Rewriting Abstract Structures: Materialization Explained Categorically

The paper develops an abstract (over-approximating) semantics for double-pushout rewriting of graphs and graph-like objects. The focus is on the so-called materialization of left-hand sides from abstract graphs, a central concept in previous work. The first contribution is an accessible, general explanation of how materializations arise from universal properties and categorical constructions, in particular partial map classifiers, in a topos. Second, we introduce an extension by enriching objects with annotations and give a precise characterization of strongest post-conditions, which are effectively computable under certain assumptions

Micropillars with a controlled number of site-controlled quantum dots

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Appl. Phys. Lett. 112, 071101 (2018) and may be found at https://doi.org/10.1063/1.5017692.We report on the realization of micropillars with site-controlled quantum dots (SCQDs) in the active layer. The SCQDs are grown via the buried stressor approach which allows for the positioned growth and device integration of a controllable number of QDs with high optical quality. This concept is very powerful as the number and the position of SCQDs in the cavity can be simultaneously controlled by the design of the buried-stressor. The fabricated micropillars exhibit a high degree of position control for the QDs above the buried stressor and Q-factors of up to 12 000 at an emission wavelength of around 930 nm. We experimentally analyze and numerically model the cavity Q-factor, the mode volume, the Purcell factor, and the photon-extraction efficiency as a function of the aperture diameter of the buried stressor. Exploiting these SCQD micropillars, we experimentally observe a Purcell enhancement in the single-QD regime with FP = 4.3 ± 0.3.EC/FP7/615613/EU/External Quantum Control of Photonic Semiconductor Nanostructures/EXQUISITEDFG, SFB 787, Halbleiter - Nanophotonik: Materialien, Modelle, Bauelement

A Category Theoretical Approach to the Concurrent Semantics of Rewriting: Adhesive Categories and Related Concepts

This thesis studies formal semantics for a family of rewriting formalisms that have arisen as category theoretical abstractions of the so-called algebraic approaches to graph rewriting. The latter in turn generalize and combine features of term rewriting and Petri nets. Two salient features of (the abstract versions of) graph rewriting are a suitable class of categories which captures the structure of the objects of rewriting, and a notion of independence or concurrency of rewriting steps – as in the theory of Petri nets. Category theoretical abstractions of graph rewriting such as double pushout rewriting encapsulate the complex details of the structures that are to be rewritten by considering them as objects of a suitable abstract category, for example an adhesive one. The main difficulty of the development of appropriate categorical frameworks is the identification of the essential properties of the category of graphs which allow to develop the theory of graph rewriting in an abstract framework. The motivations for such an endeavor are twofold: to arrive at a succint description of the fundamental principles of rewriting systems in general, and to apply well-established verification and analysis techniques of the theory of Petri nets (and also term rewriting systems) to a wide range of distributed and concurrent systems in which states have a "graph-like" structure. The contributions of this thesis thus can be considered as two sides of the same coin: on the one side, concepts and results for Petri nets (and graph grammars) are generalized to an abstract category theoretical setting; on the other side, suitable classes of "graph-like" categories which capture the essential properties of the category of graphs are identified. Two central results are the following: first, (concatenable) processes are faithful partial order representations of equivalence classes of system runs which only differ w.r.t. the rescheduling of causally independent events; second, the unfolding of a system is established as the canonical partial order representation of all possible events (following the work of Winskel). Weakly ω-adhesive categories are introduced as the theoretical foundation for the corresponding formal theorems about processes and unfoldings. The main result states that an unfolding procedure for systems which are given as single pushout grammars in weakly ω-adhesive categories exists and can be characetrised as a right adjoint functor from a category of grammars to the subcategory of occurrence grammars. This result specializes to and improves upon existing results concerning the coreflective semantics of the unfolding of graph grammars and Petri nets (under an individual token interpretation). Moreover, the unfolding procedure is in principle usable as the starting point for static analysis techniques such as McMillan’s finite complete prefix method. Finally, the adequacy of weakly ω-adhesive categories as a categorical framework is argued for by providing a comparison with the notion of topos, which is a standard abstraction of the categories of sets (and graphs)

