Computational Complexity Studies of Synchronous Boolean Finite Dynamical Systems
Abstract

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 \(t\)?
 1.
The three problems are each \(\mathrm {PSPACE}\)complete if the boolean function basis contains \(\mathrm {NAND}\), \(\mathrm {NOR}\) or both \(\mathrm {AND}\) and \(\mathrm {OR}\).
 2.
The Convergence Problem is solvable in polynomial time if the set \(B\) is one of \(\{\mathrm {AND}\}\), \(\{\mathrm {OR}\}\) and \(\{\mathrm {XOR}, \mathrm {NXOR}\}\).
 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 \(\mathrm {UP}\cap \mathrm {coUP}\); thus, these are unlikely to be \(\mathrm {NP}\)hard.
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