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 \({\Pi }^{p}_{2}\)-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.

在本文中，我们研究了在同步布尔有限动力系统上定义的许多计算问题的复杂性，其中更新函数是从异或及其否定的模板集中选择的。我们首先表明，如果对象执行排列，则可达性和路径交叉问题在对数空间均匀 AC ^{1中是可解的，而已知可达性问题在 P 中，路径交叉问题通常在 UP 中。}我们还探讨了在对象的子集上测试可达性或相交的情况，并表明这加剧了问题的复杂性：这两个问题都变成了 NP 完全问题，甚至 \({\Pi }^{p}_{ 2 }\)-如果我们进一步需要交叉点的普遍性，则完成。我们接下来考虑精确的循环长度问题，即确定是否存在在配置空间中产生具有恰好给定长度的循环的初始配置，并证明该问题是 NP 完全问题。最后，我们考虑t -predecessor 和t -Garden of Eden 问题，并证明即使t的值也作为实例的一部分以二进制给出，并且这两个问题都在对数空间中，这些问题在多项式时间内是可解的 - uniform NC ^2如果t的值作为实例的一部分以一元形式给出。