Abstract. Looking for a theory of communication complexity for P systems, we consider here so-called evolution-communication (EC for short) P systems, where objects evolve by multiset rewriting rules without target commands and pass through membranes by means of symport/antiport rules. (Actually, in most cases below we use only symport rules.) We first propose a way to measure the communication costs by means of “quanta of energy ” (produced by evolution rules and) consumed by communication rules. EC P systems with such costs are proved to be Turing complete in all three cases with respect to the relation between evolution and communication operations: priority of communication, mixing the rules without priority for any type, priority of evolution (with the cost of communication increasing in this ordering in the universality proofs). More appropriate measures of communication complexity are then defined, as dynamical parameters, counting the communication steps or the number (and the weight) of communication rules used during a computation. Such parameters can be used in three ways: as properties of P systems (considering the families of sets of numbers generated by systems with a given communication complexity), as conditions to be imposed on computations (accepting only those computations with a communication complexity bounded by a given threshold), and as standard complexity measures (defining the class of problems which can be solved by P systems with a bounded complexity). Because we ignore the evolution steps, in all three cases it makes sense to consider hierarchies starting with finite complexity thresholds. We only give some preliminary results about these hierarchies (for instance, proving that already their lower levels contain complex – e.g., non-semilinear – sets), and we leave open many problems and research issues. 1 1

Abstract. We introduce a new variant of membrane systems where the rules are directly assigned to membranes (and not to the regions as this is usually observed in the area of membrane systems) and, moreover, every membrane carries an energy value that can be changed during a computation by objects passing through the membrane. For the application of rules leading from one configuration of the system to the succeeding configuration we consider a sequential model and do not use the model of maximal parallelism. The result of a successful computation is considered to be the distribution of energy values carried by the membranes. We will show that for such systems using a kind of priority relation on the rules we already obtain universal computational power. When omitting the priority relation, we obtain a characterization of the family of Parikh sets generated by context-free matrix grammars (with λ−rules). 1

Energy–based P systems have been recently defined as P systems in which the amount of energy manipulated and/or consumed during computations is taken into account. In this paper we propose two quantum versions of energy–based P systems. Both versions are defined just like classical energy–based P systems, but for objects and rules. Objects are represented as pure states in the Hilbert space C d, whereas the definition of rules differs between the two models. In the former, rules are defined as bijective functions — implemented as unitary operators — which transform the objects from the alphabet. In the latter, rules are defined as generic functions which map the alphabet into itself. Such functions are implemented using a generalization of the Conditional Quantum Control technique, and may yield to non– unitary operators. Finally, we address some problems and outline some directions for future work. 1

P systems are considered in the dynamical perspective of biological and biochemical systems. In this sense, the focus of computational processes is in their behavioral patterns rather than in their final states encoding answers to initial inputs. The framework of "state transition dynamics" is outlined where general dynamical concepts are formulated in completely discrete terms. A metabolic algorithm is defined which computes the evolution of P systems when initial states and reaction parameters are given. This algorithm is applied to the analysis of important oscillatory phenomena of biological interest.

Summary. We propose three quantum algorithms to solve the 3-SAT NP-complete decision problem. The first algorithm builds, for any instance φ of 3-SAT, a quantum Fredkin circuit that computes a superposition of all classical evaluations of φ in a given output line. Similarly, the second and third algorithms compute the same superposition on a given register of a quantum register machine, and as the energy of a given membrane in a quantum P system, respectively. Assuming that a specific non–unitary operator, built using the well known creation and annihilation operators, can be realized as a quantum gate, as an instruction of the quantum register machine, and as a rule of the quantum P system, respectively, we show how to decide whether φ is a positive instance of 3-SAT. The construction relies also upon the assumption that an external observer is able to distinguish, as the result of a measurement, between a null and a non–null vector. 1

Summary. Energy plays an important role in many theoretical computational models. In this paper we review some results we have obtained in the last few years concerning the computational power of two variants of P systems that manipulate energy while performing their computations: energy-based and UREM P systems. In the former, a fixed amount of energy is associated to each object, and the rules transform objects by manipulating their energy. We show that if we assign local priorities to the rules, then energy–based P systems are as powerful as Turing machines, otherwise they can be simulated by vector addition systems and hence are not universal. We also discuss the simulation of conservative and reversible circuits of Fredkin gates by means of (self)– reversible energy–based P systems. On the other side, UREM P systems are membrane systems in which a given amount of energy is associated to each membrane. The rules transform and move single objects among the regions. When an object crosses a membrane, it may modify the associated energy value. Also in this case, we show that UREM P systems reach the power of Turing machines if we assign a sort of local priorities to the rules, whereas without priorities they characterize the class P sMAT λ, and hence are not universal. 1