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Mechanised reasoning systems and computer algebra systems have apparentlydifferent objectives. Their integration is, however, highly desirable, since in manyformal proofs both of the two different tasks, proving and calculating, have to beperformed. Even more importantly, proof and computation are often interwoven andnot easily separable. In the context of producing reliable proofs, the question howto ensure correctness when integrating a computer algebra system into a mechanisedreasoning system is crucial. In this contribution, we discuss the correctness prob-lems that arise from such an integration and advocate an approach in which thecalculations of the computer algebra system are checked at the calculus level of themechanised reasoning system. This can be achieved by adding a verbose mode to thecomputer algebra system which produces high-level protocol information that can beprocessed by an interface to derive proof plans. Such a proof plan in turn can beexpanded to proofs at different levels of abstraction, so the approach is well-suited forproducing a high-level verbalised explication as well as for a low-level machine check-able calculus-level proof. We present an implementation of our ideas and exemplifythem using an automatically solved extended example.
We propose a specification language for the formalization of data types with par-tial or non-terminating operations as part of a rewrite-based logical frameworkfor inductive theorem proving. The language requires constructors for designat-ing data items and admits positive/negative conditional equations as axioms inspecifications. The (total algebra) semantics for such specifications is based onso-called data models. We present admissibility conditions that guarantee theunique existence of a distinguished data model with properties similar to thoseof the initial model of a usual equational specification. Since admissibility of aspecification requires confluence of the induced rewrite relation, we provide aneffectively testable confluence criterion which does not presuppose termination.
To prove difficult theorems in a mathematical field requires substantial know-ledge of that field. In this paper a frame-based knowledge representation formalismis presented, which supports a conceptual representation and to a large extent guar-antees the consistency of the built-up knowledge bases. We define a semantics ofthe representation by giving a translation into the underlaying logic.
The amount of user interaction is the prime cause of costs in interactiveprogram verification. This paper describes an internal analogy techniquethat reuses subproofs in the verification of state-based specifications. Itidentifies common patterns of subproofs and their justifications in orderto reuse these subproofs; thus significant savings on the number of userinteractions in a verification proof are achievable.
We present an empirical study of mathematical proofs by diagonalization, the aim istheir mechanization based on proof planning techniques. We show that these proofs canbe constructed according to a strategy that (i) finds an indexing relation, (ii) constructsa diagonal element, and (iii) makes the implicit contradiction of the diagonal elementexplicit. Moreover we suggest how diagonal elements can be represented.
The problem of providing connectivity for a collection of applications is largely one of data integration: the communicating parties must agree on thesemantics and syntax of the data being exchanged. In earlier papers [#!mp:jsc1!#,#!sg:BSG1!#], it was proposed that dictionaries of definitions foroperators, functions, and symbolic constants can effectively address the problem of semantic data integration. In this paper we extend that earlier work todiscuss the important issues in data integration at the syntactic level and propose a set of solutions that are both general, supporting a wide range of dataobjects with typing information, and efficient, supporting fast transmission and parsing.
Chains of Recurrences (CRs) are a tool for expediting the evaluation of elementary expressions over regular grids. CR based evaluations of elementaryexpressions consist of 3 major stages: CR construction, simplification, and evaluation. This paper addresses CR simplifications. The goal of CRsimplifications is to manipulate a CR such that the resulting expression is more efficiently to evaluate. We develop CR simplification strategies which takethe computational context of CR evaluations into account. Realizing that it is infeasible to always optimally simplify a CR expression, we give heuristicstrategies which, in most cases, result in a optimal, or close-to-optimal expressions. The motivations behind our proposed strategies are discussed and theresults are illustrated by various examples.
Algorithms in Singular
(1999)
Top-down and bottom-up theorem proving approaches have each specific ad-vantages and disadvantages. Bottom-up provers profit from strong redundancycontrol and suffer from the lack of goal-orientation, whereas top-down provers aregoal-oriented but have weak calculi when their proof lengths are considered. Inorder to integrate both approaches our method is to achieve cooperation betweena top-down and a bottom-up prover: The top-down prover generates subgoalclauses, then they are processed by a bottom-up prover. We discuss theoreticaspects of this methodology and we introduce techniques for a relevancy-basedfiltering of generated subgoal clauses. Experiments with a model eliminationand a superposition-based prover reveal the high potential of our cooperation approach.The author was supported by the Deutsche Forschungsgemeinschaft (DFG).
