TY - GEN

T1 - Synchronous boolean finite dynamical systems on directed graphs over XOR functions

AU - Ogihara, Mitsunori

AU - Uchizawa, Kei

PY - 2020/8/1

Y1 - 2020/8/1

N2 - 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 Πp2-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.

AB - 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 Πp2-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.

KW - Computational complexity

KW - Dynamical systems

KW - Garden of Eden

KW - Predecessor

KW - Reachability

UR - http://www.scopus.com/inward/record.url?scp=85090505423&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85090505423&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.MFCS.2020.76

DO - 10.4230/LIPIcs.MFCS.2020.76

M3 - Conference contribution

AN - SCOPUS:85090505423

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020

A2 - Esparza, Javier

A2 - Kral�, Daniel

A2 - Kral�, Daniel

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

T2 - 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020

Y2 - 25 August 2020 through 26 August 2020

ER -