Structural Decomposition of Reactions of Graph-Like Objects

Inspired by decomposition problems in rule-based formalisms in Computational Systems Biology and recent work on compositionality in graph transformation, this paper proposes to use arbitrary colimits to "deconstruct" models of reactions in which states are represented as objects of adhesive categories. The fundamental problem is the decomposition of complex reactions of large states into simpler reactions of smaller states. The paper defines the local decomposition problem for transformations. To solve this problem means to "reconstruct" a given transformation as the colimit of "smaller" ones where the shape of the colimit and the decomposition of the source object of the transformation are fixed in advance. The first result is the soundness of colimit decomposition for arbitrary double pushout transformations in any category, which roughly means that several "local" transformations can be combined into a single "global" one. Moreover, a solution for a certain class of local decomposition problems is given, which generalizes and clarifies recent work on compositionality in graph transformation. Introduction Compositional methods for the synthesis and analysis of computational systems remain a fruitful research topic with potential applications in practice. Though compositionality is most clearly exhibited in semantics for process calculi where structural operational semantics (SOS) can be found in its "pure" form, a slightly broader perspective is appropriate to make use of the fundamental ideas of SOS in interdisciplinary research. The first source of inspiration of the present paper is the κ-calculus [6], which is an influential modelling framework in Computational Systems Biology. The κ-calculus allows to give abstract, formal descriptions of biological systems that can be used to explain the reaction (rate) of complex systems, so-called complexes, in terms of the reaction (rate) of each of its subsystems, which are called partial complexes. Leaving quantitative aspects as a topic for future research, we concentrate on a specific sub-problem, namely the "purely structural" decomposition of reactions. In the κ-calculus, system states are composed of partial complexes and they have an intuitive, graphical representation. Hence, it is natural to investigate the decomposition of (reactions of) system states using concepts from graph transformation. In its simplest form, the idea of composition of graph transformations is by means of coproducts. Intuitively, the coproduct of two graphs models the assembly of two states put side by side and the two (sub-)states react independently of each other. A well-known, related theorem about graph transformations is the so-called Parallelism Theorem (see e.g. [5, Theorem 17]). A more general formalism of compositionality that is based on pushouts has been (re-)considered in In this paper, we shall remove the restriction to pushouts as a composition mechanism and generalize the results of [18] from pushouts to (pullback stable) colimits of arbitrary shape. This considerably enlarges the set of available gluing patterns. As a simple example, we can now equip each sub-state with several interfaces; this would be appropriate for the model of a cell in an organism that is in direct contact with each of its neighbouring cells with some part of its membrane; each area of contact would be modelled by a different interface. Content of the paper After reviewing some basic category theoretical concepts and the definition of adhesive categories in Section 1, we begin Section 2 with the "deconstruction" of models of system states; more precisely, we explain in Section 2.1 how suitably finite objects in adhesive categories arise as the colimit of a diagram of "atomic" objects, namely irreducible objects in the sense of The main problem, which is concerned with the decomposition of a "global" transformation into a family of "local" ones, is addressed in Section 3. We give a formal description of local decomposition problems, which consist of a given decomposition of a state (as a colimit of a certain shape) and a rule that describes a possible reaction of the state; to solve such a problem means to extend the decomposition of the state to a decomposition of the whole reaction (using colimits of the same shape). Section 3.1 presents a "global" solution, which first constructs the whole transformation "globally"; a "more local" solution of the problem is possible if we are given extra information that involve a generalization of the accommodations o

Updating Probabilistic Knowledge on Condition/Event Nets using Bayesian Networks

The paper extends Bayesian networks (BNs) by a mechanism for dynamic changes to the probability distributions represented by BNs. One application scenario is the process of knowledge acquisition of an observer interacting with a system. In particular, the paper considers condition/event nets where the observer\u27s knowledge about the current marking is a probability distribution over markings. The observer can interact with the net to deduce information about the marking by requesting certain transitions to fire and observing their success or failure. Aiming for an efficient implementation of dynamic changes to probability distributions of BNs, we consider a modular form of networks that form the arrows of a free PROP with a commutative comonoid structure, also known as term graphs. The algebraic structure of such PROPs supplies us with a compositional semantics that functorially maps BNs to their underlying probability distribution and, in particular, it provides a convenient means to describe structural updates of networks

Pattern Graphs and Rule-Based Models: The Semantics of Kappa

International audienceDomain-specific rule-based languages to represent the systems of reactions that occur inside cells, such as Kappa and BioNetGen, have attracted significant recent interest. For these models, powerful simulation and static analysis techniques have been developed to understand the behaviour of the systems that they represent, and these techniques can be transferred to other fields. The languages can be understood intuitively as transforming graph-like structures, but due to their expressivity these are difficult to model in 'traditional' graph rewriting frameworks. In this paper, we introduce pattern graphs and closed morphisms as a more abstract graph-like model and show how Kappa can be encoded in them by connecting its single-pushout semantics to that for Kappa. This level of abstraction elucidates the earlier single-pushout result for Kappa, teasing apart the proof and guiding the way to richer languages, for example the introduction of compartments within cells

Cesium‐vapor‐based delay of single photons emitted by deterministically fabricated quantum dot microlenses

Quantum light sources are key building blocks of photonic quantum technologies. For many applications, it is of interest to control the arrival time of single photons emitted by such quantum devices, or even to store single photons in quantum memories. In situ electron beam lithography is applied to realize InGaAs quantum dot (QD)‐based single‐photon sources, which are interfaced with cesium (Cs) vapor to control the time delay of emitted photons. Via numerical simulations of the light–matter interaction in realistic QD‐Cs‐vapor configurations, the influence of the vapor temperature and spectral QD‐atom detuning is explored to maximize the achievable delay in experimental studies. As a result, this hybrid quantum system allows to trigger the emission of single photons with a linewidth as low as 1.54 GHz even under non‐resonant optical excitation and to delay the emission pulses by up to (15.71 ± 0.01) ns for an effective cell length of 150 mm. This work can pave the way for scalable quantum systems relying on a well‐controlled delay of single photons on a time scale of up to a few tens of nanoseconds.BMBF, 03V0630TIB, Entwicklung einer Halbleiterbasierten Einzelphotonenquelle für die QuanteninformationstechnologieBMBF, 13N14876, Quantenkommunikations-Systeme auf Basis von Einzelphotonenquellen (QuSecure)DFG, 43659573, SFB 787: Halbleiter - Nanophotonik: Materialien, Modelle, BauelementeTU Berlin, Open-Access-Mittel - 201