Deterministic catalytic systems are not universal

Abstract

We look at a 1-membrane catalytic P system with evolution rules of the form Ca → Cv or a → v , where C is a catalyst, a is a noncatalyst symbol, and v is a (possibly null) string representing a multiset of noncatalyst symbols. (Note that we are only interested in the multiplicities of the symbols.) A catalytic system (CS) can be regarded as a language acceptor in the following sense. Given an input alphabet Σ consisting of noncatalyst symbols, the system starts with an initial configuration wz, where w is a fixed string of catalysts and noncatalysts not containing any symbol in z, and z = a 1 n 1 ⋯ a k n k for some nonnegative integers n 1 , … , n k , with { a 1 , … , a k } ⊆ Σ . At each step, a maximal multiset of rules is nondeterministically selected and applied in parallel to the current configuration to derive the next configuration (note that the next configuration is not unique, in general). The string z is accepted if the system eventually halts. It is known that a 1-membrane CS is universal in the sense that any unary recursively enumerable language can be accepted by a 1-membrane CS (even by purely CSs, i.e., when all rules are of the form Ca → Cv ). A CS is said to be deterministic if at each step there is a unique maximally parallel multiset of rules applicable. It has been an open problem whether deterministic systems of this kind are universal. We answer this question negatively. We show that the membership problem for deterministic CSs is decidable. In fact, we show that the Parikh map of the language ( ⊆ a 1 * ⋯ a k * ) accepted by any deterministic CS is a simple semilinear set which can be effectively constructed. Since nondeterministic 1-membrane CS acceptors (with two catalysts) are universal, our result gives the first example of a variant of P systems for which the nondeterministic version is universal, but the deterministic version is not. We also show that for a deterministic 1-membrane CS using only rules of type Ca → Cv , the set of reachable configurations from a given initial configuration is an effective semilinear set. The application of rules of type a → v , however, is sufficient to render the reachability set nonsemilinear. Our results generalize to multimembrane deterministic CSs. We also consider deterministic CSs which allow rules to be prioritized and investigate three classes of such systems, depending on how priority in the application of the rules is interpreted. For these three prioritized systems, we obtain contrasting results: two are universal and one only accepts semilinear sets.