Spiking neural P systems with extended rules: universality and languages

We consider spiking neural P systems with rules allowed to introduce zero, one, or more spikes at the same time. The motivation comes both from constructing small universal systems and from generating strings; previous results from these areas are briefly recalled. Then, the computing power of the obtained systems is investigated, when considering them as number generating and as language generating devices. In the first case, a simpler proof of universality is obtained, while in the latter case we find characterizations of finite and recursively enumerable languages (without using any squeezing mechanism, as it was necessary in the case of standard rules). The relationships with regular languages are also investigated.     

Unexpected universality results for three classes of P systems with symport/antiport

Symport and antiport are biological ways of transporting molecules through membranesin ``collaborating' pairs; in the case of symport the two molecules pass in the same direction, in the case of antiport the two molecules pass in opposite directions. Here we first survey the results about the computing power of membrane systems (P systems) using only symport/antiport rules (hence these systems compute by communication only), then we consider a recently introduced, way of defining the result of a computation in a membrane system: looking for the trace of certain objects in their movement through membranes. Rather unexpected, in this way we get characterizations of recursively enumerable languages by means of membrane systems with symport/antiport which work with multisets of objects (note the qualitative difference between the data structure used by computations – multisets: no ordering– and the data structure of the output – strings: linear ordering). A similar remark holds true for the case of analysing P systems, which work in an automata-like manner: the sequence of certain distinguished objects taken from the environment during acomputation is the string recognized by the computation. We also survey universality results from this area, with sketched proofs. Some open problems are also formulated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Further remarks on P systems with active membranes, separation, merging, and release rules

The P systems are a class of distributed parallel computing devices of a biochemical type. In this note, we show that by using membrane separation to obtain exponential workspace, SAT problem can be solved in linear time in a uniform and confluent way by active P systems without polarizations. This improves some results already obtained by A. Alhazov, T.-O. Isdorj. A universality result related to membrane separation is also obtained.Dedicated to Prof. Zhang Kemin for his 70th birthday     

About P systems with symport/antiport

It is proved that four membranes suffice to a variant of P systems with symport/antiport with maximal parallelism to generate all recursively enumerable sets of numbers. P systems with symport/antiport without maximal parallelism are also studied, considering two termination criteria.     

Spiking neural P systems: An improved normal form

Spiking neural P systems (in short, SN P systems) are computing devices based on the way the neurons communicate through electrical impulses (spikes). These systems involve various ingredients; among them, we mention forgetting rules and the delay in firing rules. However, it is known that the universality can be obtained without using these two features. In this paper we improve this result in two respects: (i) each neuron contains at most two rules (which is optimal for systems used in the generative mode), and (ii) the rules in the neurons using two rules have the same regular expression which controls their firing. This result answers a problem left open in the literature, and, in this context, an incompleteness in some previous proofs related to the elimination of forgetting rules is removed. Moreover, this result shows a somewhat surprising uniformity of the neurons in the SN P systems able to simulate Turing machines, which is both of a theoretical interest and it seems to correspond to a biological reality. When a bound is imposed on the number of spikes present in a neuron at any step of a computation (such SN P systems are called finite), two surprising results are obtained. First, a characterization of finite sets of numbers is obtained in the generative case (this contrasts the case of other classes of SN P systems, where characterizations of semilinear sets of numbers are obtained for finite SN P systems). Second, the accepting case is strictly more powerful than the generative one: all finite sets and also certain arithmetical progressions can be accepted. A precise characterization of the power of accepting finite SN P systems without forgetting rules and delay remains to be found.     

Membrane systems with carriers

A membrane system is a model of computation which is inspired by some basic features of biological membranes. In this paper we consider another biologically inspired notion, viz., the notion of a carrier (or vehicle), as, e.g., used in gene cloning. We investigate the power of membrane systems where the rules for the evolving of objects are replaced by the rules that carry objects (by vehicles) through membranes. It turns out that these systems (even with a small number of membranes, a small number of carriers, and a small number of passengers taken by carriers) are computationally universal.     

