Assembling molecules in ATOMIX is hard

  • Authors:

  • Markus Holzer;Stefan Schwoon

  • Affiliations:

  • Institut für Informatik, Technische Universität München, Arcisstraße 21, München D-80290, Germany;Institut für Formele Methoden der Informatik, Universität Stuttgart, Universität sstr. 38, 70569 Stuttgart, Germany

  • Venue:
  • Theoretical Computer Science - Algorithmic combinatorial game theory
  • Year:
  • 2004

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Abstract

It is shown that assembling molecules in the ATOMIX game can be used to simulate finite automata. In particular, an instance of ATOMIX is constructed that has a solution if and only if the non-emptiness intersection problem for finite automata is solvable. This shows that the game under consideration is PSPACE-complete, improving a recent result of Hüffner et al. (Lecture Notes in Computer Science, Vol. 2174, Springer, Vienna, Austria, 2001, pp. 229-243). Moreover, the given reduction shows that there are ATOMIX games which have exponentially long optimal solutions. We also give an easy construction of ATOMIX game levels whose optimal solutions meet the worst case.