A sofic measure is the stationary law of a finite-alphabet stochastic process obtained from a stationary finite-state Markov chain by a finite block map. Such a map determines each output symbol from a fixed finite window of input symbols. Sofic measures are also called hidden Markov measures (Béal et al. 2026).
Equivalently, the probability of every finite word 
has a representation 
by nonnegative finite matrices
and vectors, where 
is the ordered product of the matrices
assigned to its letters. A sofic measure need not have finite Markov
order.
If the Markov order is finite and the least dimension of a linear representation is 
, then its order is at most 
. This bound is sharp even
on an alphabet with three symbols (Béal et
al. 2026). Béal et al. (2026) credit Gemini with an extremal-set-theoretic
approach and GPT-6 Astra with the exterior-algebra argument.
