### 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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Title of host publication | 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017 |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Volume | 83 |

ISBN (Electronic) | 9783959770460 |

DOIs | |

State | Published - Nov 1 2017 |

Event | 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017 - Aalborg, Denmark Duration: Aug 21 2017 → Aug 25 2017 |

### Other

Other | 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017 |
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Country | Denmark |

City | Aalborg |

Period | 8/21/17 → 8/25/17 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Software

### Cite this

*42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017*(Vol. 83). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.MFCS.2017.8

**Generalized predecessor existence problems for boolean finite dynamical systems.** / Kawachi, Akinori; Ogihara, Mitsunori; Uchizawa, Kei.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017.*vol. 83, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017, Aalborg, Denmark, 8/21/17. https://doi.org/10.4230/LIPIcs.MFCS.2017.8

}

TY - GEN

T1 - Generalized predecessor existence problems for boolean finite dynamical systems

AU - Kawachi, Akinori

AU - Ogihara, Mitsunori

AU - Uchizawa, Kei

PY - 2017/11/1

Y1 - 2017/11/1

N2 - 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.

AB - 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.

KW - Computational complexity

KW - Dynamical systems

KW - Garden of Eden

KW - Predecessor

UR - http://www.scopus.com/inward/record.url?scp=85038443054&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85038443054&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.MFCS.2017.8

DO - 10.4230/LIPIcs.MFCS.2017.8

M3 - Conference contribution

VL - 83

BT - 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -