Counting Membrane Systems

A decision problem is one that has a yes/no answer, while a counting problem asks how many possible solutions exist associated with each instance. Every decision problem X has associated a counting problem, denoted by #X, in a natural way by replacing the question “is there a solution?” with “how many solutions are there?”. Counting problems are very attractive from a computational complexity point of view: if X is an NP-complete problem then the counting version #X is NP-hard, but the counting version of some problems in class P can also be NP-hard. In this paper, a new class of membrane systems is presented in order to provide a natural framework to solve counting problems. The class is inspired by a special kind of non-deterministic Turing machines, called counting Turing machines, introduced by L. Valiant. A polynomial-time and uniform solution to the counting version of the SAT problem (a well-known #P-complete problem) is also provided, by using a family of counting polarizationless P systems with active membranes, without dissolution rules and division rules for non-elementary membranes but where only very restrictive cooperation (minimal cooperation and minimal production) in object evolution rules is allowed

Ultra-Broadband Coherence-Domain Imaging Using Parametric Downconversion and Superconducting Single-Photon Detectors at 1064 nm

Coherence-domain imaging systems can be operated in a single-photon counting mode, offering low detector noise; this in turn leads to increased sensitivity for weak light sources and weakly reflecting samples. We have demonstrated that excellent axial resolution can be obtained in a photon-counting coherence domain imaging (CDI) system that uses light generated via spontaneous parametric down-conversion (SPDC) in a chirped periodically poled stoichiometric lithium tantalate (chirped-PPSLT) structure, in conjunction with a niobium nitride superconducting single-photon detector (SSPD). The bandwidth of the light generated via SPDC, as well as the bandwidth over which the SSPD is sensitive, can extend over a wavelength region that stretches from 700 to 1500 nm. This ultra-broad wavelength band offers a near-ideal combination of deep penetration and ultra-high axial resolution for the imaging of biological tissue. The generation of SPDC light of adjustable bandwidth in the vicinity of 1064 nm, via the use of chirped-PPSLT structures, had not been previously achieved. To demonstrate the usefulness of this technique, we have constructed images for a hierarchy of samples of increasing complexity: a mirror, a nitrocellulose membrane, and a biological sample comprising onion-skin cells

A static analysis for Brane Calculi providing global occurrence counting information

In this paper we propose a static analysis for Brane Calculi [1], based on Abstract Interpretation [2] techniques. Our analysis statically approximates the dynamic behaviour of Brane systems, by providing a description of the possible hierarchical structure of membranes and of the processes possibly associated to each membrane, together with global occurrence counting information. Our analysis can be computed in polynomial time. We apply it to investigate several biological systems in which occurrence counting information plays a crucial role. In particular, our case study concerns the formation of the haemoglobin polymer in presence of alterations and investigate the influence that such alterations have on the ability of the haemoglobin polymer to bind oxygen molecules

A Global Occurrence Counting Analysis for Brane Calculi

We propose a polynomial static analysis for Brane Calculi, based on Abstract Interpretation techniques. The analysis provides a description of the possible hierarchical structure of membranes and of the processes possibly associated to each membrane, together with global occurrence counting information. Our analysis can be applied in the biological setting to investigate systems in which the information on the number of membranes occurring in the system plays a crucial role

Simulating counting oracles with cooperation

We prove that monodirectional shallow chargeless P systems with active membranes and minimal cooperation working in polynomial time precisely characterise P#P k , the complexity class of problems solved in polynomial time by deterministic Turing machines with a polynomial number of parallel queries to an oracle for a counting problem

Phonon counting thermometry of an ultracoherent membrane resonator near its motional ground state

Generation of non-Gaussian quantum states of macroscopic mechanical objects is key to a number of challenges in quantum information science, ranging from fundamental tests of decoherence to quantum communication and sensing. Heralded generation of single-phonon states of mechanical motion is an attractive way towards this goal, as it is, in principle, not limited by the object size. Here we demonstrate a technique which allows for generation and detection of a quantum state of motion by phonon counting measurements near the ground state of a 1.5 MHz micromechanical oscillator. We detect scattered photons from a membrane-in-the-middle optomechanical system using an ultra-narrowband optical filter, and perform Raman-ratio thermometry and second-order intensity interferometry near the motional ground state ($\bar{n}=0.23\pm0.02$ phonons). With an effective mass in the nanogram range, our system lends itself for studies of long-lived non-Gaussian motional states with some of the heaviest objects to date.Comment: 11 pages, 10 figure

Generation of mechanical interference fringes by multi-photon counting

Exploring the quantum behaviour of macroscopic objects provides an intriguing avenue to study the foundations of physics and to develop a suite of quantum-enhanced technologies. One prominent path of study is provided by quantum optomechanics which utilizes the tools of quantum optics to control the motion of macroscopic mechanical resonators. Despite excellent recent progress, the preparation of mechanical quantum superposition states remains outstanding due to weak coupling and thermal decoherence. Here we present a novel optomechanical scheme that significantly relaxes these requirements allowing the preparation of quantum superposition states of motion of a mechanical resonator by exploiting the nonlinearity of multi-photon quantum measurements. Our method is capable of generating non-classical mechanical states without the need for strong single photon coupling, is resilient against optical loss, and offers more favourable scaling against initial mechanical thermal occupation than existing schemes. Moreover, our approach allows the generation of larger superposition states by projecting the optical field onto NOON states. We experimentally demonstrate this multi-photon-counting technique on a mechanical thermal state in the classical limit and observe interference fringes in the mechanical position distribution that show phase superresolution. This opens a feasible route to explore and exploit quantum phenomena at a macroscopic scale.Comment: 16 pages, 4 figures. v1: submitted for review on 28 Jan 2016. v2: significantly revised manuscript. v3: some further revisions and some extra results included. v3: new results added, extra author added, close to published version, supplementary material available with published versio

Correlation functions quantify super-resolution images and estimate apparent clustering due to over-counting

We present an analytical method to quantify clustering in super-resolution localization images of static surfaces in two dimensions. The method also describes how over-counting of labeled molecules contributes to apparent self-clustering and how the effective lateral resolution of an image can be determined. This treatment applies to clustering of proteins and lipids in membranes, where there is significant interest in using super-resolution localization techniques to probe membrane heterogeneity. When images are quantified using pair correlation functions, the magnitude of apparent clustering due to over-counting will vary inversely with the surface density of labeled molecules and does not depend on the number of times an average molecule is counted. Over-counting does not yield apparent co-clustering in double label experiments when pair cross-correlation functions are measured. We apply our analytical method to quantify the distribution of the IgE receptor (Fc{\epsilon}RI) on the plasma membranes of chemically fixed RBL-2H3 mast cells from images acquired using stochastic optical reconstruction microscopy (STORM) and scanning electron microscopy (SEM). We find that apparent clustering of labeled IgE bound to Fc{\epsilon}RI detected with both methods arises from over-counting of individual complexes. Thus our results indicate that these receptors are randomly distributed within the resolution and sensitivity limits of these experiments.Comment: 22 pages, 5 figure