# Computational complexity studies of synchronous boolean finite dynamical systems

Mitsunori Ogihara, Kei Uchizawa

Research output: Chapter in Book/Report/Conference proceedingConference contribution

3 Citations (Scopus)

### Abstract

The finite dynamical system is a system consisting of some finite number of objects that take upon a value from some domain as a state, in which after initialization the states of the objects are updated based upon the states of the other objects and themselves according to a certain update schedule. This paper studies the subclass of finite dynamical systems the synchronous boolean finite dynamical system (synchronous BFDS, for short), where the states are boolean and the state update takes place in discrete time and at the same on all objects. The present paper is concerned with some problems regarding the behavior of synchronous BFDS in which the state update functions (or the local state transition functions) are chosen from a predetermined finite basis of boolean functions B. Specifically the following three behaviors are studied: - Convergence. Does a system at hand converge on a given initial state configuration? - Path Intersection. Will a system starting in given two state configurations produce a common configuration? - Cycle Length. Since the state space is finite, every BFDS on a given initial state configuration either converges or enters a cycle having length greater than 1. If the latter is the case, what is the length of the loop? Or put more simply, for an integer t, is the length of loop greater than t? The paper studies these questions in terms of computational complexity (in the case of Cycle Length using the decision version of the problem) and shows the following: 1. The three problems are each PSPACE-complete if the boolean function basis contains NAND, NOR or both AND and OR. 2. The Convergence Problem is solvable in polynomial time if the set B is one of {AND}, {OR} and {XOR,NXOR}. 3. If the set B is chosen from the three sets as in the case of the Convergence Problem, the Path Intersection Problem is in UP, and the Cycle Length Problem is in UP∩coUP; thus, these are unlikely to be NP-hard.

Original language English (US) Theory and Applications of Models of Computation - 12th Annual Conference, TAMC 2015, Proceedings Springer Verlag 87-98 12 9076 9783319171418 10.1007/978-3-319-17142-5_9 Published - 2015 12th Annual Conference on Theory and Applications of Models of Computation, TAMC 2015 - Singapore, SingaporeDuration: May 18 2015 → May 20 2015

### Publication series

Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 9076 03029743 16113349

### Other

Other 12th Annual Conference on Theory and Applications of Models of Computation, TAMC 2015 Singapore Singapore 5/18/15 → 5/20/15

### Fingerprint

Computational complexity
Dynamical systems
Computational Complexity
Boolean functions
Dynamical system
Cycle Length
Configuration
Update
Boolean Functions
Polynomials
Intersection
Converge
NAND
Path
State Transition
Initialization
Polynomial time
State Space
Discrete-time
Schedule

### ASJC Scopus subject areas

• Computer Science(all)
• Theoretical Computer Science

### Cite this

Ogihara, M., & Uchizawa, K. (2015). Computational complexity studies of synchronous boolean finite dynamical systems. In Theory and Applications of Models of Computation - 12th Annual Conference, TAMC 2015, Proceedings (Vol. 9076, pp. 87-98). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9076). Springer Verlag. https://doi.org/10.1007/978-3-319-17142-5_9
Theory and Applications of Models of Computation - 12th Annual Conference, TAMC 2015, Proceedings. Vol. 9076 Springer Verlag, 2015. p. 87-98 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9076).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Ogihara, M & Uchizawa, K 2015, Computational complexity studies of synchronous boolean finite dynamical systems. in Theory and Applications of Models of Computation - 12th Annual Conference, TAMC 2015, Proceedings. vol. 9076, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 9076, Springer Verlag, pp. 87-98, 12th Annual Conference on Theory and Applications of Models of Computation, TAMC 2015, Singapore, Singapore, 5/18/15. https://doi.org/10.1007/978-3-319-17142-5_9
Ogihara M, Uchizawa K. Computational complexity studies of synchronous boolean finite dynamical systems. In Theory and Applications of Models of Computation - 12th Annual Conference, TAMC 2015, Proceedings. Vol. 9076. Springer Verlag. 2015. p. 87-98. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-319-17142-5_9
Ogihara, Mitsunori ; Uchizawa, Kei. / Computational complexity studies of synchronous boolean finite dynamical systems. Theory and Applications of Models of Computation - 12th Annual Conference, TAMC 2015, Proceedings. Vol. 9076 Springer Verlag, 2015. pp. 87-98 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
@inproceedings{ea41f38f5502408ab2915f7b1dc141bc,
title = "Computational complexity studies of synchronous boolean finite dynamical systems",
abstract = "The finite dynamical system is a system consisting of some finite number of objects that take upon a value from some domain as a state, in which after initialization the states of the objects are updated based upon the states of the other objects and themselves according to a certain update schedule. This paper studies the subclass of finite dynamical systems the synchronous boolean finite dynamical system (synchronous BFDS, for short), where the states are boolean and the state update takes place in discrete time and at the same on all objects. The present paper is concerned with some problems regarding the behavior of synchronous BFDS in which the state update functions (or the local state transition functions) are chosen from a predetermined finite basis of boolean functions B. Specifically the following three behaviors are studied: - Convergence. Does a system at hand converge on a given initial state configuration? - Path Intersection. Will a system starting in given two state configurations produce a common configuration? - Cycle Length. Since the state space is finite, every BFDS on a given initial state configuration either converges or enters a cycle having length greater than 1. If the latter is the case, what is the length of the loop? Or put more simply, for an integer t, is the length of loop greater than t? The paper studies these questions in terms of computational complexity (in the case of Cycle Length using the decision version of the problem) and shows the following: 1. The three problems are each PSPACE-complete if the boolean function basis contains NAND, NOR or both AND and OR. 2. The Convergence Problem is solvable in polynomial time if the set B is one of {AND}, {OR} and {XOR,NXOR}. 3. If the set B is chosen from the three sets as in the case of the Convergence Problem, the Path Intersection Problem is in UP, and the Cycle Length Problem is in UP∩coUP; thus, these are unlikely to be NP-hard.",
author = "Mitsunori Ogihara and Kei Uchizawa",
year = "2015",
doi = "10.1007/978-3-319-17142-5_9",
language = "English (US)",
isbn = "9783319171418",
volume = "9076",
series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
publisher = "Springer Verlag",
pages = "87--98",
booktitle = "Theory and Applications of Models of Computation - 12th Annual Conference, TAMC 2015, Proceedings",