P systems without multiplicities of symbol-objects

In this paper we investigate P systems whose compartments contain sets of symbol-objects rather than multisets of objects, as it is common in membrane computing. If the number of membranes cannot grow, then in this framework we can characterize exactly the regular languages. If membrane creation or membrane division is allowed, then the Parikh sets of recursively enumerable languages can be generated. The last result also implies the universality of P systems with active membranes (with multisets of symbol-objects) without polarizations.     

PX systems = P systems + X machines

The paper is a survey of the main features of P systems, X machines and of a new computational device called PX system. The sequential and the parallel PX systems are presented. Results reflecting the computational power of these models and their effectiveness in solving NP-complete problems are briefly mentioned.     

Array-rewriting P systems

We consider array languages (sets of picturesconsisting of symbols placed in the lattice points of the 2D grid) and thepossibility to handle them with P systems. After proving binary normal formsfor array matrix grammars (which, even in the case when no appearance checking isused, are known to generate the array languages of arbitrary array grammars), weprove that the P systems with context-free rules (with three membranes and no control on the communication or the use of rules) are computationally universal, able togenerate all computable array languages. Some open problems are also formulated.     

P Systems with Mobile Membranes

P systems with active membranes are among the central ones in membrane computing, and they were shown to be both computationally universal (able to simulate Turing machines) and computationally efficient (able to solve hard problems in polynomial time). However, in all cases, these results were obtained by making use of several powerful features, such as membrane polarization, label changing, division of non-elementary membranes, priorities, or cooperative rules. This paper contributes to the research effort of introducing a class of P systems with active membranes having none of the features mentioned above, but still preserving the power and the efficiency. The additional feature we consider instead are the operations of endocytosis and exocytosis: moving a membrane inside a neighboring membrane, or outside the membrane where it is placed. We investigate the power and the efficiency of these systems (also using membrane division) by first proving that they can simulate (with a linear slowdown and without introducing non-determinism) rewriting P systems with 2-replication, for which the universality and the possibility of solving NP-complete problems in polynomial time are known. In this way, the universality and efficiency are also obtained for our systems. We also give a direct and simple proof for the universality result – without using division rules (the proof uses nine membranes, but we do not know whether this number can be decreased).     

P systems with energy accounting ∗

We consider P systems where each evolution rule “Produces” or “Consumes” some quantity of energy, in amounts which are expressed as integer numbers. In each moment and in each membrane the total energy involved in an evolution step should be positive, but if “Soo much” energy is present in a membrane, then the membrane will be destroyed (dissolved). We show that this feature is rather powerful. In the case of multisets of symbol-objects we find that systems with two membranes and arbitrary energy associated with rules, or with arbitrarily many membranes and a bounded energy associated with rules characterize the recursively enumerable sets of vectors of natural numbers (catalysts and priorities are used). In the case of string-objects we have only proved that the recursively enumerable languages can be generated by systems with arbitrarily many membranes and bounded energy; when bounding the number of membranes and leaving free the quantity of energy associated with each rule we have only generated all matrix languages. Several research topics are also pointed out.     

Computing using signals: from cells to P systems

In cell biology a fundamental topic is the study of how biological signals are managed by cells. Signals can arise from inside the cell or from the external environment and the correct answer to certain signals is essential for bacteria to survive in a certain environment. Starting from these biological motivations we consider a model of P systems where the computation is controlled by signals which move across the regions. In particular, we consider signals-based P systems where the symbol-objects cannot be moved and the evolution rules can be activated/inactivated using a finite number of signals (signal-promoters) moved across the membranes; differently from standard P systems using promoters, in our case signal-promoters cannot be created during the computation. After discussing the biological motivations we show how this model becomes universal when it uses one catalyst and a bounded number of signal-promoters. Also results concerning signals-based P systems using non cooperative rules together with several open problems are presented.     

Non-confluence in divisionless P systems with active membranes

We describe a solution to the SAT problem via non-confluent P systems with active membranes, without using membrane division rules. Furthermore, we provide an algorithm for simulating such devices on a nondeterministic Turing machine with a polynomial slowdown. Together, these results prove that the complexity class of problems solvable non-confluently and in polynomial time by this kind of P system is exactly the class .     

