Coinduction for preordered algebra

We develop a combination, called hidden preordered algebra, between preordered algebra, which is an algebraic framework supporting specification and reasoning about transitions, and hidden algebra, which is the algebraic framework for behavioural specification. This combination arises naturally within the heterogeneous framework of the modern formal specification language CafeOBJ. The novel specification concept arising from this combination, and which constitutes its single unique feature, is that of behavioural transition. We extend the coinduction proof method for behavioural equivalence to coinduction for proving behavioural transitions.     

Coinduction for Exact Real Number Computation

This paper studies coinductive representations of real numbers by signed digit streams and fast Cauchy sequences. It is shown how the associated coinductive principle can be used to give straightforward and easily implementable proofs of the equivalence of the two representations as well as the correctness of various corecursive exact real number algorithms. The basic framework is the classical theory of coinductive sets as greatest fixed points of monotone operators and hence is different from (though related to) the type theoretic approach by Ciaffaglione and Gianantonio.     

Coinduction in Control of Partially Observed Discrete-Event Systems

Coalgebra and coinduction provide new results and insights for the supervisory control of discrete-event systems (DES) with partial observations. In the case of full observations, coinduction has been used to define a new operation on languages called supervised product, which represents the tuple of languages of the supervised system. The first language acts as a supervisor and the second as an open-loop system (plant). We show first that the supervised product is equal to the infimal controllable superlanguage of the supervisor's (specification) language with respect to the plant language. This can be generalized to the partial observation case, where the supervised product is shown to be equal to the infimal controllable and observable superlanguage. A modification on the supervised product is presented, which corresponds to the control policy for with the issue of observability is separated from the issue of controllability. The operation defined by coinduction is shown to be equal to the infimal observable superlanguage.     

Control of Discrete-Event Systems with Partial Observations Using Coalgebra and Coinduction

Control of discrete-event systems with partial observations is treated by concepts and results of coalgebra and coinduction. Coalgebra is part of abstract algebra and enables a generalization of the computer science concept of bisimulation. It can be applied to automata theory and then provides a powerful algebraic tool to treat problems of supervisory control. A framework for control of discrete-event systems with partial observations is formulated in terms of coalgebra. The contributions to control theory are besides the framework, algorithms for supremal normal and supremal normal and controllable sublanguages of the plant.     

Structural Induction and Coinduction in a Fibrational Setting

We present a categorical logic formulation of induction and coinduction principles for reasoning about inductively and coinductively defined types. Our main results provide sufficient criteria for the validity of such principles: in the presence of comprehension, the induction principle for initial algebras is admissible, and dually, in the presence of quotient types, the coinduction principle for terminal coalgebras is admissible. After giving an alternative formulation of induction in terms of binary relations, we combine both principles and obtain a mixed induction/coinduction principle which allows us to reason about minimal solutionsX≅σ(X) whereXmay occur both positively and negatively in the type constructor σ. We further strengthen these logical principles to deal with contexts and prove that such strengthening is valid when the (abstract) logic we consider is contextually/functionally complete. All the main results follow from a basic result about adjunctions between “categories of algebras” (inserters).     

Generalised Epipolar Constraints

In this paper we will discuss structure and motion problems for curved surfaces. These will be studied using the silhouettes or apparent contours in the images. The problem of determining camera motion from the apparent contours of curved three-dimensional surfaces, is studied. It will be shown how special points, called epipolar tangency points or frontier points, can be used to solve this problem. A generalised epipolar constraint is introduced, which applies to points, curves, as well as to apparent contours of surfaces. The theory is developed for both continuous and discrete motion, known and unknown orientation, calibrated and uncalibrated, perspective, weak perspective and orthographic cameras. Results of an iterative scheme to recover the epipolar line structure from real image sequences using only the outlines of curved surfaces, is presented. A statistical evaluation is performed to estimate the stability of the solution. It is also shown how the motion of the camera from a sequence of images can be obtained from the relative motion between image pairs.     

Generalised quantum weakest preconditions

Generalisation of the quantum weakest precondition result of D’Hondt and Panangaden is presented. In particular the most general notion of quantum predicate as positive operator valued measure (termed POVM) is introduced. The previously known quantum weakest precondition result has been extended to cover the case of POVM playing the role of a quantum predicate. Additionally, our result is valid in infinite dimension case and also holds for a quantum programs defined as a positive but not necessary completely positive transformations of a quantum states.     

Generalised chebyshev basis functions

The construction of generalised Chebyshev basis functions in one dimension is carried out for both linear and quadratic cases. The optimal selection of the point of reflection of the required Chebyshev Polynomial (s) is identified.     

Generalised Nonlocal Image Smoothing

We propose a discrete variational approach for image smoothing consisting of nonlocal data and smoothness constraints that penalise general dissimilarity measures defined on image patches. One of such dissimilarity measures is the weighted L ₂ distance between patches. In such a case we derive an iterative neighbourhood filter that induces a new similarity measure in the photometric domain. It can be regarded as an extended patch similarity measure that evaluates not only the patch similarity of two chosen pixels, but also the similarity of their corresponding neighbours. This leads to a more robust smoothing process since the pixels selected for averaging are more coherent with the local image structure. By slightly modifying the way the similarities are computed we obtain two related filters: The NL-means filter of Buades et al. (SIAM Multiscale Model. Simul. 4(2):490–530, 2005b) and the NDS filter of Mrázek et al. (Geometric Properties for Incomplete Data, Computational Imaging and Vision, vol. 31, pp. 335–352, Springer, Dordrecht, 2006). In fact, the proposed approach can be considered as a generalisation of the latter filter to the space of patches. We also provide novel insights into relations of the NDS filter with diffusion/regularisation methods as well as with some recently proposed graph regularisation techniques. We evaluate our method for the task of denoising greyscale and colour images degraded with Gaussian and salt-and-pepper noise, demonstrating that it compares very well to other more sophisticated approaches.     