We examine an approach for demand-driven cooperative theorem proving.We briefly point out the problems arising from the use of common success-driven cooperation methods, and we propose the application of our approachof requirement-based cooperative theorem proving. This approach allows for abetter orientation on current needs of provers in comparison with conventional co-operation concepts. We introduce an abstract framework for requirement-basedcooperation and describe two instantiations of it: Requirement-based exchangeof facts and sub-problem division and transfer via requests. Finally, we reporton experimental studies conducted in the areas superposition and unfailing com-pletion.The author was supported by the Deutsche Forschungsgemeinschaft (DFG).
HOT is an automated higher-order theorem prover based on HTE, an extensional higher-order tableaux calculus (Kohlhase 95). The first part of the paper introduces a variant of the calculus which closely corresponds to the proof procedure implemented in HOT. The second part discusses HOT's design that can be characterized as a concurrent Blackboard architecture. We show the usefulness of the implementation by including benchmark results for over one hundred solved problems from logic and set theory.
Orderings on polynomial interpretations of operators represent a powerful technique for proving thetermination of rewriting systems. One of the main problems of polynomial orderings concerns thechoice of the right interpretation for a given rewriting system. It is very difficult to develop techniquesfor solving this problem. Here, we present three new heuristic approaches: (i) guidelines for dealingwith special classes of rewriting systems, (ii) an algorithm for choosing appropriate special polynomialsas well as (iii) an extension of the original polynomial ordering which supports the generation ofsuitable interpretations. All these heuristics will be applied to examples in order to illustrate theirpractical relevance.
We study families V of curves in P2(C) of degree d having exactly r singular points of given topological or analytic types. We derive new sufficient conditions for V to be T-smooth (smooth of the expected dimension), respectively to be irreducible. For T-smoothness these conditions involve new invariants of curve singularities and are conjectured to be asymptotically proper, i.e., optimal up to a constant factor. To obtain the results, we study the Castelnuovo function, prove the irreducibility of the Hilbert scheme of zero-dimensional schemes associated to a cluster of infinitely near points of the singularities and deduce new vanishing theorems for ideal sheaves of zero-dimensional schemes in P2. Moreover, we give a series of examples of cuspidal curves where the family V is reducible, but where ss1(P2nC) coincides (and is abelian) for all C 2 V .
After the notion of Gröbner bases and an algorithm for constructing them was introduced by Buchberger [Bu1, Bu2] algebraic geometers have used Gröbner bases as the main computational tool for many years, either to prove a theorem or to disprove a conjecture or just to experiment with examples in order to obtain a feeling about the structure of an algebraic variety. Nontrivial problems coming either from logic, mathematics or applications usually lead to nontrivial Gröbner basis computations, which is the reason why several improvements have been provided by many people and have been implemented in general purpose systems like Axiom, Maple, Mathematica, Reduce, etc., and systems specialized for use in algebraic geometry and commutative algebra like CoCoA, Macaulay and Singular. The present paper starts with an introduction to some concepts of algebraic geometry which should be understood by people with (almost) no knowledge in this field. In the second chapter we introduce standard bases (generalization of Gr"obner bases to non-well-orderings), which are needed for applications to local algebraic geometry (singularity theory), and a method for computing syzygies and free resolutions. The last chapter describes a new algorithm for computing the normalization of a reduced affine ring and gives an elementary introduction to singularity theory. Then we describe algorithms, using standard bases, to compute infinitesimal deformations and obstructions, which are basic for the deformation theory of isolated singularities. It is impossible to list all papers where Gr"obner bases have been used in local and global algebraic geometry, and even more impossible to give an overview about these contributions. We have, therefore, included only references to papers mentioned in this tutorial paper. The interested reader will find many more in the other contributions of this volume and in the literature cited there.
Algorithmic ideal theory
(1999)
Algebraic geometers have used Gröbner bases as the main computational tool for many years, either to prove a theorem or to disprove a conjecture or just to experiment with examples in order to obtain a feeling about the structure of an algebraic variety. Non-trivial mathematical problems usually lead to non-trivial Gröbner basis computations, which is the reason why several improvements and efficient implementations have been provided by algebraic geometers (for example, the systems CoCoA, Macaulay and SINGULAR). The present paper starts with an introduction to some concepts of algebraic geometry which should be understood by people with (almost) no knowledge in this field. In the second chapter we introduce standard bases (generalization of Gröbner bases to non-well-orderings), which are needed for applications to local algebraic geometry (singularity theory), and a method for computing syzygies and free resolutions. In the third chapter several algorithms for primary decomposition of polynomial ideals are presented, together with a discussion of improvements and preferable choices. We also describe a newly invented algorithm for computing the normalization of a reduced affine ring. The last chapter gives an elementary introduction to singularity theory and then describes algorithms, using standard bases, to compute infinitesimal deformations and obstructions, which are basic for the deformation theory of isolated singularities. It is impossible to list all papers where Gröbner basis have been used in local and global algebraic geometry, and even more impossible to give an overview about these contributions. We have, therefore, included only a few references to papers which contain interesting applications and which are not mentioned in this tutorial paper. The interested reader will find many more in the other contributions of this volume and in the literature cited there.
