A solution to the only one object problem with dissolution rules

In the framework of membrane computing, (non-)uniform families of recognizer membrane systems are usually defined to solve abstract decision problems. In this sense, the use of finite resources for each member of the family makes the difference with respect to Turing machines solving these problems. While keeping the finite nature of these systems, it is interesting to know which type of problems can be solved by means of a single membrane system. For this purpose, the complexity class \(\textbf{PMC}^{1p}_{\mathcal {R}}\) ${\mathbf{PMC}}_{\mathcal{R}}^{1p}$ was defined as the class of problems that can be solved by means of a single membrane system in polynomial time. Due to the polynomial-time encoding of the input, at least all the problems from P can be solved with a trivial system. To go below P, the class \(\textbf{PMC}^{1f}_{\mathcal {R}}\) ${\mathbf{PMC}}_{\mathcal{R}}^{1f}$ restricts the definition of this encoding. In this work, we study the capability of different types of membrane systems to solve the ONLY-ONE-OBJECT problem, while having the encoding restriction.