22 research outputs found

    Natural Factors of the Medvedev Lattice Capturing IPC

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    Skvortsova showed that there is a factor of the Medvedev lattice which captures intuitionistic propositional logic (IPC). However, her factor is unnatural in the sense that it is constructed in an ad hoc manner. We present a more natural example of such a factor. We also show that for every non-trivial factor of the Medvedev lattice its theory is contained in Jankov's logic, the deductive closure of IPC plus the weak law of the excluded middle. This answers a question by Sorbi and Terwijn

    Intuitionistic logic and Muchnik degrees

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    We prove that there is a factor of the Muchnik lattice that captures intuitionistic propositional logic. This complements a now classic result of Skvortsova for the Medvedev lattice

    Mass problems and intuitionistic higher-order logic

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    In this paper we study a model of intuitionistic higher-order logic which we call \emph{the Muchnik topos}. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass problems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, \emph{the Muchnik reals}, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a \emph{choice principle} (xyA(x,y))wxA(x,wx)(\forall x\,\exists y\,A(x,y))\Rightarrow\exists w\,\forall x\,A(x,wx) and a \emph{bounding principle} (xyA(x,y))zxy(yT(x,z)A(x,y))(\forall x\,\exists y\,A(x,y))\Rightarrow\exists z\,\forall x\,\exists y\,(y\le_{\mathrm{T}}(x,z)\land A(x,y)) where x,y,zx,y,z range over Muchnik reals, ww ranges over functions from Muchnik reals to Muchnik reals, and A(x,y)A(x,y) is a formula not containing ww or zz. For the convenience of the reader, we explain all of the essential background material on intuitionism, sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems, Muchnik degrees, and Kolmogorov's calculus of problems. We also provide an English translation of Muchnik's 1963 paper on Muchnik degrees.Comment: 44 page

    Comparing the degrees of enumerability and the closed Medvedev degrees

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    We compare the degrees of enumerability and the closed Medvedev degrees and find that many situations occur. There are nonzero closed degrees that do not bound nonzero degrees of enumerability, there are nonzero degrees of enumerability that do not bound nonzero closed degrees, and there are degrees that are nontrivially both degrees of enumerability and closed degrees. We also show that the compact degrees of enumerability exactly correspond to the cototal enumeration degrees

    Polyatomic Logics and Generalised Blok-Esakia Theory

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    This paper presents a novel concept of a Polyatomic Logic and initiates its systematic study. This approach, inspired by Inquisitive semantics, is obtained by taking a variant of a given logic, obtained by looking at the fragment covered by a selector term. We introduce an algebraic semantics for these logics and prove algebraic completeness. These logics are then related to translations, through the introduction of a number of classes of translations involving selector terms, which are noted to be ubiquitous in algebraic logic. In this setting, we also introduce a generalised Blok-Esakia theory which can be developed for special classes of translations. We conclude by showing some systematic connections between the theory of Polyatomic Logics and the general Blok-Esakia theory for a wide class of interesting translations.Comment: 48 pages, 2 figure

    Inkjet printing digital image generation and compensation for surface chemistry effects

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    Additive manufacturing (AM) of electronic materials using digital inkjet printing (DIJP) is of research interests nowadays because of its potential benefits in the semiconductor industry. Current trends in manufacturing electronics feature DIJP as a key technology to enable the production of customised and microscale functional devices. However, the fabrication of microelectronic components at large scale demands fast printing of tight features with high dimensional accuracy on substrates with varied surface topography which push inkjet printing process to its limits. To understand the DIJP droplet deposition on such substrates, generally requires computational fluid dynamics modelling which is limited in its physics approximation of surface interactions. Otherwise, a kind of “trial and error” approach to determining how the ink spreads, coalesce and solidifies over the substrate is used, often a very time-consuming process. Consequently, this thesis aims to develop new modelling techniques to predict fast and accurately the surface morphology of inkjet-printed features, enabling the optimisation of DIJP control parameters and the compensation of images for better dimensional accuracy of printed electronics devices. This investigation explored three categories of modelling techniques to predict the surface morphology of inkjet-printed features: physics-based, data-driven and hybrid physics-based and data-driven. Two physics-based numerical models were developed to reproduce the inkjet printing droplet deposition and solidification processes using a lattice Boltzmann (LB) multiphase flow model and a finite element (FE) chemo-mechanical model, respectively. The LB model was limited to the simulation of single tracks and small square films and the FE model was mainly employed for the distortion prediction of multilayer objects. Alternatively, two data-driven models were implemented to reconstruct the surface morphology of single tracks and free-form films using images from experiments: image analysis (IA) and shape from shading (SFS). IA assumed volume conservation and minimal energy drop shape to reconstruct the surface while SFS resolved the height of the image using a reflection model. Finally, a hybrid physics-based and data-driven approach was generated which incorporates the uncertainty of droplet landing position and footprint, hydrostatic analytical models, empirical correlations derived from experiments, and relationships derived from physics-based models to predict fast and accurately any free-form layer pattern as a function of physical properties, printing parameters and wetting characteristics. Depending on the selection of the modelling technique to predict the deformed geometry, further considerations were required. For the purely physics-based and data-driven models, a surrogate model using response surface equations was employed to create a transfer function between printing parameters, substrate wetting characteristics and the resulting surface morphology. The development of a transfer function significantly decreased the computational time required by purely physics-based models and enabled the parameter optimisation using a multi-objective genetic algorithm approach to attain the best film dimensional accuracy. Additionally, for multilayer printing applications, a layer compensation approach was achieved utilizing a convolutional neural network trained by the predicted (deformed) geometry to reduce the out of plane error to target shape. The optimal combination of printing parameters and input image compensation helped with the generation of fine features that are traditionally difficult for inkjet, improved resolution of edges and corners by reducing the amount of overflow from material, accounted for varied topography and capillary effects thereof on the substrate surface and considered the effect of multiple layers built up on each other. This study revealed for the first time to the best of our knowledge the role of the droplet location and footprint diameter uncertainty in the stability and uniformity of printed features. Using a droplet overlap map which was proposed as a universal technique to assess the effect of printing parameters on pattern geometry, it was shown that reliable limits for break-up and bulging of printed features were obtained. Considering droplet position and diameter size uncertainties, predicted optimal printing parameters improved the quality of printed films on substrates with different wettability. Finally, a stability diagram illustrating the onset of bulging and separation for lines and films as well as the optimal drop spacing, printing frequency and stand-off distance was generated to inform visually the results. This investigation has developed a predictive physics-based model of the surface morphology of DIJP features on heterogeneous substrates and a methodology to find the printing parameters and compensate the layer geometry required for optimum part dimensional accuracy. The simplicity of the proposed technique makes it a promising tool for model driven inkjet printing process optimization, including real time process control and paves the way for better quality devices in the printed electronics industry
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