Minimizing evolution-communication P systems and automata

Evolution-communication P systems are a variant of P systems allowing both rewriting rules and symport/antiport rules, thus having separated the rewriting and the communication. The purpose of this paper is to solve an open problem stated in Reference,¹⁾ namely generating the family of Turing computable sets of vectors of natural numbers instead of the family of Turing computable sets of natural numbers. The same construction also reduces the 3-membrane non-cooperative case and the 2-membrane 1-catalyst case to the 2-membrane non-cooperative case. Also, EC P automata are introduced and it is proved that 2-membrane EC P automata with a promoter can accept all recursively enumerable languages. Finally, a definition of an extended system is given, and its universality is proved using the rules of more restricted types. Artiom Alhazov: He has graduated from Mathematics and Computer Science, State University of Moldova in 2001, and is currently a Ph.D. student in Chişinăm, Moldova, and Tarragona, Spain. He has won prizes at 3 National Olympiads in Informatics and in Mathematics (1995 and 1996), participated at 8th International Olympiad in Informatics (Veszprem, Hungary, 1996). He has experience in programming and teaching, and has published 18 papers, mostly in Membrane Computing. His interests are Origami, Mathematics, Programming, Theoretical Computer Science, Formal Linguistics and Biocomputing.     

P systems with active membranes and without polarizations

P systems with active membranes but without using electrical charges (polarizations) are shown to be complete for generating recursively enumerable string languages when working on string objects and using only rules with membrane transitions as well as rules with membrane dissolving and elementary membrane division, but also when using various other kinds of rules, even including a new type of rules allowing for membrane generation. In particular, allowing for changing membrane labels turns out to be a very powerful control feature.     

The power of communication: P systems with symport/antiport

In the attempt to have a framework where the computation is done by communication only, we consider the biological phenomenon of trans-membrane transport of couples of chemicals (one say symport when two chemicals pass together through a membrane, in the same direction, and antiport when two chemicals pass simultaneously through a membrane, in opposite directions). Surprisingly enough, membrane systems without changing (evolving) the used objects and with the communication based on rules of this type are computationally complete, and this result is achieved even for pairs of communicated objects (as encountered in biology). Five membranes are used; the number of membranes is reduced to two if more than two chemicals may collaborate when passing through membranes. Andrei Paun: He graduated the Faculty of Mathematics of Bucharest University in 1998, received his M.Sc. degree from The University of Western Ontario in 1999, and since then he is a PhD student in the Computer Science Department of University of Western Ontario, London, Canada (under the guidance of prof. Sheng Yu). The topic of his thesis is Molecular Computing (especially, DNA and Membrane Computing), but his research interests also include neural networks, implementing automata, combinatorics on words. Gheorghe Paun: (the proud father of two sons, including the first author of this paper) He is a member of the Romanian Academy, working as a senior researcher in the Institute of Mathematics of the Romanian Academy, Bucharest, and as a Ramon y Cajal researcher in Rovira i Virgili University of Tarragona, Spain. He is one of the most active authors in (the theory of) DNA Computing, (co)author of many papers in this area, (co)author and (co)editor of several books. In 1998 he has initiated the area of Membrane Computing. Other research interests: regulated rewriting, grammar systems, contextual grammars, combinatorics on words, computational linguistics.     

