Local Optimality Does Not Compose: Invariant Simplices and the Complexity of Shared Probabilistic Automata

Abstract

Probabilistic models are often constructed separately for different test suites and then expected to share a single latent dynamical system. We study bounded-state shared realization for probabilistic automata from finite partial trace specifications. A shared realization uses one normalized automaton with named initial distributions, a common family of row-stochastic letter-transition matrices, and one final output vector. We compare it with separate local realizations and a labelled query-specific static relaxation. Local and labelled query-specific static optima need not compose. For every $k,L\ge 1$, we construct finite partial specifications for which each individual specification and the labelled query-specific static relaxation have minimum state count $k$, whereas every exact shared realization requires exactly $k(L+1)$ states. The factor $L+1$ reflects the prefix progress that must be tracked by a single shared transition family. In the linear-rank-tight calibrated regime, the existence of an exact $k$-state shared realization is equivalent to finding a $(k-1)$-simplex containing the root responses, contained in the response polytope, and forward-invariant under every single-letter affine update. For finite rational specifications, exact shared realization with at most $K$ states is $\exists \mathbb{R}$-complete, while distinguishing zero minimax defect from inverse-polynomial defect is $\mathsf{PromiseNP}$-complete. Both classifications hold over a fixed five-letter alphabet when the local and labelled query-specific static optima already equal $K$. They remain valid with Boolean final output vectors, and the robust gap persists over the full cube $[0,1{]}^d$. Thus the hardness arises from shared word composition.