Specification and verification of database dynamics

Summary A framework is proposed for the structured specification and verification of database dynamics. In this framework, the conceptual model of a database is a many sorted first order linear tense theory whose proper axioms specify the update and the triggering behaviour of the database. The use of conceptual modelling approaches for structuring such a theory is analysed. Semantic primitives based on the notions of event and process are adopted for modelling the dynamic aspects. Events are used to model both atomic database operations and communication actions (input/output). Nonatomic operations to be performed on the database (transactions) are modelled by processes in terms of trigger/reaction patterns of behaviour. The correctness of the specification is verified by proving that the desired requirements on the evolution of the database are theorems of the conceptual model. Besides the traditional data integrity constraints, requirements of the form Under condition W, it is guaranteed that the database operation Z will be successfully performed are also considered. Such liveness requirements have been ignored in the database literature, although they are essential to a complete definition of the database dynamics.

Notation

Classical Logic Symbols (Appendix 1) for all (universal quantifier) - exists at least once (existential quantifier) - ¬ no (negation) - implies (implication) - is equivalent to (equivalence) - and (conjunction) - or (disjunction) Tense Logic Symbols (Appendix 1) G always in the future - G ₀ always in the future and now - F sometime in the future - F ₀ sometime in the future or now - H always in the past - H ₀ always in the past and now - P sometime in the past - P ₀ sometime in the past or now - X in the next moment - Y in the previous moment - L always - M sometime Event Specification Symbols (Sects. 3 and 4.1) (x) means immediately after the occurrence of x - (x) means immediately before the occurrence of x - (x) means x is enabled, i.e., x may occur next - { } ({w ₁} x{w ₂}) states that if w ₁ holds before the occurrence of x, then w ₂ will hold after the occurrence of x (change rule) - [ ] ([oa₁, ..., oa_n]x) states that only the object attributes oa₁, ..., oa _n are modifiable by x (scope rule) - {{ }} ({{w}}x) states that if x may occur next, then w holds (enabling rule) Process Specification Symbols (Sects. 5.3 and 5.4) :: for causal rules - for behavioural rules Transition-Pattern Composition Symbols (Sects. 5.2 and 5.3) ; sequential composition - ¦ choice composition - parallel composition - :| guarded alternative composition Location Predicates (Sect. 5.2) (z) means immediately after the occurrence of the last event of z (after) - (z) means immediately before the occurrence of the first event of z (before) - (z) means after the beginning of z and before the end of z (during) - ( z) means before the occurrence of an event of z (at)     

Probabilistic analysis of the application of the cross ratio to model based vision: Misclassification

The cross ratio of four colinear points is of fundamental importance in model based vision, because it is the simplest numerical property of an object that is invariant under projection to an image. It provides a basis for algorithms to recognise objects from images without first estimating the position and orientation of the camera.A quantitative analysis of the effectiveness of the cross ratio in model based vision is made. A given imageI of four colinear points is classified by making comparisons between the measured cross ratio of the four image points and the cross ratios stored in the model database. The imageI is accepted as a projection of an objectO with cross ratio if |–|ntu, wheren is the standard deviation of the image noise,t is a threshold andu=. The performance of the cross ratio is described quantitatively by the probability of rejectionR, the probability of false alarmF and the probability of misclassificationp (), defined for two model cross ratios , . The trade off between these different probabilities is determined byt.It is assumed that in the absence of an object the image points have identical Gaussian distributions, and that in the presence of an object the image points have the appropriate conditional densities. The measurements of the image points are subject to small random Gaussian perturbations. Under these assumptions the trade offs betweenR,F andp () are given to a good approximation byR=2(1–(t)),F=r _F t, t|–|^–1, where is the relative noise level, is cumulative distribution function for the normal distribution,r _F is constant, ande is a function of only. The trade off betweenR andF is obtained in Maybank (1994). In this paper the trade off betweenR andp () is obtained.It is conjectured that the general form of the above trade offs betweenR,F andp () is the same for a range of invariants useful in model based vision. The conjecture prompts the following definition: an invariant which has trade offs betweenR,F,p () of the above form is said to benon-degenerate for model based vision.The consequences of the trade off betweenR andp () are examined. In particular, it is shown that for a fixed overall probability of misclassification there is a maximum possible model cross ratio _m , and there is a maximum possible numberN of models. Approximate expressions for _m andN are obtained. They indicate that in practice a model database containing only cross ratio values can have a size of order at most ten, for a physically plausible level of image noise, and for a probability of misclassification of the order 0.1.     

Three-valued representative systems

A representative system defined onn voters or propositionsi = 1,,n is a functionF: {1,0, -1} ⁿ {1,0, -1} which is monotonic (D E F(D) F(E)), unanimous (F(1,, 1) = 1), dual (F(-D) = -F(D)), and satisfies a positivity property which says that the set of all non-zero vectors in {1, 0, -1} ⁿ for whichF(D) = 0 can be partitioned into two dual subsets each of which has the property that ifD andE are in the subset thenD _i+E _i > 0 for somei. Representative systems can be defined recursively from the coordinate projectionsS _i (D) = D _i using sign functions, and in this format they are interpreted as hierarchical voting systems in which outcomes of votes in lower levels act as votes in higher levels of the system. For each positive integern, (n) is defined as the smallest positive integer such that all representative systems defined on {1, 0, -1} ⁿ can be characterized by(n) or fewer hierarchical levels. The function is nondecreasing inn, unbounded above, and satisfies(n) n–1 for alln. In addition,(n) = n–1 forn {1, 2, 3, 4}, and it is conjectured that does not continue to grow linearly asn increases.     

The plane with parallel coordinates

By means ofParallel Coordinates planar graphs of multivariate relations are obtained. Certain properties of the relationship correspond tothe geometrical properties of its graph. On the plane a point line duality with several interesting properties is induced. A new duality betweenbounded and unbounded convex sets and hstars (a generalization of hyperbolas) and between Convex Unions and Intersections is found. This motivates some efficient Convexity algorithms and other results inComputational Geometry. There is also a suprising cusp inflection point duality. The narrative ends with a preview of the corresponding results inR ^N .     

A criterion for a-stability of linear multistep integration formulae

Summary Two easy-to-check conditions are given which, together, are sufficient and almost necessary forA-stability of linear multistep integration formulae. As an example, these conditions are applied to the two-parameter family of all two-step methods which are at least second-order accurate.

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Zusammenfassung Es werden zwei leicht nachprüfbare Bedingungen hergeleitet, welche fürA-Stabilität von linearen Mehrschrittverfahren hinreichend und fast notwendig sind. Als Beispiel werden diese Bedingungen auf die zweiparametrige Klasse aller Zweischrittverfahren von mindestens zweiter Ordnung angewendet.

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With 1 Figure     

Spectral Versus Pseudospectral Solutions of the Wave Equation by Waveform Relaxation Methods

We examine spectral and pseudospectral methods as well as waveform relaxation methods for the wave equation in one space dimension. Our goal is to study block Gauss–Jacobi waveform relaxation schemes which can be efficiently implemented in a parallel computing environment. These schemes are applied to semidiscrete systems written in terms of sparse or dense matrices. It is demonstrated that the spectral formulations lead to the implicit system of ordinary differential equations Wã = Sã + g(t)w, with sparse matrices W and S which can be effectively solved by direct application of any Runge–Kutta method. We also examine waveform relaxation iterations based on splittings W = W ₁–W ₂ and S = S ₁ + S ₂ and demonstrate that these iterations are only linearly convergent on finite time windows. Waveform relaxation methods applied to the explicit system ã = W ^–1 Sã + g(t)W ^–1 w are somewhat faster but less convenient to implement since the matrix W ^–1 S is no longer sparse. The pseudospectral methods lead to the system = D + g(t)w with a differentiation matrix D of order one and the corresponding waveform relaxation iterations are much faster than the iterations corresponding to the spectral cases (both implicit and explicit).     

A combinatorial property of EOL languages

Let be an alphabet and II its nontrivial binary partition. Each word over can uniquely be decomposed in subwords (called blocks) consisting of letters of II _i only,i {1,2}. LetK ^*.K has a long block property (with respect to II), abbreviated asLB-property, if there exists a functionf:N ⁺ N ⁺ such that for everyw K and every positive integerm the number of blocks of length at mostm inw is bounded byf(m). K has a clustered block property (with respect to II), abbreviated asCB-property, if there exists a positive integern _o and a growing functiong:N ⁺ N ⁺ such that for everyw K and for every positive integerm the blocks of length at mostm can be covered by at mostn _o segments of length at mostg(m).It is proved that aCB-property always implies aLB-property but not necessarily other way around. It is proved that an EOL language has aLB-property if and only if it has aCB-property.     

Sound and Complete Qualitative Simulation Needs “Quantitative” Filtering

The AI methodology of qualitative reasoning furnishes useful tools to scientists and engineers who need to deal with incomplete system knowledge during design, analysis, or diagnosis tasks. Qualitative simulators have a theoretical soundness guarantee; they cannot overlook any concrete equation implied by their input. On the other hand, the basic qualitative simulation algorithms have been shown to suffer from the incompleteness problem; they may allow non-solutions of the input equation to appear in their output. The question of whether a simulator with purely qualitative input which never predicts spurious behaviors can ever be achieved by adding new filters to the existing algorithm has remained unanswered. In this paper, we show that, if such a sound and complete simulator exists, it will have to be able to handle numerical distinctions with such a high precision that it must contain a component that would better be called a quantitative, rather than qualitative reasoner. This is due to the ability of the pure qualitative format to allow the exact representation of the members of a rich set of numbers.     

The mixing properties of certain processes related to Markov chains

Summary Let {Y _n ,– < n < } be either a function of a stationary Markov chain with countable state space, or a finitary process in the sense of Heller [3]. The purpose of this note is to prove that if {Y _n ,– < n < } is mixing, then it is aK-shift. (Definitions will be given below.)IfT is a measure-preserving transformation on a probability space, then the following implications relevant to the present paper are known: (1)T is aK-shift T is (r + 1)-mixing T isr-mixing T is totally ergodic T is ergodic, and (2)T is ergodic T is totally ergodic T isr-mixing T is aK-shift.It is not known if the classes ofr-mixing and (r + 1)-mixing transformations are distinct. (1-mixing is also called mixing.) The results of this note then imply that for the classes of transformation that we are consideringr-mixing and (r + 1)-mixing are equivalent.This research was partially supported by NSF Grant GP-19660.     

A Solution for the N-bit Parity Problem Using a Single Translated Multiplicative Neuron

A solution to the N-bit parity problem employing a single multiplicative neuron model, called translated multiplicative neuron ( _t -neuron), is proposed. The _t -neuron presents the following advantages: (a) N1, only 1 _t -neuron is necessary, with a threshold activation function and parameters defined within a specific interval; (b) no learning procedures are required; and (c) the computational cost is the same as the one associated with a simple McCulloch-Pitts neuron. Therefore, the _t -neuron solution to the N-bit parity problem has the lowest computational cost among the neural solutions presented to date.     

Mesh Selection for a Nearly Singular Boundary Value Problem

In this paper, we investigate the numerical solution of a model equation u _xx = exp(– ) (and several slightly more general problems) when 1 using the standard central difference scheme on nonuniform grids. In particular, we are interested in the error behaviour in two limiting cases: (i) the total mesh point number N is fixed when the regularization parameter 0, and (ii) is fixed when N. Using a formal analysis, we show that a generalized version of a special piecewise uniform mesh 12 and an adaptive grid based on the equidistribution principle share some common features. And the optimal meshes give rates of convergence bounded by |log()| as 0 and N is given, which are shown to be sharp by numerical tests.     

3-D interpretation of optical flow by renormalization

This article studies 3-D interpretation of optical flow induced by a general camera motion relative to a surface of general shape. First, we describe, using the image sphere representation, an analytical procedure that yields an exact solution when the data are exact: we solve theepipolar equation written in terms of theessential parameters and thetwisted optical flow. Introducing a simple model of noise, we then show that the solution is statistically biased. In order to remove the statistical bias, we propose an algorithm calledrenormalization, which automatically adjusts to unknown image noise. A brief discussion is also given to thecritical surface that yields ambiguous 3-D interpretations and the use of theimage plane representation.     

A Representation of the Interval Hull of a Tolerance Polyhedron Describing Inclusions of Function Values and Slopes

Given a nonempty set of functions

Transformation of programs for fault-tolerance

In this paper we describe how a program constructed for afault-free system can be transformed into afault-tolerant program for execution on a system which is susceptible to failures. A program is described by a set of atomic actions which perform transformations from states to states. We assume that a fault environment is represented by a programF. Interference by the fault environmentF on the execution of a programP can then be described as afault-transformation which transformsP into a program (P). This is proved to be equivalent to the programPP _F , whereP _F is derived fromP andF, and defines the union of the sets of actions ofP andF _P . A recovery transformation transformsP into a program (P) =PR by adding a set ofrecovery actions R, called arecovery program. If the system isfailstop and faults do not affect recovery actions, we have ((P))=(P)R=PP _F R We illustrate this approach to fault-tolerant programming by considering the problem of designing a protocol that guarantees reliable communication from a sender to a receiver in spite of faults in the communication channel between them.     

Directional differentiation of marginal functions in mathematical programming problems

We consider regular mathematical programming problems of the form f(x, y) inf, y F(x), x Rⁿ, where F(x) = {y R^m h_i| (x, y) 0, , h_i (x, y) = 0, . The directional derivatives offunctions (x) = inf{f(x, y)|y F(x)} are estimated.Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 70–77, November–December, 1991.     

Best Huffman trees

Summary Given a sequence of positive weights, W=w ₁...w _n >0, there is a Huffman tree, T (T-up) which minimizes the following functions: max{d(w_i)}; d(w_i); f(d(w_i)) w _i(here d(w _i) represents the distance of a leaf of weight w _i to the root and f is a function defined for nonnegative integers having the property that g(x) = f(x + 1) – f(x) is monotone increasing) over the set of all trees for W having minimal expected length. Minimizing the first two functions was first done by Schwartz [5]. In the case of codes where W is a sequence of probabilities, this implies that the codes based on T have all their absolute central moments minimal. In particular, they are the least variance codes which were also described by Kou [3]. Furthermore, there exists a Huffman tree T, (T-down) which maximizes the functions considered above.However, if g(x) is monotone decreasing, T and T, respectively maximize and minimize f(d(w_i) w _i) over the set of all trees for W having minimal expected length. In addition, we derive a number of interesting results about the distribution of labels within Huffman trees. By suitable modifications of the usual Huffman tree construction, (see [1]) T and T can also be constructed in time O(n log n).     

Computer-Aided Analysis of Transformation Formulas for Appel and Horn Functions

Simultaneous use of general and special linear transformations allows 36 transformations for the Appel function F ₄ to be obtained. This result is important because F ₄ is widely used in physics and mathematics. The transformations express F ₄ in terms of the Appel functions F ₁, F ₂, and F ₃; the Horn function H ₂; and non-Hornian series G, K, and . Until now, it was not possible to obtain linear multiplets having such a large dimensionality. A program based on Maple V4 was developed for formula generation. Details of program operation and possible applications of the results obtained are discussed.     

Eine bedingte Form der Integral-Ungleichung von Gronwall,leichte Verallgemeinerung eines Stabilitätssatzes von Perron und überlauffreie Skalierung von Analogrechenschaltungen

Zusammenfassung Es geht in dieser Arbeit in der Hauptsache darum, ein vorgelegtes Differentialgleichungssystem so zu skalieren, daß in der zugehörigen Analogrechnerschaltung die Spannungen an den Ausgängen der Integratoren die durch die Referenzspannung einerseits und durch das Auflösevermögen andererseits gesetzten Schranken nicht über- bzw. unterschreiten. Es werden Abschätzungssätze hergeleitet, die diese Frage im Apriori-Sinn, also ohne die Lösung des Differentialgleichungssystems zu kennen, zu lösen gestatten. Zur Abschätzung werden zunächst Normen, dannKamke-Normen verwendet. Der im Titel erwähnte Satz vonPerron ergibt sich durch spezielle Normengebung und Verzicht auf Abschätzung nach unten. Erschwert werden die Betrachtungen durch die relative Schwäche der Forderung, daß die rechte Seite des Systemsdx/dt=f(x,t) der Bedingung aus xa folgt f(x,t)v(t)x genüge (...:=Norm,a positiv reell). Dadurch scheint es bei Abschätzungen mitKamke-Normen nicht mehr möglich, von den in der Literatur über Existenzbeweise und Abschätzungssätze üblichen Methoden Gebrauch zu machen. Zur Lösung dieser Frage wird eine bedingte Form des bekannten Satzes vonGronwall (auch Satz vonBellman genannt) entwickelt.

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A conditional version of the integral inequality of gronwall, a slight generalization of a stability theorem of perron, and overflow-free scaling of analogue computer set-ups

Summary The main subject of this paper is the scaling of a given set of differential equations in such a way that the output voltages of the integrators of the associated analogue computer set-up do not exceed certain upper and lower bounds imposed by the reference voltage and the limited power of resolution of the elements of the analogue computer. The paper gives a priori bounds on the solution of the differential set. Some of these bounds work with norms, others withKamke-norms.Perron's stability theorem mentioned in the title of this paper results by inserting special norms and neglecting lower bounds. A difficulty arises by the relative weakness of the condition xa implies f(x,t)v(t)x on the right hand side of the setdx/dt=f(x,t), where ... is any norm anda is a positive real constant. As a consequence of this, it seems no longer possible to use the usual techniques known from the literature on existence theorems and bounds for the solution of differential equations. To cope with this situation, a conditional version of the well-known theorem ofGronwall (also known by the name of Lemma ofBellman) will be derived.

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Diese Arbeit ist Teil einer am Institut für Angewandte Mathematik der Technischen Hochschule München unter Anleitung von Herrn o. Prof. Dr. rer. nat. habil.J. Heinhold angefertigten Dissertation.     

A Delay-Averaged Logistic Model

For the equation x(t) = x(t) (1-(1/) _t-- ^t- x(u)du), > 0, > 0, > 0, conditions for the stability of a nonzero stationary solution under small perturbations are determined.     

Duality in Quantum Information Geometry

Let H be a separable Hilbert space. We consider the manifold M consisting of density operators on H such that ^p is of trace class for some p (0, 1). We say M is nearby if there exists C > 1 such that C ^–1C. We show that the space of nearby points to can be furnished with the two flat connections known as the (±)-affine structures, which are dual relative to the BKM metric. We furnish M with a norm making it into a Banach manifold.