We study a polynomial-time decision problem in which \emph{causal execution} is part of the instance specification. Each input describes a depth-$N$ process in which a single non-duplicable token must traverse an ordered sequence of steps, revealing only $O(1)$ bits of routing information at each step. The accept/reject outcome is defined solely by completion of this prescribed execution, rather than by order-free evaluation of the input. A deterministic Turing machine executes the process in $Θ(N)$ time. Using standard information-theoretic tools - specifically cut-set bounds for relay channels and Fano's inequality - we show that any execution respecting the causal constraints requires $Ω(N)$ units of causal time. Information about the delivery path can advance by at most one hop per unit of causal time, so the process admits no asymptotic parallel speedup. We further show that no classical $\mathbf{NC}$ circuit family can implement the process when circuit depth is interpreted as realizable parallel time. This identifies a class of polynomial-time problems with intrinsic causal structure and highlights a gap between logical parallelism and causal executability.

翻译：我们研究一个多项式时间判定问题，其中\emph{因果执行}是实例规约的一部分。每个输入描述了一个深度为$N$的过程，其中单个不可复制的令牌必须遍历一个有序的步骤序列，每一步仅揭示$O(1)$比特的路由信息。接受/拒绝结果完全由这一规定执行的完成情况定义，而非通过对输入进行无序评估。确定性图灵机以$Θ(N)$时间执行该过程。利用标准信息论工具——特别是中继信道的割集界和Fano不等式——我们证明任何遵循因果约束的执行都需要$Ω(N)$单位的因果时间。关于传递路径的信息每单位因果时间最多只能前进一跳，因此该过程不允许渐近并行加速。我们进一步证明，当电路深度被解释为可实现的并行时间时，没有经典的$\mathbf{NC}$电路族可以实现该过程。这识别了一类具有内在因果结构的多项式时间问题，并凸显了逻辑并行性与因果可执行性之间的差距。