Singular algebraic curves, their existence, deformation, families (from the local and global point of view) attract continuous attention of algebraic geometers since the last century. The aim of our paper is to give an account of results, new trends and bibliography related to the geometry of equisingular families of algebraic curves on smooth algebraic surfaces over an algebraically closed field of characteristic zero. This theory is founded in basic works of Plücker, Severi, Segre, Zariski, and has tight links and finds important applications in singularity theory, topology of complex algebraic curves and surfaces, and in real algebraic geometry.
If \(A\) generates a bounded cosine function on a Banach space \(X\) then the negative square root \(B\) of \(A\) generates a holomorphic semigroup, and this semigroup is the conjugate potential transform of the cosine function. This connection is studied in detail, and it is used for a characterization of cosine function generators in terms of growth conditions on the semigroup generated by \(B\). This characterization relies on new results on the inversion of the vector-valued conjugate potential transform.
Let \(X\) be a Banach lattice. Necessary and sufficient conditions for a linear operator \(A:D(A) \to X\), \(D(A)\subseteq X\), to be of positive \(C^0\)-scalar type are given. In addition, the question is discussed which conditions on the Banach lattice imply that every operator of positive \(C^0\)-scalar type is necessarily of positive scalar type.
We consider regularizing iterative procedures for ill-possed problems with random and nonrandom additive errors. The rate of square-mean convergence for iterative procedures with random errors is studied. The comparison theorem is established for the convergence of procedures with and without additive errors.
Vigenere-Verschlüsselung
(1999)
The computational complexity of combinatorial multiple objective programming problems is investigated. NP-completeness and #P-completeness results are presented. Using two definitions of approximability, general results are presented, which outline limits for approximation algorithms. The performance of the well known tree and Christofides' heuristics for the TSP is investigated in the multicriteria case with respect to the two definitions of approximability.
Convex Operators in Vector Optimization: Directional Derivatives and the Cone of Decrease Directions
(1999)
The paper is devoted to the investigation of directional derivatives and the cone of decrease directions for convex operators on Banach spaces. We prove a condition for the existence of directional derivatives which does not assume regularity of the ordering cone K. This result is then used to prove that for continuous convex operators the cone of decrease directions can be represented in terms of the directional derivatices . Decrease directions are those for which the directional derivative lies in the negative interior of the ordering cone K. Finally, we show that the continuity of the convex operator can be replaced by its K-boundedness.
The notion of the balance number introduced in [3,page 139] through a certain set contraction procedure for nonscalarized multiobjective global optimization is represented via a min-max operation on the data of the problem. This representation yields a different computational procedure for the calculation of the balance number and allows us to generalize the approach for problems with countably many performance criteria.
In this paper we consider the problem of locating one new facility in the plane with respect to a given set of existing facility where a set of polygonal barriers restricts traveling. This non-convex optimization problem can be reduced to a finite set of convex subproblems if the objective function is a convex function of the travel distances between the new and the existing facilities (like e.g. the Median and Center objective functions). An exact Algorithm and a heuristic solution procedure based on this reduction result are developed.
Let rC and rD be two convexdistance funtions in the plane with convex unit balls C and D. Given two points, p and q, we investigate the bisector, B(p,q), of p and q, where distance from p is measured by rC and distance from q by rD. We provide the following results. B(p,q) may consist of many connected components whose precise number can be derived from the intersection of the unit balls, C nd D. The bisector can contain bounded or unbounded 2-dimensional areas. Even more surprising, pieces of the bisector may appear inside the region of all points closer to p than to q. If C and D are convex polygons over m and m vertices, respectively, the bisector B(p,q) can consist of at most min(m,n) connected components which contain at most 2(m+n) vertices altogether. The former bound is tight, the latter is tight up to an additive constant. We also present an optimal O(m+n) time algorithm for computing the bisector.