}

TY - GEN

T1 - Computational complexity studies of synchronous boolean finite dynamical systems

AU - Ogihara, Mitsunori

AU - Uchizawa, Kei

PY - 2015

Y1 - 2015

N2 - The finite dynamical system is a system consisting of some finite number of objects that take upon a value from some domain as a state, in which after initialization the states of the objects are updated based upon the states of the other objects and themselves according to a certain update schedule. This paper studies the subclass of finite dynamical systems the synchronous boolean finite dynamical system (synchronous BFDS, for short), where the states are boolean and the state update takes place in discrete time and at the same on all objects. The present paper is concerned with some problems regarding the behavior of synchronous BFDS in which the state update functions (or the local state transition functions) are chosen from a predetermined finite basis of boolean functions B. Specifically the following three behaviors are studied: - Convergence. Does a system at hand converge on a given initial state configuration? - Path Intersection. Will a system starting in given two state configurations produce a common configuration? - Cycle Length. Since the state space is finite, every BFDS on a given initial state configuration either converges or enters a cycle having length greater than 1. If the latter is the case, what is the length of the loop? Or put more simply, for an integer t, is the length of loop greater than t? The paper studies these questions in terms of computational complexity (in the case of Cycle Length using the decision version of the problem) and shows the following: 1. The three problems are each PSPACE-complete if the boolean function basis contains NAND, NOR or both AND and OR. 2. The Convergence Problem is solvable in polynomial time if the set B is one of {AND}, {OR} and {XOR,NXOR}. 3. If the set B is chosen from the three sets as in the case of the Convergence Problem, the Path Intersection Problem is in UP, and the Cycle Length Problem is in UP∩coUP; thus, these are unlikely to be NP-hard.

AB - The finite dynamical system is a system consisting of some finite number of objects that take upon a value from some domain as a state, in which after initialization the states of the objects are updated based upon the states of the other objects and themselves according to a certain update schedule. This paper studies the subclass of finite dynamical systems the synchronous boolean finite dynamical system (synchronous BFDS, for short), where the states are boolean and the state update takes place in discrete time and at the same on all objects. The present paper is concerned with some problems regarding the behavior of synchronous BFDS in which the state update functions (or the local state transition functions) are chosen from a predetermined finite basis of boolean functions B. Specifically the following three behaviors are studied: - Convergence. Does a system at hand converge on a given initial state configuration? - Path Intersection. Will a system starting in given two state configurations produce a common configuration? - Cycle Length. Since the state space is finite, every BFDS on a given initial state configuration either converges or enters a cycle having length greater than 1. If the latter is the case, what is the length of the loop? Or put more simply, for an integer t, is the length of loop greater than t? The paper studies these questions in terms of computational complexity (in the case of Cycle Length using the decision version of the problem) and shows the following: 1. The three problems are each PSPACE-complete if the boolean function basis contains NAND, NOR or both AND and OR. 2. The Convergence Problem is solvable in polynomial time if the set B is one of {AND}, {OR} and {XOR,NXOR}. 3. If the set B is chosen from the three sets as in the case of the Convergence Problem, the Path Intersection Problem is in UP, and the Cycle Length Problem is in UP∩coUP; thus, these are unlikely to be NP-hard.

UR - http://www.scopus.com/inward/record.url?scp=84929620665&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84929620665&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-17142-5_9

DO - 10.1007/978-3-319-17142-5_9

M3 - Conference contribution

SN - 9783319171418

VL - 9076

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 87

EP - 98

BT - Theory and Applications of Models of Computation - 12th Annual Conference, TAMC 2015, Proceedings

PB - Springer Verlag

ER -