## Final Activity Report Summary - MOLCIP (Molecular computing and intractable problems)

This research project considered new computational possibilities provided by computing with biomolecules (biocomputing). Biocomputing (also known as molecular computing) is a new interdisciplinary area of science where computer science, chemistry, biology, and physics meet, that promises revolutionary changes in computations in the near future. It aims at developing new devices and computing systems to manipulate information operating at atomic or molecular scale on the basis of biological, chemical, electronic, photonic and/or mechanical principles. We considered both theoretical and practical aspects of some important models of biocomputing.

In particularly we considered networks of evolutionary processors (NEPs) and hybrid networks of evolutionary processors (HNEPs), P systems with active membranes, symport / antiport P systems, insertion-deletion systems and insertion-deletion P systems with small parameters. We considered the frontier of intractability (in our case undecidability) in problems for them, in order to better understand their computational possibilities (universality, efficiency, complexity and so on). Thus, we showed that every recursively enumerable language can be generated by a complete HNEP with seven nodes. So, we positively answer the question posed by E.Csuhaj-Varjú, C.Martín-Vide and V.Mitrana (2005) about the possibility to generate an arbitrary recursively enumerable language over an alphabet V with a complete HNEP of a size smaller than 27 +3*card(V). We also showed that the family of HNEPs with two nodes is not computationally complete.

Membrane computing is a formal framework of distributed computing and cellular computing. Due to massive parallelism and exponential space some intractable computational problems can be solved by P systems with active membranes in polynomial number of steps. We generalise this approach from decisional problems to the computational ones, by providing a solution of a #P-Complete problem, namely to compute the permanent of a binary matrix. We also considered application of P systems with active membranes in mathematical linguistics. We implemented the work with the prefix tree by P systems with strings and active membranes. We presented the algorithms of searching in a dictionary and updating it implemented as membrane systems. The systems are constructed as reusable modules, so they are suitable for using as sub-algorithms for solving more complicated problems.

At last we considered model of biocomputing based on insertion and deletion operation (InsDel systems). The insertion and the deletion operations originate from the language theory, where they were introduced mainly with linguistic motivation. In the last years, the study InsDel systems has received a new motivation from molecular computing, because, from the biological point of view, insertion-deletion operations correspond to mismatched annealing of DNA sequences. We considered some of these systems with small size of parameters (lengths of inserted and deleted words, and the left and the right contexts) in framework of P systems and obtained some unexpected results. Thus, we showed that P systems with context-free insertion and deletion rules of one symbol generate languages strictly included in MAT, however some non-context-free languages may be generated by these systems. If Parikh vectors are considered, then the corresponding family equals to PsMAT. When a priority of deletion over insertion is introduced, PsRE can be characterised, but in terms of language generation such systems cannot generate a lot of languages because there is no control on the position of an inserted symbol. If one-sided contextual insertion or deletion rules are used, then this can be controlled and all recursively enumerable languages can be generated. The same result holds if a context-free deletion of two symbols is allowed.

A lot of problems are opened and the investigation of InsDel systems is very interesting and promising task.

In particularly we considered networks of evolutionary processors (NEPs) and hybrid networks of evolutionary processors (HNEPs), P systems with active membranes, symport / antiport P systems, insertion-deletion systems and insertion-deletion P systems with small parameters. We considered the frontier of intractability (in our case undecidability) in problems for them, in order to better understand their computational possibilities (universality, efficiency, complexity and so on). Thus, we showed that every recursively enumerable language can be generated by a complete HNEP with seven nodes. So, we positively answer the question posed by E.Csuhaj-Varjú, C.Martín-Vide and V.Mitrana (2005) about the possibility to generate an arbitrary recursively enumerable language over an alphabet V with a complete HNEP of a size smaller than 27 +3*card(V). We also showed that the family of HNEPs with two nodes is not computationally complete.

Membrane computing is a formal framework of distributed computing and cellular computing. Due to massive parallelism and exponential space some intractable computational problems can be solved by P systems with active membranes in polynomial number of steps. We generalise this approach from decisional problems to the computational ones, by providing a solution of a #P-Complete problem, namely to compute the permanent of a binary matrix. We also considered application of P systems with active membranes in mathematical linguistics. We implemented the work with the prefix tree by P systems with strings and active membranes. We presented the algorithms of searching in a dictionary and updating it implemented as membrane systems. The systems are constructed as reusable modules, so they are suitable for using as sub-algorithms for solving more complicated problems.

At last we considered model of biocomputing based on insertion and deletion operation (InsDel systems). The insertion and the deletion operations originate from the language theory, where they were introduced mainly with linguistic motivation. In the last years, the study InsDel systems has received a new motivation from molecular computing, because, from the biological point of view, insertion-deletion operations correspond to mismatched annealing of DNA sequences. We considered some of these systems with small size of parameters (lengths of inserted and deleted words, and the left and the right contexts) in framework of P systems and obtained some unexpected results. Thus, we showed that P systems with context-free insertion and deletion rules of one symbol generate languages strictly included in MAT, however some non-context-free languages may be generated by these systems. If Parikh vectors are considered, then the corresponding family equals to PsMAT. When a priority of deletion over insertion is introduced, PsRE can be characterised, but in terms of language generation such systems cannot generate a lot of languages because there is no control on the position of an inserted symbol. If one-sided contextual insertion or deletion rules are used, then this can be controlled and all recursively enumerable languages can be generated. The same result holds if a context-free deletion of two symbols is allowed.

A lot of problems are opened and the investigation of InsDel systems is very interesting and promising task.