### Abstract

A Boolean Finite Synchronous Dynamical System (BFDS, for short) consists of a finite number of objects that each maintains a boolean state, where after individually receiving state assignments, the objects update their state with respect to object-specific time-independent boolean functions synchronously in discrete time steps. The present paper studies the computational complexity of determining, given a boolean finite synchronous dynamical system, a configuration, which is a boolean vector representing the states of the objects, and a positive integer t, whether there exists another configuration from which the given configuration can be reached in t steps. It was previously shown that this problem, which we call the t-Predecessor Problem, is NP-complete even for t=1 if the update function of an object is either the conjunction of arbitrary fan-in or the disjunction of arbitrary fan-in. This paper studies the computational complexity of the t-Predecessor Problem for a variety of sets of permissible update functions as well as for polynomially bounded t. It also studies the t-Garden-Of-Eden Problem, a variant of the t-Predecessor Problem that asks whether a configuration has a t-predecessor, which itself has no predecessor. The paper obtains complexity theoretical characterizations of all but one of these problems.

Original language | English (US) |
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Journal | Theoretical Computer Science |

DOIs | |

State | Accepted/In press - Jan 1 2018 |

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### Keywords

- Computational complexity
- Dynamical systems
- Garden of Eden
- Predecessor

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Theoretical Computer Science*. https://doi.org/10.1016/j.tcs.2018.08.026