In planar location problems with barriers one considers regions which are forbidden for the siting of new facilities as well as for trespassing. These problems areimportant since they reflect various real-world situations.The resulting mathematical models have a non-convex objectivefunction and are therefore difficult to tackle using standardmethods of location theory even in the case of simple barriershapes and distance funtions.For the case of center objectives with barrier distancesobtained from the rectilinear or Manhattan metric it is shown that the problem can be solved by identifying a finitedominating set (FDS) the cardinality of which is bounded bya polynomial in the size of the problem input. The resultinggenuinely polynomial algorithm can be combined with bound computations which are derived from solving closely connectedrestricted location and network location problems.It is shown that the results can be extended to barrier center problems with respect to arbitrary block norms having fourfundamental directions.
In this paper we deal with an NP-hard combinatorial optimization problem, the k-cardinality tree problem in node weighted graphs. This problem has several applications , which justify the need for efficient methods to obtain good solutions. We review existing literature on the problem. Then we prove that under the condition that the graph contains exactly one trough, the problem can be solved in ploynomial time. For the general NP-hard problem we implemented several local search methods to obtain heuristics solutions, which are qualitatively better than solutions found by constructive heuristics and which require significantly less time than needed to obtain optimal solutions. We used the well known concepts of genetic algorithms and tabu search with useful extensions. We show that all the methods find optimal solutions for the class of graphs containing exactly one trough. The general performance of our methods as compared to other heuristics is illustrated by numerical results.
Given a finite set of points in the plane and a forbidden region R, we want to find a point X not an element of int(R), such that the weighted sum to all given points is minimized. This location problem is a variant of the well-known Weber Problem, where we measure the distance by polyhedral gauges and allow each of the weights to be positive or negative. The unit ball of a polyhedral gauge may be any convex polyhedron containing the origin. This large class of distance functions allows very general (practical) settings - such as asymmetry - to be modeled. Each given point is allowed to have its own gauge and the forbidden region R enables us to include negative information in the model. Additionally the use of negative and positive weights allows to include the level of attraction or dislikeness of a new facility. Polynomial algorithms and structural properties for this global optimization problem (d.c. objective function and a non-convex feasible set) based on combinatorial and geometrical methods are presented.
In this paper a new trend is introduced into the field of multicriteria location problems. We combine the robustness approach using the minmax regret criterion together with Pareto-optimality. We consider the multicriteria Weber location problem which consists of simultaneously minimizing a number of weighted sum-distance functions and the set of Pareto-optimal locations as its solution concept. For this problem, we characterize the Pareto-optimal solutions within the set of robust locations for the original weighted sum-distance functions. These locations have both the properties of stability and non-domination which are required in robust and multicriteria programming.
Complete presentations provide a natural solution to the word problem in monoids and groups. Here we give a simple way to construct complete presentations for the direct product of groups, when such presentations are available for the factors. Actually, the construction we are referring to is just the classical construction for direct products of groups, which has been known for a long time, but whose completeness-preserving properties had not been detected. Using this result and some known facts about Coxeter groups, we sketch an algorithm to obtain the complete presentation of any finite Coxeter group. A similar application to Abelian and Hamiltonian groups is mentioned.
Compared to standard numerical methods for hyperbolic systems of conservation laws, Kinetic Schemes model propagation of information by particles instead of waves. In this article, the wave and the particle concept are shown to be closely related. Moreover, a general approach to the construction of Kinetic Schemes for hyperbolic conservation laws is given which summarizes several approaches discussed by other authors. The approach also demonstrates why Kinetic Schemes are particularly well suited for scalar conservation laws and why extensions to general systems are less natural.
Nonlinear dissipativity, asymptotical stability, and contractivity of (ordinary) stochastic differential equations (SDEs) with some dissipative structure and their discretizations are studied in terms of their moments in the spirit of Pliss (1977). For this purpose, we introduce the notions and discuss related concepts of dissipativity, growth- bounded and monotone coefficient systems, asymptotical stability and contractivity in wide and narrow sense, nonlinear A-stability, AN-stability, B-stability and BN-stability for stochastic dynamical systems - more or less as stochastic counterparts to deterministic concepts. The test class of in a broad sense interpreted dissipative SDEs as natural analogon to dissipative deterministic differential systems is suggested for stochastic-numerical methods. Then, in particular, a kind of mean square calculus is developed, although most of ideas and analysis can be carried over to general "stochastic Lp-case" (p * 1). By this natural restriction, the new stochastic concepts are theoretically meaningful, as in deterministic analysis. Since the choice of step sizes then plays no essential role in related proofs, we even obtain nonlinear A-stability, AN-stability, B-stability and BN-stability in the mean square sense for this implicit method with respect to appropriate test classes of moment-dissipative SDEs.