P transducers

We introduce in this paper four classes of P transducers: arbitrary, initial, isolated arbitrary, isolated and initial. The first two classes are universal, they can compute the same word functions as Turing machines, the latter two are incomparable with finite state sequential transducers, generalized or not. We study the effect of the composition, and show that iteration increases the power of these latter classes, also leading to a new characterization of recursively enumerable languages. The “Sevilla carpet” of a computation is defined for P transducers, giving a representation of the control part for these P transducers. Gabriel Ciobanu, Ph.D.: He has graduated from the Faculty of Mathematics, “A.I.Cuza” University of Iasi, and received his Ph.D. from the same university. He is a senior researcher at the Institute of Computer Science of the Romanian Academy. He has wide-ranging interests in computing including distributed systems and concurrency, computational methods in biology, membrane computing, and theory of programming (semantics, formal methods, logics, verification). He has published around 90 papers in computer science and mathematics, a book on programming semantics and a book on network programming. He is a co-editor of three volumes. He has visited various universities in Europe, Asia, and North America, giving lectures and invited talks. His webpage is http://www.info.uaic.ro/gabriel Gheorghe Păun, Ph.D.: He has graduated from the Faculty of Mathematics, University of Bucharest, in 1974 and received his Ph.D. from the same university in 1977. Curently he works as senior researcher in the Institute of Mathematics of the Romanian Academy, as well as a Ramon y Cajal researcher in Sevilla University, Spain. He has repeatedly visited numerous universities in Europe, Asia, and North America. His main research areas are formal language theory and its applications, computational linguistics, DNA computing, and membrane computing (a research area initiated by him). He has published over 400 research papers (collaborating with many researchers worldwide), has lectured at over 100 universities, and gave numerous invited talks at recognized international conferences. He has published 11 books in mathematics and computer science, has edited about 30 collective volumes, and also published many popular science books and books on recreational mathematics (games). He is on the editorial boards of fourteen international journals in mathematics, computer science, and linguistics, and was/is involved in the program/steering/organizing commitees for many recognized conferences and workshops. In 1997 he was elected a member of the Romanian Academy. Gheorghe Ştefănescu, Ph.D.: He received his B.Sc./M.Sc./Ph.D. degrees in Computer Science from the University of Bucharest. Currently, he is a Professor of Computer Science at the University of Bucharest and a Senior Fellow at the National University of Singapore. Previously, he was a researcher at the Institute of Mathematics of the Romanian Academy and has held visiting positions in The Netherlands, Germany, and Japan. His current research focuses on formal methods in computer science, particularly on process and network algebras, formal methods for interactive, real-time, and object-oriented systems. Some of his results may be found in his book on “Network Algebra,” Springer, 2000.     

On the power of elementary features in spiking neural P systems

Since their first publication in 2006, spiking neural (SN) P systems have already attracted the attention of a lot of researchers. This might be owing to the fact that this abstract computing device follows basic principles known from spiking neural nets, but its implementation is discrete, using membrane computing background. Among the elementary properties which confer SN P systems their computational power one can count the unbounded fan-in (indegree) and fan-out (outdegree) of each “neuron”, synchronicity of the whole system, the possibility of delaying and/or removing spikes in neurons, the capability of evaluating arbitrary regular expressions in neurons in constant time and some others. In this paper we focus on the power of these elementary features. Particularly, we study the power of the model when some of these features are disabled. Rather surprisingly, even very restricted SN P systems keep their universal computational power. Certain important questions regarding this topic still remain open.     

P systems with minimal parallelism

A current research topic in membrane computing is to find more realistic P systems from a biological point of view, and one target in this respect is to relax the condition of using the rules in a maximally parallel way. We contribute in this paper to this issue by considering the minimal parallelism of using the rules: if at least a rule from a set of rules associated with a membrane or a region can be used, then at least one rule from that membrane or region must be used, without any other restriction (e.g., more rules can be used, but we do not care how many). Weak as it might look, this minimal parallelism still leads to universality. We first prove this for the case of symport/antiport rules. The result is obtained both for generating and accepting P systems, in the latter case also for systems working deterministically. Then, we consider P systems with active membranes, and again the usual results are obtained: universality and the possibility to solve NP-complete problems in polynomial time (by trading space for time).     

On languages generated by asynchronous spiking neural P systems

In this paper, we investigate the languages generated by asynchronous spiking neural P systems. Characterizations of finite languages and recursively enumerable languages are obtained by asynchronous spiking neural P systems with extended rules. The relationships of the languages generated by asynchronous spiking neural P systems with regular and non-semilinear languages are also investigated.