The concept of a matter object being annihilated when meeting its corresponding anti-matter object is investigated in the context of membrane systems, i.e., of (distributed) multiset rewriting systems applying rules in the maximally parallel way. Computational completeness can be obtained with using only non-cooperative rules besides these matter/anti-matter annihilation rules if these annihilation rules have priority over the other rules. Without this priority condition, in addition catalytic rules with one single catalyst are needed to get computational completeness. Even deterministic systems are obtained in the accepting case. Universal P systems with a rather small number of rules – 57 for computing systems, 59 for generating and 52 for accepting systems – can be constructed when using non-cooperative rules together with matter/anti-matter annihilation rules having weak priority. Allowing anti-matter objects as input and/or output, we even get a computationally complete computing model for computations on integer numbers. Interpreting sequences of symbols taken in from and/or sent out to the environment as strings, we get a model for computations on strings, which can even be interpreted as representations of elements of a group based on a computable finite presentation
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