We study reversibility and determinism aspects of sequential multiset processing systems, and the strong versions of these properties. Syntactic criteria are established for both strong determinism and for strong reversibility. It also shown that without control all four classes - deterministic, strongly deterministic, reversible, strongly reversible -are not universal, while allowing priorities or inhibitors the first and the third class become universal. Moreover, strongly deterministic multiset rewriting systems with priorities are also universal.
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