Intrinsic Sequentiality in P: Causal Limits of Parallel Computation

TL;DR

This paper demonstrates that certain polynomial-time problems with intrinsic causal structures cannot be sped up using parallel computation due to fundamental causal constraints, establishing a limit on parallelism.

Contribution

It introduces a new class of problems with inherent causal sequentiality, proving they cannot be efficiently parallelized using classical circuit models.

Findings

Any causal execution requires linear time, ruling out asymptotic parallel speedup.

Classical NC circuits cannot implement the process within their depth constraints.

Identifies a gap between logical parallelism and causal executability.

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.