Abstract In this paper, we present an encoding of the $$\lambda $$ λ -calculus in a multiset rewriting system and provide a few applications of the construction. For this purpose, we choose the calculus named String MultiSet Rewriting, which was introduced in Barbuti et al. (Electron Notes Theor Comput Sci 194:19–34, 2008) by Barbuti et al. With the help of our encoding, we give alternative proofs for the standardization and the finiteness of developments theorems in the $$\lambda $$ λ -calculus.

Abstract In this paper, it is investigated how different features of membrane systems can be simulated by the boundary rule application. Firstly, it is discussed how the effect of maximally parallel mode can be obtained by non-cooperative boundary rules applied only in sequential mode, then it is also demonstrated how membrane dissolution, or the application of using promoters and inhibitors can be simulated.

Abstract In this paper, in connection with the program of extending the Curry–Howard isomorphism to classical logic, we study the $\lambda \mu$ -calculus of Parigot emphasizing the difference between the original version of Parigot and the version of de Groote in terms of normalization properties. In order to talk about a satisfactory representation of the integers, besides the usual $\beta$ -, $\mu$ -, and $\mu '$ -reductions, we consider the $\lambda \mu$ -calculus augmented with the reduction rules $\rho$ , $\theta$ and $\varepsilon$ . We show that we need all of these rules for this purpose. Then we prove that, with the syntax of Parigot, the calculus enjoys the strong normalization property even when we add the rules $\rho$ , $\theta$ , and $\epsilon$ , while the $\lambda \mu$ -calculus presented with the more flexible de Groote-style syntax, in contrast, has only the weak normalization property. In particular, we present a normalization algorithm for the $\beta \mu \mu '\rho \theta \varepsilon$ -reduction in the de Groote-style calculus.

Membrane systems are nature motivated computational models inspired by certain basic features of biological cells and their membranes. They are examples of the chemical computational paradigm which describes computation in terms of chemical solutions