Generalised fuzzy soft sets

In this paper, we define generalised fuzzy soft sets and study some of their properties. Application of generalised fuzzy soft sets in decision making problem and medical diagnosis problem has been shown.     

Generalised multivariable supertwisting algorithm

The purpose of this paper is to present an extension of the generalised supertwisting algorithm (STA) to the multivariable framework. We begin by introducing an algorithm that may be deemed as a linear, quasicontinuous, or discontinuous multivariable system, depending on the functions that define them. For the class represented by such an algorithm we prove the robust, Lyapunov stability of the origin and characterise the perturbations that preserve its stability. In particular, when its vector field is discontinuous or quasicontinuous our algorithm is endowed with finite‐time stability. Due to its resemblance to the scalar case, we denote such finite‐time stable systems as generalised multivariable STA. Furthermore, the class of finite‐time stable systems comprise the currently available versions of STAs. To finalise, by means of simulation examples, we show that our proposed finite‐time stable algorithms are well suited for signals online differentiation and highlight their dynamical traits.     

A Coinduction Principle for Recursive Data Types Based on Bisimulation

The concept of bisimulation from concurrency theory is used to reason about recursively defined data types. From two strong-extensionality theorems stating that the equality (resp. inequality) relation is maximal among all bisimulations, a proof principle for the final coalgebra of an endofunctor on a category of data types (resp. domains) is obtained. As an application of the theory developed, an internal full abstraction result (in the sense of S. Abramsky and C.-H. L. Ong [Inform. and Comput.105, 159–267 (1993)] for the canonical model of the untyped call-by-valueλ-calculus is proved. Also, the operational notion of bisimulation and the denotational notion of final semantics are related by means of conditions under which both coincide.     

Generalised folds for nested datatypes

Nested datatypes generalise regular datatypes in much the same way that context-free languages generalise regular ones. Although the categorical semantics of nested types turns out to be similar to the regular case, the fold functions are more limited because they can only describe natural transformations. Practical considerations therefore dictate the introduction of a generalised fold function in which this limitation can be overcome. In the paper we show how to construct generalised folds systematically for each nested datatype, and show that they possess a uniqueness property analogous to that of ordinary folds. As a consequence, generalised folds satisfy fusion properties similar to those developed for regular datatypes. Such properties form the core of an effective calculational theory of inductive datatypes. Received September 1998 / Accepted in revised form July 1999     

Generalised correlation for multi-feature correspondence

Computing correspondences between pairs of images is fundamental to all structures from motion algorithms. Correlation is a popular method to estimate similarity between patches of images. In the standard formulation, the correlation function uses only one feature such as the gray level values of a small neighbourhood. Research has shown that different features—such as colour, edge strength, corners, texture measures—work better under different conditions. We propose a framework of generalized correlation that can compute a real valued similarity measure using a feature vector whose components can be dissimilar. The framework can combine the effects of different image features, such as multi-spectral features, edges, corners, texture measures, etc., into a single similarity measure in a flexible manner. Additionally, it can combine results of different window sizes used for correlation with proper weighting for each. Relative importances of the features can be estimated from the image itself for accurate correspondence. In this paper, we present the framework of generalised correlation, provide a few examples demonstrating its power, as well as discuss the implementation issues.     

Upper and Lower Bounds on the Complexity of Generalised Resolution and Generalised Constraint Satisfaction Problems

Capturing propositional logic, constraint satisfaction problems and systems of polynomial equations, we introduce the notion of systems with finite instantiation by partial assignments, fipa-systems for short, which are independent of special representations of problem instances, but which are based on an axiomatic approach with instantiation (or substitution) by partial assignments as the fundamental notion. Fipa-systems seem to constitute the most general framework allowing for a theory of resolution with nontrivial upper and lower bounds. For every fipa-system we generalise relativised hierarchies originating from generalised Horn formulas [14,26,33,43], and obtain hierarchies of problem instances with recognition and satisfiability decision in polynomial time and linear space, quasi-automatising relativised and generalised tree resolution and utilising a general “quasi-tight” lower bound for tree resolution. And generalising width-restricted resolution from [7,14,25,33], for every fipa-system a (stronger) family of hierarchies of unsatisfiable instances with polynomial time recognition is introduced, weakly automatising relativised and generalised full resolution and utilising a general lower bound for full resolution generalising [7,17,25,33].     

Generalised Gaussian radial basis function neural networks

The mixed use of different shapes of radial basis functions (RBFs) in radial basis functions neural networks (RBFNNs) is investigated in this paper. For this purpose, we propose the use of a generalised version of the standard RBFNN, based on the generalised Gaussian distribution. The generalised radial basis function (GRBF) proposed in this paper is able to reproduce other different radial basis functions (RBFs) by changing a real parameter τ. In the proposed methodology, a hybrid evolutionary algorithm (HEA) is employed to estimate the number of hidden neuron, the centres, type and width of each RBF associated with each radial unit. In order to test the performance of the proposed methodology, an experimental study is presented with 20 datasets from the UCI repository. The GRBF neural network (GRBFNN) was compared to RBFNNs with Gaussian, Cauchy and inverse multiquadratic RBFs in the hidden layer and to other classifiers, including different RBFNN design methods, support vector machines (SVMs), a sparse probabilistic classifier (sparse multinominal logistic regression, SMLR) and other non-sparse (but regularised) probabilistic classifiers (regularised multinominal logistic regression, RMLR). The GRBFNN models were found to be better than the alternative RBFNNs for almost all datasets, producing the highest mean accuracy rank.