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Synchronous Boolean Finite Dynamical Systems on Directed Graphs over XOR Functions

Authors Mitsunori Ogihara, Kei Uchizawa

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Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
Keywords
  • Computational complexity
  • dynamical systems
  • Garden of Eden
  • predecessor
  • reachability

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Abstract

In this paper, we investigate the complexity of a number of computational problems defined on a synchronous boolean finite dynamical system, where update functions are chosen from a template set of exclusive-or and its negation. We first show that the reachability and path-intersection problems are solvable in logarithmic space-uniform AC¹ if the objects execute permutations, while the reachability problem is known to be in P and the path-intersection problem to be in UP in general. We also explore the case where the reachability or intersection are tested on a subset of objects, and show that this hardens complexity of the problems: both problems become NP-complete, and even Π^p₂-complete if we further require universality of the intersection. We next consider the exact cycle length problem, that is, determining whether there exists an initial configuration that yields a cycle in the configuration space having exactly a given length, and show that this problem is NP-complete. Lastly, we consider the t-predecessor and t-Garden of Eden problem, and prove that these are solvable in polynomial time even if the value of t is also given in binary as part of instance, and the two problems are in logarithmic space-uniform NC² if the value of t is given in unary as part of instance.

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Mitsunori Ogihara and Kei Uchizawa. Synchronous Boolean Finite Dynamical Systems on Directed Graphs over XOR Functions. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 76:1-76:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.MFCS.2020.76

Author Details

Mitsunori Ogihara
  • Department of Computer Science, University of Miami, FL, USA
Kei Uchizawa
  • Graduate School of Science and Engineering, Yamagata University, Japan

Funding

This work was supported by JSPS KAKENHI Grant Numbers JP19K11817.

Acknowledgements

We thank the reviewers for their constructive criticisms and many valuable suggestions.

References

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