Nonlinear stochastic dynamical systems as ordinary stochastic differential equations and stochastic difference methods are in the center of this presentation in view of the asymptotical behaviour of their moments. We study the exponential p-th mean growth behaviour of their solutions as integration time tends to infinity. For this purpose, the concepts of nonlinear contractivity and stability exponents for moments are introduced as generalizations of well-known moment Lyapunov exponents of linear systems. Under appropriate monotonicity assumptions we gain uniform estimates of these exponents from above and below. Eventually, these concepts are generalized to describe the exponential growth behaviour along certain Lyapunov-type functionals.
In this paper we discuss a special class of regularization methods for solving the satellite gravity gradiometry problem in a spherical framework based on band-limited spherical regularization wavelets. Considering such wavelets as a reesult of a combination of some regularization methods with Galerkin discretization based on the spherical harmonic system we obtain the error estimates of regularized solutions as well as the estimates for regularization parameters and parameters of band-limitation.
Seinen Versuch, den Begriff der negativen Größen in die Weltweisheit einzuführen beginnt der neununddreißigjährige Immanuel Kant mit einer grundsätzlichen Erörterung über einen etwaigen Gebrauch, den man in der Weltweisheit von der Mathematik ma-chen kann. Dabei stellt er die These auf, daß Mathematik grundsätzlich nur auf zweierlei Art in die Philosophie eingreifen könne. Eine erste Möglichkeit sieht Kant in der Nachahmung mathematischer Methoden bei der Darstellung von Philosophie, die andere Möglichkeit besteht für ihn in der konkreten Anwendung mathematischer Theorien in der Naturlehre. Die zuerst genannte Möglichkeit beurteilt Kant ausgesprochen negativ; seine Kritik an dem von Comenius zunächst ganz allgemein formulierten und dann von Christian Wolff insbesondere für die Philosophie favorisierten Programm einer Präsentation der Philosophie nach mathematischem Vorbild einer Darstellung more geometrico demonstrata ist hinlänglich bekannt. Die Verwendung von Mathematik in der Naturlehre sieht Kant zwar durchaus positiv; in den Metaphysischen Anfangsgründen der Naturwissenschaft wird er gut zwei Jahrzehnte später sogar jene berühmte Behauptung hinzufügen, daß in jeder besonderen Naturlehre nur so viel eigentliche Wissenschaft angetroffen werden könne, als darin Mathematik anzutreffen ist. Dennoch weist Kant mit aller Deutlichkeit auf die engen Grenzen des Wirkungsbereichs solcher Anwendungen von Mathematik hin, denn seiner Meinung nach würden aber auch nur die zur Naturlehre gehörigen Einsichten von derartigem mathematischem Zugriff profitieren.
We consider the "representation type" of the classification problem of vector bundles on a projective curve. We prove that this problem is always either finite, or tame, or wild and we completely describe those curves which are of finite, resp. tame, vector bundle type. We also give a complete list of indecomposable vector bundles for the finite and tame cases.
Let P2r be the projective plane blown up at r generic points. Denote by E0; E1; : : : ; Er the strict transform of a generic straight line on P2 and the exceptional divisors of the blown-up points on P2r respectively. We consider the variety Virr of all irreducible curves C with k nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness, irreducibility and non-emptiness. Moreover, we extend our conditions for the smoothness and the irreducibility on families of reducible curves. For r ^ 9 we give the complete answer concerning the existence of nodal curves in Virr.
Many discrepancy principles are known for choosing the parameter \(\alpha\) in the regularized operator equation \((T^*T+ \alpha I)x_\alpha^\delta = T^*y^\delta\), \(||y-y^d||\leq \delta\), in order to approximate the minimal norm least-squares solution of the operator equation \(Tx=y\). In this paper we consider a class of discrepancy principles for choosing the regularization parameter when \(T^*T\) and \(T^*y^\delta\) are approximated by \(A_n\) and \(z_n^\delta\) respectively with \(A_n\) not necessarily self - adjoint. Thisprocedure generalizes the work of Engl and Neubauer (1985),and particular cases of the results are applicable to the regularized projection method as well as to a degenerate kernel method considered by Groetsch (1990).
