Elementary Fixed Point Theorems * Theorem of Borsuk and Topological Transversality * Homology and Fixed Points * Leray-Schauder Degree and Fixed Point Index * The Lefschetz-Hopf Theory * Selected Topics * Index

Fixed Point Theory

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Hilbert's tenth problem

Tissue-like P systems with evolutional symport/antiport rules

Membrane computing and complexity theory: A characterization of PSPACE

A quick introduction to membrane computing

We define space complexity classes in the framework of membrane computing, giving some initial results about their mutual relations and their connection with time complexity classes, and identifying some potentially interesting problems which require further research

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Introducing a Space Complexity Measure for P Systems

We investigate the computational complexity of deciding the occurrence of many different dynamical behaviours in reaction systems, with an emphasis on biologically relevant problems (i.e., existence of fixed points and fixed point attractors). We show that the decision problems of recognising these dynamical behaviours span a number of complexity classes ranging from FO-uniform AC 0 to \({\Pi_2^{\rm P}}\)-completeness with several intermediate problems being either NP or coNP-complete.

Fixed Points and Attractors of Reaction Systems

Polynomial-time P systems with active membranes characterise PSPACE by exploiting membranes nested to a polynomial depth, which may be subject to membrane division rules. When only elementary leaf membrane division rules are allowed, the computing power decreases to PPP = P#P, the class of problems solvable in polynomial time by deterministic Turing machines equipped with oracles for counting or majority problems. In this paper we investigate a variant of intermediate power, limiting membrane nesting hence membrane division to constant depth, and we prove that the resulting P systems can solve all problems in the counting hierarchy CH, which is located between PPP and PSPACE. In particular, for each integer k ≥ 0 we provide a lower bound to the computing power of P systems of depth k.

Membrane Division, Oracles, and the Counting Hierarchy

The decision problems solved in polynomial time by P systems with elementary active membranes are known to include the class \(\mathbf{P}^{\# \mathbf{P}}\). This consists of all the problems solved by polynomial-time deterministic Turing machines with polynomial-time counting oracles. In this paper we prove the reverse inclusion by simulating P systems with this kind of machines: this proves that the two complexity classes coincide, finally solving an open problem by Paun on the power of elementary division. The equivalence holds for both uniform and semi-uniform families of P systems, with or without membrane dissolution rules. Furthermore, the inclusion in \(\mathbf{P}^{\# \mathbf{P}}\) also holds for the P systems involved in the P conjecture (with elementary division and dissolution but no charges), which improves the previously known upper bound \(\mathbf{PSPACE}\).

Simulating Elementary Active Membranes

For many models of P systems and tissue P systems, the main behavior of a specific system can be simulated by a corresponding system with only one membrane or cell, respectively; this effective construction is called flattening. In this paper we describe the main procedure of flattening for specific variants of static (tissue) P systems as well as for classes of dynamic (tissue) P systems with a bounded number of possible membrane structures or a bounded number of cells during any computation.

Antonio E. Porreca

Papers

Introducing a Space Complexity Measure for P Systems

Fixed Points and Attractors of Reaction Systems

Membrane Division, Oracles, and the Counting Hierarchy

Simulating Elementary Active Membranes

Flattening in (Tissue) P Systems