Pseudo-inversion on Formal Languages

We consider the pseudo-inversion operation inspired by a biological event as a result of the partial inversion. We define the pseudoinversion of a string w = uxv to consist of all strings vxu, where uv = λ and consider the operation from a formal language theoretic viewpoint. We show that regular languages are closed under the pseudoinversion operation whereas context-free languages are not. Furthermore, we consider the iterated pseudo-inversion operation and establish the basic properties. Finally, we introduce the pseudo-inversion-freeness and examine closure properties and decidability problems for regular and context-free languages. We establish that pseudo-inversion-freeness is decidable in polynomial time for regular languages and undecidable for context-free languages.