Spektralsequenzen
(1999)
In the scalar case one knows that a complex normalized function of boundedvariation \(\phi\) on \([0,1]\) defines a unique complex regular Borel measure\(\mu\) on \([0,1]\). In this note we show that this is no longer true in generalin the vector valued case, even if \(\phi\) is assumed to be continuous. Moreover, the functions \(\phi\) which determine a countably additive vectormeasure \(\mu\) are characterized.
A compact subset E of the complex plane is called removable if all bounded analytic functions on its complement are constant or, equivalently, i f its analytic capacity vanishes. The problem of finding a geometric characterization of the removable sets is more than a hundred years old and still not comp letely solved.
Let (Epsilon_k) be a sequence of experiments with the same finite parameter set. Suppose only that identification of the parameter is possible asymptotically. For large classes of information functionals we show that their exponential rates of convergence towards complete information coincide. As a special case we obtain the rate of the Shannon capacity of product experiments.
In 1979, J.M. Bernardo argued heuristically that in the case of regular product experiments his information theoretic reference prior is equal to Jeffreys' prior. In this context, B.S. Clarke and A.R. Barron showed in 1994, that in the same class of experiments Jeffreys' prior is asymptotically optimal in the sense of Shannon, or, in Bayesian terms, Jeffreys' prior is asymptotically least favorable under Kullback Leibler risk. In the present paper, we prove, based on Clarke and Barron's results, that every sequence of Shannon optimal priors on a sequence of regular iid product experiments converges weakly to Jeffreys' prior. This means that for increasing sample size Kullback Leibler least favorable priors tend to Jeffreys' prior.
The following two norms for holomorphic functions \(F\), defined on the right complex half-plane \(\{z \in C:\Re(z)\gt 0\}\) with values in a Banach space \(X\), are equivalent:
\[\begin{eqnarray*} \lVert F \rVert _{H_p(C_+)} &=& \sup_{a\gt0}\left( \int_{-\infty}^\infty \lVert F(a+ib) \rVert ^p \ db \right)^{1/p}
\mbox{, and} \\ \lVert F \rVert_{H_p(\Sigma_{\pi/2})} &=& \sup_{\lvert \theta \lvert \lt \pi/2}\left( \int_0^\infty \left \lVert F(re^{i \theta}) \right \rVert ^p\ dr \right)^{1/p}.\end{eqnarray*}\] As a consequence, we derive a description of boundary values ofsectorial holomorphic functions, and a theorem of Paley-Wiener typefor sectorial holomorphic functions.
We compare different notions of differentiability of a measure along a vector field on a locally convex space. We consider in the L2-space of a differ entiable measure the analoga of the classical concepts of gradient, divergence and Laplacian (which coincides with the OrnsteinUhlenbeck operator in the Gaussian case). We use these operators for the extension of the basic results of Malliavin and Stroock on the smoothness of finite dimensional image measures under certain nonsmooth mappings to the case of non-Gaussian measures. The proof of this extension is quite direct and does not use any Chaos-decomposition. Finally, the role of this Laplacian in the procedure of quantization of anharmonic oscillators is discussed.
Starting from the uniqueness question for mixtures of distributions this review centers around the question under which formally weaker assumptions one can prove the existence of SPLIFs, in other words perfect statistics and tests. We mention a couple of positive and negative results which complement the basic contribution of David Blackwell in 1980. Typically the answers depend on the choice of the set theoretic axioms and on the particular concepts of measurability.
A large set of criteria to evaluate formal methods for reactive systems is presented. To make this set more comprehensible, it is structured according to a Concept-Model of formal methods. It is made clear that it is necessary to make the catalogue more specific before applying it. Some of the steps needed to do so are explained. As an example the catalogue is applied within the context of the application domain building automation systems to three different formal methods: SDL, statecharts, and a temporallogic.
Im Bereich der Expertensysteme ist das Problemlösen auf der Basis von bekannten Fallbeispielen ein derzeit sehr aktuelles Thema. Auch für Diagnoseaufgaben gewinnt der fallbasierte Ansatz immer mehr an Bedeutung. In diesem Papier soll der im Rahmen des Moltke -Projektes1 an der Universität Kaiserslautern entwickelte fallbasierte Problemlöser Patdex/22 vorgestellt werden. Ein erster Prototyp, Patdex/1, wurde bereits 1988 entwickelt.
