWA. This paper establishes lower bounds on the state blow-up of these transformations: it is shown that making an n-state GWA traversing k-ary graphs return to the initial node requires at least 2(n-1)(k-3) states in the worst case; the same lower bound holds for the transformation to halting automata. Automata satisfying both properties at once must have at least 4(n-1)(k-3) states. A reversible automaton must have at least 4(n-1)(k-3)-1 states. These bounds are asymptotically tight to the upper bounds proved using the methods from the literature.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Theory of computation → Models of computation
  • Finite automata
  • graph-walking automata
  • halting
  • reversibility


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