This is a comprehensive (and friendly) introduction to membrane computing (MC), meant to offer both computer scientists and non-computer scientists an up-to-date overview of the field. That is why the set of notions introduced here is rather large, but the presentation is informal, without proofs and with rigorous definitions given only for the basic types of P systems — symbol object P systems with multiset rewriting rules, systems with symport/antiport rules, systems with string objects, tissue-like P systems, and neural-like P systems. Besides a list of (biologically inspired or mathematically motivated) ingredients/features which can be used in systems of these types, we also mention a series of results, as well as a series of research trends and topics.

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Introduction to Membrane Computing

Commutative grammars are introduced, and various classes of commutative grammars are defined. The complexity of uniform word problems for commutative grammars is investigated.

Commutative grammars: the complexity of uniform word problems

We propose a characterization of a Chomsky-like hierarchy of multiset grammars in terms of multiset automata. We also present the deterministic variants of all the automata investigated and prove that, unlike the string case, most of them are strictly less powerful than the nondeterministic variants. Some open problems and further directions of research are briefly discussed.

Multiset Automata

https://eecs.wsu.edu/~zdang/papers/catalytic.pdf

Catalytic P systems, semilinear sets, and vector addition systems

A macroset is a (finite or infinite) set of multisets over a finite alphabet. We introduce a Chomsky-like hierarchy of multiset rewriting devices which, therefore, generate macrosets. Some results are proved about the power of these devices and some open problems are formulated. We also present an algebraic characterization of some of the macroset families as least fixed point solutions of algebraic systems of equations.

Toward a Formal Macroset Theory

Commutative grammars are a formalism for generating bags, equivalent to vector addition systems and Petri nets. Known results are recalled and new one are presented on reachability and boundedness. In particular some subclasses of commutative grammars are introduced which admit a positive answer to these problems, and generate semilinear languages. Finally the equivalence, via Parikh mapping, of commutative and matrix grammars is proven.

D. Mandrioli

Papers

Commutative grammars