We present a mathematical knowledge base containing the factual know-ledge of the first of three parts of a textbook on semi-groups and automata,namely "P. Deussen: Halbgruppen und Automaten". Like almost all math-ematical textbooks this textbook is not self-contained, but there are somealgebraic and set-theoretical concepts not being explained. These concepts areadded to the knowledge base. Furthermore there is knowledge about the nat-ural numbers, which is formalized following the first paragraph of "E. Landau:Grundlagen der Analysis".The data base is written in a sorted higher-order logic, a variant of POST ,the working language of the proof development environment OmegaGamma mkrp. We dis-tinguish three different types of knowledge: axioms, definitions, and theorems.Up to now, there are only 2 axioms (natural numbers and cardinality), 149definitions (like that for a semi-group), and 165 theorems. The consistency ofsuch knowledge bases cannot be proved in general, but inconsistencies may beimported only by the axioms. Definitions and theorems should not lead to anyinconsistency since definitions form conservative extensions and theorems areproved to be consequences.
Das System ART (ASF RRL Translation) stellt im wesentlichen eine Umgebung dar,in welcher die Modularisierbarkeit von Beweisen (Induktionsbeweisen über Gleichungs-spezifikationen) untersucht werden kann. Es wurde die bereits bestehende Spezifikati-onsprache ASF (siehe [BeHeKl89]), in welcher modularisierte Spezifikationen möglichsind, so erweitert, daß zusätzlich auch Beweisaufgaben spezifiziert werden können. Imfolgenden wird diese erweiterte Spezifikationsprache auch ASF genannt. Als Bewei-ser für die Beweisaufgaben einer Spezifikation wurde RRL (siehe [KaZh89]) gewählt.RRL kann sowohl Kommandos aus einem File abarbeiten, wie auch Sitzungsprotokolleanfertigen, mit deren Hilfe sich die Beweisverläufe und Benutzereingaben der entspre-chenden RRL-Sitzung rekonstruieren lassen. In ART kann nun eine ASF-Spezifikation,die Beweisaufgaben umfassen kann, in ein File übersetzt werden, welches von RRLabgearbeitet werden kann. Dies wird im folgenden kurz mit 'Übersetzung von ASF nach RRL' bezeichnet. Bei der Abarbeitung eines solchen Files wird von RRL ein Sit-zungsprotokoll angelegt. ART kann dieses Sitzungsprotokoll dazu heranziehen, neueErgebnisse, wie etwa den erfolgreichen Beweis einer Beweisaufgabe, zu ermitteln, umdiese Ergebnisse der ursprüngliche Spezifikation hinzuzufügen. Dies wird im folgendenkurz mit 'Rückübersetzung von RRL nach ASF' bezeichnet. Im Kern besteht ART alsoaus einer Komponente zur Übersetzung von ASF nach RRL und aus einer Komponentezur Rückübersetzung von RRL nach ASF.
The paper shows that characterizing the causal relationship between significant events is an important but non-trivial aspect for understanding the behavior of distributed programs. An introduction to the notion of causality and its relation to logical time is given; some fundamental results concerning the characterization of causality are pre- sented. Recent work on the detection of causal relationships in distributed computations is surveyed. The relative merits and limitations of the different approaches are discussed, and their general feasibility is analyzed.
Die systematische Verbesserung von Techniken zur Entwicklung und Betreuung von Software setzt eine explizite Darstellung der in einem Projekt ablaufenden Vorgnge (Prozesse) voraus. Diese Darstellungen (Prozemodelle) werden durch Software- Prozemodellierung gewonnen. Eine Sprache zur Beschreibung solcher Modelle ist MVP-L. Verschiedene Standard-Prozemodelle existieren bereits. Bisher gibt es jedoch kaum dokumentierte Software-Entwicklungsprozesse, die speziell fr die Entwicklung reaktiver Systeme entworfen worden sind, d. h. auf die besonderen Anfordernisse bei der Entwicklung reaktiver Systeme zugeschnitten sind. Auch ist bisher nur wenig Erfahrung dokumentiert, fr welche Art von Projektkontexten diese Prozesse gltig sind. Eine Software- Entwicklungsmethode, die - mit Einschrnkungen - zur Entwicklung reaktiver Systeme geeignet ist, ist SOMT (SDL-oriented Object Modeling Technique). Dieser Bericht beschreibt die erfahrungsbasierte Modellierung der Software-Entwicklungsprozesse von SOMT mit MVP-L. Zunchst werden inhaltliche Grundlagen der Software-Entwicklungsmethode SOMT beschrieben. Insbesondere wird auf die eingesetzten Techniken und deren Kombination eingegangen. Anschlieend werden mgliche Projektkontexte charakterisiert, in denen das SOMT-Modell im Sinne eines Erfahrungselements Gltigkeit hat. Darauf werden der Modellierungsvorgang sowie hierbei gemachte Erfahrungen dokumentiert. Eine vollstndige Darstellung des Modells in grafischer MVP-L-Notation befindet sich im Anhang. Die Darstellung des Modells in textueller Notation kann der SFB-Erfahrungsdatenbank entnommen werden.
