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We investigate the decidability of observational equivalence and approximation in “Syntactic Control of Interference” (SCI). By associating denotations of terms in an inequationally fully abstract model of finitary basic SCI with multitape finite state automata, we show that observational approximation is not decidable (even at first order), but that observational equivalence is decidable for all terms. We then consider the same problems for basic SCI extended with non-local control in the form of backwards jumps. We show that both observational approximation and observational equivalence are decidable in this language by describing a fully abstract games model in which strategies are regular languages.