Monomial Dynamical Systems in # P-complete

Jang-Woo Park; Shuhong Gao

doi:10.15415/mjis.2012.11008

Authors

Jang-Woo Park School of Arts & Sciences, University of Houston - Victoria, Victoria, TX 77901, USA
Shuhong Gao Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975, USA

DOI:

https://doi.org/10.15415/mjis.2012.11008

Keywords:

preperiodic, monomial dynamics, directed path

Abstract

In this paper, we study boolean monomial dynamical systems. Colón-Reyes, Jarrah, Laubenbacher, and Sturmfels(2006) studied fixed point structure of boolean monomial dynamical systems of f by associating the dynamical systems of f with its dependency graph χ_f and Jarrah, Laubenbacher, and Veliz-Cuba(2010) extended it and presented lower and upper bound for the number of cycles of a given length for general boolean monomial dynamics. But, it is even difficult to determine the exact number of fixed points of boolean monomial dynamics. We show that the problem of counting fixed points of a boolean monomial dynamical systems is #P-complete, for which no efficient algorithm is known. This is proved by a 1-1 correspondence between fixed points of f sand antichains of the poset of strongly connected components of χf..

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