A Gentle Introduction to Membrane Systems and Their Computational Properties

Abstract

The theory of computation investigates the nature and properties of algorithmic procedures. This field emerged in the 20s and 30s of the 20th century from the work on the philosophy and the foundations of mathematics. Of great importance and inspiration was David Hilbert’s ambitious program to “dispose of the foundational questions in mathematics once and for all” [34], that led to fundamental results in logic such as Gdel’s incompleteness theorems [6], and ultimately to the birth of recursion theory (nowadays mostly referred to as computability theory) and computer science itself. The formal notion of computability that is almost universally adopted today is due to Alan Turing, who introduced in his ground-breaking paper On computable numbers, with an application to the Entscheidungsproblem [32] a simple, elegant and convincing mathematical formalisation of the notion of computation, as it is carried out by a human executor equipped with enough scratch paper. Turing’s work showed that, as long as we accept his notion of computation, there exist well-formed mathematical questions whose answer cannot be computed. In particular, one of the main challenges of Hilbert’s program, the Entscheidungsproblem (finding a decision procedure for the validity of statements in first-order logic) was proved to be unsolvable. This formalisation, that rapidly became known as the Turing machine, is still the reference model for computing devices in theoretical computer science, as it also enjoys the property of being a good model of actual electronic computers; this is also due to the fact that it was itself an inspiration for the design of automatic computing machinery [5]. With the development of computers as a technology, being able to solve a particular problem proved not to be satisfying: fast, efficient solutions are needed. This led to the development of computational complexity theory, pioneered [9] by Hartmanis and Stearns in the paper On the computational complexity of algorithms [12], that also gives the name to the field. Identifying