Exact Local Optimality Does Not Compose: Invariant Simplices and the Complexity of Chronological Stochastic Realization
Year: 2026
Type: preprint
Abstract: Probabilistic models constructed separately for different families of experiments are often expected to share one latent dynamical system. We formalize the resulting shared finite partial-realization problem as \emph{chronological stochastic realization}: deciding whether finite partial response specifications admit a \emph{chronological shared stochastic model} (CSSM) within a prescribed state budget. A CSSM is a normalized probabilistic automaton in which multiple named preparations share stochastic transitions and a single terminal effect. Local and tagged-static optimality do not compose at fixed dimension. For every $k,L\ge1$, we construct finite menus with local and tagged-static widths $k$, yet chronological width exactly $k(L+1)$, isolating compatibility with word composition as an independent state resource. In the ordinary-rank-tight calibrated regime, an exact $k$-state CSSM exists if and only if a $(k-1)$-simplex contains the root responses, lies in the response polytope, and is forward invariant under every primitive affine update. For finite rational menus, deciding whether an exact CSSM with at most $K$ states exists is $\exists\mathbb{R}$-complete. The promise problem distinguishing zero minimax defect from defect at least $ε$ is $\mathsf{PromiseNP}$-complete, with hardness already holding for inverse-polynomial $ε$. Both classifications hold over a fixed five-letter alphabet on instances where every individual menu and the tagged independent-query static relaxation have exact minimum state count $K$. They also hold with Boolean terminal effects, and the robust soundness gap remains valid over all effects in $[0,1]^d$. Thus chronological compatibility remains computationally hard even when the local and tagged-static optima are fixed at $K$.
Source: arXiv (Cornell University)
Author: Zhao, Yixin
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213593707
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