TY - JOUR

T1 - Generalized predecessor existence problems for Boolean finite dynamical systems on directed graphs

AU - Kawachi, Akinori

AU - Ogihara, Mitsunori

AU - Uchizawa, Kei

N1 - Funding Information:
We thank the reviewers for their constructive criticisms and many valuable suggestions. This work was supported by JSPS KAKENHI Grant Numbers JP16H01705 , JP16K00006 , JP17H01695 , JP17K12640 , ELC project ( MEXT KAKENHI Grant Numbers JP24106009 , JP24106010 ), and National Science Foundation , NSF-SMA-1747631 .

PY - 2019/3/1

Y1 - 2019/3/1

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

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U2 - 10.1016/j.tcs.2018.08.026

DO - 10.1016/j.tcs.2018.08.026

M3 - Article

AN - SCOPUS:85052972332

VL - 762

SP - 25

EP - 40

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -