Interpretations of open default theories in non-monotonic logics

It is shown that the translation of an open default into a modal formula x(L(x)LM ₁ (x)...LM _m (x)w(x)) gives rise to an embedding of open default systems into non-monotonic logics.     

Optimal multilayer channel routing with overlap

This paper presents algorithms for multiterminal net channel routing where multiple interconnect layers are available. Major improvements are possible if wires are able to overlap, and our generalized main algorithm allows overlap, but only on everyKth (K 2) layer. Our algorithm will, for a problem with densityd onL layers,L K + 3,provably use at most three tracks more than optimal: (d + 1)/L/K + 2 tracks, compared with the lower bound of d/L/K. Our algorithm is simple, has few vias, tends to minimize wire length, and could be used if different layers have different grid sizes. Finally, we extend our algorithm in order to obtain improved results for adjacent (K = 1) overlap: (d + 2)/2L/3 + 5 forL 7.This work was supported by the Semiconductor Research Corporation under Contract 83-01-035, by a grant from the General Electric Corporation, and by a grant at the University of the Saarland.     

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.     

Global optimization methods for engineering applications: A review

A review of the methods for global optimization reveals that most methods have been developed for unconstrained problems. They need to be extended to general constrained problems because most of the engineering applications have constraints. Some of the methods can be easily extended while others need further work. It is also possible to transform a constrained problem to an unconstrained one by using penalty or augmented Lagrangian methods and solve the problem that way. Some of the global optimization methods find all the local minimum points while others find only a few of them. In any case, all the methods require a very large number of calculations. Therefore, the computational effort to obtain a global solution is generally substantial. The methods for global optimization can be divided into two broad categories: deterministic and stochastic. Some deterministic methods are based on certain assumptions on the cost function that are not easy to check. These methods are not very useful since they are not applicable to general problems. Other deterministic methods are based on certain heuristics which may not lead to the true global solution. Several stochastic methods have been developed as some variation of the pure random search. Some methods are useful for only discrete optimization problems while others can be used for both discrete and continuous problems. Main characteristics of each method are identified and discussed. The selection of a method for a particular application depends on several attributes, such as types of design variables, whether or not all local minima are desired, and availability of gradients of all the functions.Notation Number of equality constraints - () ^T A transpose of a vector - A A hypercubic cell in clustering methods - Distance between two adjacent mesh points - Probability that a uniform sample of sizeN contains at least one point in a subsetA ofS - A(v, x) Aspiration level function - A The set of points with cost function values less thanf(x _G ^* ) +. Same asA _f () - A _f () A set of points at which the cost function value is within off(x _G ^* ) - A () A set of points x with[f(x)] smaller than - A _N The set ofN random points - A _q The set of sample points with the cost function value f _q - _Q The contraction coefficient; –1 _Q 0 - _R The expansion coefficient; _E > 1 - _R The reflection coefficient; 0 < _R 1 - A _x () A set of points that are within the distance from x _G ^* - D Diagonal form of the Hessian matrix - det() Determinant of a matrix - d _j A monotonic function of the number of failed local minimizations - d _t Infinitesimal change in time - d _x Infinitesimal change in design - A small positive constant - (t) A real function called the noise coefficient - ₀ Initial value for(t) - exp() The exponential function - f ^(c) The record; smallest cost function value over X^(C) - [f(x)] Functional for calculating the volume fraction of a subset - Second-order approximation tof(x) - f(x) The cost function - An estimate of the upper bound of global minimum - f ^E The cost function value at x^E - f ^L The cost function value at x^L - f _opt The current best minimum function value - f ^P The cost function value at x ^P - f ^Q The cost function value at x ^Q - f _q A function value used to reduce the random sample - f ^R The cost function value at x ^R - f ^S The cost function value at x^S - f _{T F min} A common minimum cost function value for several trajectories - f _{TF opt} The best current minimum value found so far forf _{TF min} - f ^W The cost function value at x ^W - G Minimum number of points in a cell (A) to be considered full - The gamma function - A factor used to scale the global optimum cost in the zooming method - Minimum distance assumed to exist between two local minimum points - gi(x) Constraints of the optimization problem - H The size of the tabu list - H(x^*) The Hessian matrix of the cost function at x^* - h _j Half side length of a hypercube - h _m Minimum half side lengths of hypercubes in one row - I The unity matrix - ILIM A limit on the number of trials before the temperature is reduced - J The set of active constraints - K Estimate of total number of local minima - k Iteration counter - The number of times a clustering algorithm is executed - L Lipschitz constant, defined in Section 2 - L The number of local searches performed - _i The corresponding pole strengths - log () The natural logarithm - LS Local search procedure - M Number of local minimum points found inL searches - m Total number of constraints - m(t) Mass of a particle as a function of time - m() TheLebesgue measure of thea set - Average cost value for a number of random sample of points inS - N The number of sample points taken from a uniform random distribution - n Number of design variables - n(t) Nonconservative resistance forces - n _c Number of cells;S is divided inton _c cells - NT Number of trajectories - Pi (3.1415926) - P _i ^(j) Hypersphere approximating thej-th cluster at stagei - p(x ⁽ⁱ⁾⁾ Boltzmann-Gibbs distribution; the probability of finding the system in a particular configuration - pg A parameter corresponding to each reduced sample point, defined in (36) - Q An orthogonal matrix used to diagonalize the Hessian matrix - _i (i = 1, K) The relative size of thei-th region of attraction - r _i ^(j) Radius of thej-th hypersp here at stagei - R _x ^* Region of attraction of a local minimum x^* - r _j Radius of a hypersphere - r A critical distance; determines whether a point is linked to a cluster - R ⁿ A set ofn tuples of real numbers - A hyper rectangle set used to approximateS - S The constraint set - A user supplied parameter used to determiner - s The number of failed local minimizations - T The tabu list - t Time - T(x) The tunneling function - T _c (x) The constrained tunneling function - T _i The temperature of a system at a configurationi - TLIMIT A lower limit for the temperature - TR A factor between 0 and 1 used to reduce the temperature - u(x) A unimodal function - V(x) The set of all feasible moves at the current design - v(x) An oscillating small perturbation. - V(y⁽ⁱ⁾) Voronoi cell of the code point y⁽ⁱ⁾ - v^–1 An inverse move - v _k A move; the change from previous to current designs - w(t) Ann-dimensional standard. Wiener process - x Design variable vector of dimensionn - x^# A movable pole used in the tunneling method - x⁽⁰⁾ A starting point for a local search procedure - X^(c) A sequence of feasible points {x⁽¹⁾, x⁽²⁾,,x^(c)} - x(t) Design vector as a function of time - X^* The set of all local minimum points - x^* A local minimum point forf(x) - x^*(i) Poles used in the tunneling method - x _G ^* A global minimum point forf(x) - Transformed design space - The velocity vector of the particle as a function of time - Acceleration vector of the particle as a function of time - x ^C Centroid of the simplex excluding x ^L - x ^c A pole point used in the tunneling method - x ^E An expansion point of x ^R along the direction x ^C x ^R - x ^L The best point of a simplex - x ^P A new trial point - x ^Q A contraction point - x ^R A reflection point; reflection of x ^W on x ^C - x ^S The second worst point of a simplex - x ^W The worst point of a simplex - The reduced sample point with the smallest function value of a full cell - Y The set of code points - y ⁽ⁱ⁾ A code point; a point that represents all the points of thei-th cell - z A random number uniformly distributed in (0,1) - Z ^(c) The set of points x where [f ^(c) – ] is smaller thanf(x) - []₊ Max (0,) - | | Absolute value - The Euclidean norm - f[x(t)] The gradient of the cost function     

Shape optimization of elasto-plastic bodies under plane strains: Sensitivity analysis and numerical implementation

Optimal shape design problems for an elastic body made from physically nonlinear material are presented. Sensitivity analysis is done by differentiating the discrete equations of equilibrium. Numerical examples are included.Notation U ^ad set of admissible continuous design parameters - U _h ^ad set of admissible discrete design parameters - function fromU _h ^ad defining shape of body - _h function fromU _h ^ad defining approximated shape of body - vector of nodal values of _h - { _n} sequence of functions tending to - () domain defined by - K bulk modulus - shear modulus - penalty parameter for contact condition - V() space of virtual displacements in() - V _h(_h) finite element approximation ofV() - J cost functional - J _h discretized cost functional - J algebraic form ofJ _h - (u) stress tensor - e(u) strain tensor - K stiffness matrix - f force vector - b(q) term arising from nonlinear boundary conditions - q vector of nodal degrees of freedom - p vector of adjoint state variables - J Jacobian of isoparametric mapping - |J| determinant ofJ - N vector of shape function values on parent element - L matrix of shape function derivatives on parent element - G matrix of Cartesian derivatives of shape functions - X matrix of nodal coordinates of element - D matrix of elastic coefficients - B strain-displacement matrix - _P part of boundary where tractions are prescribed - _u part of boundary where displacements are prescribed - variable part of boundary - strain invariant     

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)     

Enhancing fault-tolerance in rate-monotonic scheduling

In this paper, we address the problem of supporting timeliness and dependability at the level of task scheduling. We consider the problem of scheduling a set of tasks, each of which, for fault-tolerance purposes, has multiple versions, onto the minimum number of processors. On each individual processor, the tasks are guaranteed their deadlines by the Rate-Monotonic algorithm. A simple online allocation heuristic is proposed. It is proven thatN2.33N ₀+, whereN is the number of processors required to feasibly schedule a set of tasks by the heuristic,N ₀ is the minimum number of processors required to feasibly schedule the same set of tasks, and is the maximum redundancy degree a task can have. The bound is also shown to be a tight upper bound. The average-case performance of the heuristic is studied through simulation. It is shown that the heuristic performs surprisingly well on the average.     

Hyperviscous shock layers and diffusion zones: Monotonicity,spectral viscosity,and pseudospectral methods for very high order differential equations

We solve two problems ofx[–, ] for arbitrary orderj. The first is to compute shock-like solutions to the hyperdiffusion equation,u1=(–1) ^j+1 u _2j,x. The second is to compute similar solutions to the stationary form of the hyper-Burgers equation, (–1) ^j u _2j.x+uu _x=0; these tanh-like solutions are asymptotic approximations to the shocks of the corresponding time dependent equation. We solve the hyperdiffusion equation with a Fourier integral and the method of steepest descents. The hyper Burgers equation is solved by a Fourier pseudospectral method with a polynomial subtraction.Except for the special case of ordinary diffusion (j=1), the jump across the shock zone is described bynonmonotonic, oscillatory functions. By smearing the front over the width of a grid spacing, it is possible to numerically resolve the shock with a weaker and weaker viscosity coefficient asj, the order of the damping, increases. This makes such hyperviscous dampings very attractive for coping with fronts since, outside the frontal zone, the impact of the artificial hyperviscosity is much smaller than with ordinary viscosity. Unfortunately, both the intensity of the oscillations and the slowness of their exponential decay from the center of the shock zone decrease asj increases so that the shock zone is muchwider than for ordinary diffusion. We also examined generalizations of Burgers equation with spectral viscosity, that is, damping which is tailored to yield exponentially small errors outside the frontal zone when combined with spectral methods. We find behavior similar to high order hyperviscosity.We conclude that high order damping, as a tool for shock-capturing, offers both advantages and drawbacks. Monotonicity, which has been the holy grail of so much recent algorithm development, is a reasonable goal only for ordinary viscosity. Hyperviscous fronts and shock zones in flows with spectral viscosity aresupposed to oscillate.     

A note on dense and nondense families of complexity classes

In a model for a measure of computational complexity, , for a partial recursive functiont, letR _t denote all partial recursive functions having the same domain ast and computable within timet. Let = {R _t |t is recursive} and let = { |_i is actually the running time function of a computation}. and are partially ordered under set-theoretic inclusion. These partial orderings have been extensively investigated by Borodin, Constable and Hopcroft in [3]. In this paper we present a simple uniform proof of some of their results. For example, we give a procedure for easily calculating a model of computational complexity for which is not dense while is dense. In our opinion, our technique is so transparent that it indicates that certain questions of density are not intrinsically interesting for general abstract measures of computational complexity, . (This is not to say that similar questions are necessarily uninteresting for specific models.)Supported by NSF Research Grants GP6120 and GJ27127.     

On generators and generative capacity of EOL forms

Summary In this paper we study the generative capacity of EOL forms from two different points of view. On the one hand, we consider the generative capacity of special EOL forms which one could call linear like and context free like, establishing the existence of a rich variety of non-regular sub-EOL language families. On the other hand, we propose the notion of a generator L of a language family We mean by this that any synchronized EOL system generating L generates — if understood as an EOL form — all languages of . We characterize the generators of the family of regular languages, and prove that other well known language families do not have generators.Partially supported under NSE Research Council of Canada, grant No. A-7700     

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.     

Representation of Algebraic Functions As Power and Fractional Power Series of Special Type

A Maple procedure is described by means of which an algebraic function given by an equation f(x y) = 0 can be expanded into a fractional power series (Puiseux series)

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.     

Long unavoidable patterns

Summary We examine long unavoidable patterns, unavoidable in the sense of Bean, Ehrenfeucht, McNulty. Zimin and independently Schmidt have shown that there is only one unavoidable pattern of length 2 ⁿ -1 on an alphabet with n letters; this pattern is a quasi-power in the sense of Schützenberger. We characterize the unavoidable words of length 2 ⁿ -2 and 2 ⁿ -3. Finally we show that every sufficiently long unavoidable word has a certain quasi-power as a subword.This work was done while the author stayed at LITP, Université Paris 6, France     

Linear Interval Tolerance Problem and Linear Programming Techniques

In this paper, we consider the linear interval tolerance problem, which consists of finding the largest interval vector included in ([A], [b]) = {x R ⁿ | A [A], b [b], Ax = b}. We describe two different polyhedrons that represent subsets of all possible interval vectors in ([A], [b]), and we provide a new definition of the optimality of an interval vector included in ([A], [b]). Finally, we show how the Simplex algorithm can be applied to find an optimal interval vector in ([A], [b]).     

A Hyperbase for Binary Lattice Hyperidentities

We define an identity to be hypersatisfied by a variety V if, whenever the operation symbols of V, are replaced by arbitrary terms (of appropriate arity) in the operations of V, the resulting identity is satisfied by V in the usual sense. Whenever the identity is hypersatisfied by a variety V, we shall say that is a V hyperidentity. For example, the identity x + x y = x (x + y) is hypersatisfied by the variety L of all lattices. A proof of this consists of a case-by-case examination of { + , } {x, y, x y, x y}, the set of all binary lattice terms. In an earlier work, we exhibited a hyperbase ^L for the set of all binary lattice (or, equivalently, quasilattice) hyperidentities of type 2, 2. In this paper we provide a greatly refined hyperbase _L . The proof that _L is a hyperbase was obtained by using the automated reasoning program Otter 3.0.4.     

Data structures and mesh modification tools for unstructured multigrid adaptive techniques

A complete set of data structures and mesh modification tools for effectively defining unstructured threedimensional multigrids on general curved domains is presented. The mesh adaptive procedures can be used for generating hierarchies of unstructured grids by means of uniform or local refinement and coarsening, while a local retriangulation algorithm is used for controlling the degradation of the quality of the mesh during adaptation. Intergrid transfer operators are efficiently realized on the fly during adaptation. The data structure allows the efficient storage and handling of multiple grids, where mesh entities belonging to multiple levels can be stored just once. The capabilities and performance of the proposed procedures are exemplified by means of examples.Nomenclature g Geometric model - m _r Mesh model at levelr - v Domain associated with modelv (v=g,m) - (v) Boundary of modelv - Closure of domain of modelv (v(v)) - G _i ^d j Topological entityi of dimensiond _j in the geometric model - M _i ^d j _(p,q) Topological entityi of dimensiond _j in the mesh model, appearing at mesh levelsp throughq - M _i ^d j _(r) Topological entityi in the mesh model of dimensiond _j , considered at mesh levelr - (M _i ^d j) Boundary of topological entity - [·] Ordered list of entities - {·} Unordered list of entities - M _i ^d j _(r) {M ^d j} Unordered group of topological entities of dimensiond _j that are adjacent toM _i ^d j _(r) at mesh levelr - Classification, i.e. association of a topological entity with a model entity     

Application of optimization methods to helicopter rotor blade design

The minimum weight design of helicopter rotor blades with constraints on multiple coupled flap-lag natural frequencies is studied in this paper. A constraint is also imposed on the minimum value of the autorotational inertia of the blade to ensure sufficient autorotational inertia to autorotate in case of an engine failure. A stress constraint is used to guard against structural failure due to blade centrifugal forces. Design variables include blade taper ratio, dimensions of the box beam located inside the airfoil and magnitudes of the nonstructural weights. The program CAMRAD is used for the blade modal analysis and the program CONMIN for the optimization. In addition, a linear approximation involving Taylor series expansion is used to reduce the analysis effort. The procedure contains a sensitivity analysis which consists of analytical derivatives of the objective function, the autorotational inertia constraint and the stress constraints. A central finite difference scheme is used for the derivatives of the frequency constraints. Optimum designs are obtained for both rectangular and tapered blades. The paper also discusses the effect of adding constraints on higher frequencies and stresses on the optimum designs. b box beam width - c chord - f ₁,f ₃,f ₄ first three lead-lag dominated frequencies (elastic modes) - f ₂,f ₅ first two flapping dominated frequencies (elastic modes) - g constraint function - h box beam height - h(z) box beam height variation along blade span - n number of blades - r _j distance from the root to the center of thej-th segment - t ₁,t ₂,t ₃ box beam wall thicknesses - x, y, z reference axes - A box beam cross-sectional area - AI autorotational inertia - E Young's modulus - F objective function - FS factor of safety - GJ torsional stiffness - I _x ,I _y total principal area moments of inertia about reference axes - L _j length ofj-th segment - M _j total mass ofj-th segment - N total number of blade segments - NDV number of design variables - R blade radius - W total blade weight - W() blade weight as a function of design variable - W _b box beam weight - W _o nonstructural blade weight (weight of skin, honeycomb, etc. along with tuning/lumped weights) - prescribed autorotational inertia - design variable increment - _h taper ratio inz direction - _i i-th design variable - _j mass density of thej-th segment - _j stress inj-th segment - _max maximum allowable stress - blade RPM - r root value - t tip value - L lower bound - U upper bound - ^ approximate value This paper is declared a work of the U.S. Government and is not subject to copyright protection in the United States     

Systems of Strings with High Mutual Complexity

Consider a binary string x ₀ of Kolmogorov complexity K(x ₀) n. The question is whether there exist two strings x ₁ and x ₂ such that the approximate equalities K(x _i x _j ) n and K(x _i x _j , x _k ) n hold for all 0 i, j, k 2, i j k, i k. We prove that the answer is positive if we require the equalities to hold up to an additive term O(log K(x ₀)). It becomes negative in the case of better accuracy, namely, O(log n).     

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.