Stochastic Languages at Sub-stochastic Cost

When does a deterministic computational model define a probability distribution? What are its properties? This work formalises and settles this stochasticity problem for weighted automata, and its generalisation cost register automata (CRA). We show that checking stochasticity is undecidable for CRAs in general. This motivates the study of the fully linear fragment, where a complete and tractable theory is established. For this class, stochasticity becomes decidable in polynomial time via spectral methods, and every stochastic linear CRA admits an equivalent model with locally sub-stochastic update functions. This provides a local syntactic characterisation of the semantics of the quantitative model. This local characterisation allows us to provide an algebraic Kleene-Schutzenberger characterisation for stochastic languages. The class of rational stochastic languages is the smallest class containing finite support distributions, which is closed under convex combination, Cauchy product, and discounted Kleene star. We also introduce Stochastic Regular Expressions as a complete and composable grammar for this class. Our framework provides the foundations for a formal theory of probabilistic computation, with immediate consequences for approximation, sampling, and distribution testing.