A Note on P Systems with Small-Size Insertion and Deletion ⋆ Artiom Alhazov 1,2 , Alexander Krassovitskiy 3 , Yurii Rogozhin 1,3 , Sergey Verlan 4 1 Institute of Mathematics and Computer Science Academy of Sciences of Moldova Academiei 5, Chi¸ sin˘au MD-2028 Moldova {artiom,rogozhin}@math.md 2 IEC, Department of Information Engineering, Graduate School of Engineering Hiroshima University, Higashi-Hiroshima 739-8527 Japan 3 Research Group on Mathematical Linguistics, Rovira i Virgili University Av. Catalunya, 35, Tarragona 43002 Spain alexander.krassovitskiy@estudiants.urv.cat 4 LACL, D´ epartement Informatique, Universit´ e Paris Est 61 av. G´ en´ eral de Gaulle, 94010 Cr´ eteil, France verlan@univ-paris12.fr Summary. We present an overview of recent results on small size insertion-deletion P systems. Together with the ordinary definition we consider systems with priority of deletion over insertion. In both cases, obtained P systems are strictly more powerful than ordinary insertion-deletion systems, and in most of the cases they are computationally complete. We list several such results. When using the priority relation, the computational completeness may be achieved by context-free insertion and deletion of one symbol only. 1 Introduction and definitions Insertion and the deletion operations originate from the language theory, being in- troduced mainly with linguistic motivation. In general form, an insertion operation means adding a substring to a given string in a specified (left and right) context, while a deletion operation means removing a substring of a given string from a specified (left and right) context. More precisely, an insertion rule (u, α, v) corre- ⋆ A.A. acknowledges the support of the Japan Society for the Promotion of Science, and the Grant-in-Aid, project 20·08364. A.A., Y.R. and S.V. acknowledge the support by the Science and Technology Center in Ukraine, project 4032. Y.R. acknowledges the support of the European Commission, project MolCIP, MIF1-CT-2006-021666. A.K. acknowledges the support PIF program of University Rovira i Virgili, and project no. MTM2007-63422 from the Ministry of Science and Education of Spain.