Asynchronous and Maximally Parallel Deterministic Controlled Non-Cooperative P Systems Characterize NFIN ∪ coNFIN Artiom Alhazov 1,2 and Rudolf Freund 3 1 Universit` a degli Studi di Milano-Bicocca Dipartimento di Informatica, Sistemistica e Comunicazione Viale Sarca 336, 20126 Milano, Italy artiom.alhazov@unimib.it 2 Institute of Mathematics and Computer Science Academy of Sciences of Moldova Academiei 5, Chi¸ sin˘au MD-2028 Moldova artiom@math.md 3 Faculty of Informatics, Vienna University of Technology Favoritenstr. 9, 1040 Vienna, Austria E-mail: rudi@emcc.at Summary. Membrane systems (with symbol objects) are distributed controlled multiset processing systems. Non-cooperative P systems with either promoters or inhibitors (of weight not restricted to one) are known to be computationally complete. In this paper we show that the power of the deterministic subclass of such systems is computationally complete in the sequential mode, but only subregular in the asynchronous mode and in the maximally parallel mode. 1 Introduction The most famous membrane computing model where determinism is a criterion of universality versus decidability is the model of catalytic P systems, see [2] and [4]. It is also known that non-cooperative rewriting P systems with either promoters or inhibitors are computationally complete, [1]. Moreover, the proof satisfies some additional properties: • Either promoters of weight 2 or inhibitors of weight 2 are enough. • The system is non-deterministic, but it restores the previous configuration if the guess is wrong, which leads to correct simulations with probability 1. The purpose of this paper is to formally prove that computational completeness cannot be achieved by deterministic systems when working in the asynchronous or in the maximally parallel mode.