A Boolean network (BN) is a discrete dynamical system defined by a Boolean function that maps to the domain itself. A trap space of a BN is a generalization of a fixed point, which is defined as the sub-hypercubes closed by the function of the BN. A trap space is minimal if it does not contain any smaller trap space. Minimal trap spaces have applications for the analysis of attractors of BNs with various update modes. This paper establishes the computational complexity results of three decision problems related to minimal trap spaces: the decision of the trap space property of a sub-hypercube, the decision of its minimality, and the decision of the membership of a given configuration to a minimal trap space. Under several cases on Boolean function representations, we investigate the computational complexity of each problem. In the general case, we demonstrate that the trap space property is coNP-complete, and the minimality and the membership properties are $\Pi_2^{\text P}$-complete. The complexities drop by one level in the polynomial hierarchy whenever the local functions of the BN are either unate, or are represented using truth-tables, binary decision diagrams, or double DNFs (Petri net encoding): the trap space property can be decided in a polynomial time, whereas deciding the minimality and the membership are coNP- complete. When the BN is given as its functional graph, all these problems are in P.

翻译：布尔网络（BN）是由映射到自身域上的布尔函数定义的离散动态系统。BN的陷阱空间是不动点的一种推广，定义为在BN函数作用下封闭的子超立方体。若陷阱空间不包含任何更小的陷阱空间，则称之为最小陷阱空间。最小陷阱空间在分析具有不同更新模式的BN吸引子中具有应用。本文建立了与最小陷阱空间相关的三个判定问题的计算复杂度结果：子超立方体的陷阱空间性质判定、其最小性判定，以及给定配置是否属于最小陷阱空间的隶属度判定。针对布尔函数表示的多种情形，我们研究了每个问题的计算复杂度。一般情况下，我们证明陷阱空间性质是coNP完全的，而最小性和隶属度性质是$\Pi_2^{\text P}$完全的。当BN的局部函数是单值函数，或使用真值表、二元决策图或双重DNF（Petri网编码）表示时，复杂度在多项式层级中降低一级：陷阱空间性质可在多项式时间内判定，而最小性和隶属度判定是coNP完全的。当BN以功能图形式给出时，所有问题均属于P类。