Computational Complexity Studies of Synchronous Boolean Finite Dynamical Systems

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. DOI: 10.1007/978-3-319-17142-5_9

Abstract

A 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 a 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 . 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 $t$ ?