Boolean networks are widely used to model biological regulatory networks and study their dynamics. Classical semantics, such as the asynchronous semantics, do not always accurately capture transient or asymptotic behaviors observed in quantitative models. To address this limitation, the Most Permissive semantics was introduced by Paulevé et al., extending Boolean dynamics with intermediate activity levels that allow components to transiently activate or inhibit their targets during transitions. In this work, we provide a Boolean encoding of the Most Permissive semantics: each component of the original network is represented by a triplet of Boolean variables, and we derive the extended logical function governing the resulting network. We prove that the asynchronous dynamics of the encoded network exactly reproduces the attainability properties of the original network under Most Permissive semantics. This encoding is implemented as a modifier within the bioLQM framework, making it directly compatible with existing tools such as GINsim. To address scalability limitations, we further extend the tool to support partial unfolding, restricted to a user-defined subset of components.

翻译：布尔网络被广泛用于对生物调控网络进行建模并研究其动力学行为。经典语义（如异步语义）并不总能准确捕捉定量模型中所观察到的瞬态或渐近行为。为解决这一局限性，Paulevé 等人提出了最容许语义，通过引入中间活动水平扩展布尔动力学，使得组件在状态转换过程中能够暂时激活或抑制其靶标。在本工作中，我们给出了最容许语义的一种布尔编码：原始网络中的每个组件由一个布尔变量三元组表示，并推导出控制结果网络的扩展逻辑函数。我们证明，编码网络的异步动力学完全再现了原始网络在最容许语义下的可达性性质。该编码已作为修饰器实现在 bioLQM 框架中，从而可直接与 GINsim 等现有工具兼容。为应对可扩展性限制，我们进一步扩展了该工具以支持部分展开，该展开可限制于用户定义的组件子集。