Intrinsic Sequentiality in P: Causal Limits of Parallel Computation

Abstract

We study a polynomial-time decision problem in which each input encodes a depth-$N$ causal execution in which a single non-duplicable token must traverse an ordered sequence of steps, revealing at most $O(1)$ bits of routing information at each step. The uncertainty in the problem lies in identifying the delivery path through the relay network rather than in the final accept/reject outcome, which is defined solely by completion of the prescribed execution. A deterministic Turing machine executes the process in $\Theta(N)$ time. Using information-theoretic tools - specifically cut-set bounds for relay channels and Fano's inequality - we prove that any execution respecting the causal constraints requires $\Omega(N)$ units of causal time, thereby ruling out 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.