In this paper we prove a reduction result for the number of criteria in convex multiobjective optimization. This result states that to decide wheter a point x in the decision space is pareto optimal it suffices to consider at most n? criteria at a time, where n is the dimension of the decision space. The main theorem is based on a geometric characterization of pareto, strict pareto and weak pareto solutions
Ramsey Numbers of K_m versus (n,k)-graphs and the Local Density of Graphs not Containing a K_m
(1999)
In this paper generalized Ramsey numbers of complete graphs K_m versus the set langle ,n,k angle of (n,k)-graphs are investigated. The value of r(K_m,langle n,k angle) is given in general for (relative to n) values of k small compared to n using a correlation with Turan numbers. These generalized Ramsey numbers con be used to determine the local densities of graphs not containing a subgraph K_m.
The Weber problem for a given finite set of existing facilities {cal E}x = {Ex_1,Ex_2, ... ,Ex_M} subset R^2 with positive weights w_m (m = 1, ... ,M) is to find a new facility X* in R^2 such that sum_{m=1}^{M} w_{m}d(X,Ex_m) is minimized for some distance function d. In this paper we consider distances defined by polyhedral gauges. A variation of this problem is obtained if barriers are introduced which are convex polygonal subsets of the plane where neither location of new facilities nor traveling is allowed. Such barriers like lakes, military regions, national parks or mountains are frequently encountered in practice.From a mathematical point of view barrier problems are difficult, since the prensence of barriers destroys the convexity of the objective function. Nevertheless, this paper establishes a descretization result: One of the grid points in the grid defined by the existing facilities and the fuundamental directions of the gauge distances can be proved to be an optimal location. Thus the barrier problem can be solved with a polynomial algorithm.
Kernel smoothing in nonparametric autoregressive schemes offers a powerful tool in modelling time series. In this paper it is shown that the bootstrap can be used for estimating the distribution of kernel smoothers. This can be done by mimicking the stochastic nature of the whole process in the bootstrap resampling or by generating a simple regression model. Consistency of these bootstrap procedures will be shown.
In this paper we consider generalizations of multifacility location problems in which as an additional constraint the new facilities are not allowed to be located in a presprcified region. We propose several different solution schemes for this non-convex optimization problem. These include a linear programming type approach, penalty approaches and barrier approaches. Moreover, structural results as well as illustratrive examples showing the difficulties of this problem are presented
To present the decision maker's (DM) preferences in multicriteria decision problems as a partially ordered set is an effective method to catch the DM's purpose and avoid misleading results. Since our paper is focused on minimal path problems, we regard the ordered set of edges (E,=). Minimal paths are defined in repect to power-ordered sets which provides an essential tool to solve such problems. An algorithm to detect minimal paths on a multicriteria minimal path problem is presented
Let P be a probability measure of the real line R such that each of the product measures P^{otimes n} assigns the value 1/2 to every half space in R^{n} having the origin as a boundary point. Then P is symmetric.Example: A strictly stable law on R is symmetric iff it has median zero. The treated symmetry problem is related to the problem of characterizing the distribution of X_1 by the distribution of (X_2 + X_1, ... ,X_n + X_1), with X_1, ... ,X_n being independent and identically distributed random variables.
In continous location problems we are given a set of existing facilities and we are looking for the location of one or several new facilities. In the classical approaches weights are assigned to existing facilities expressing the importance of the new facilities for the existing ones. In this paper, we consider a pointwise defined objective function where the weights are assigned to the existing facilities depending on the location of the new facility. This approach is shown to be a generalization of the median, center and centdian objective functions. In addition, this approach allows to formulate completely new location models. Efficient algorithms as well as structure results for this algebraic approach for location problems are presented. Extensions to the multifacility and restricted case are also considered.
In this paper we consider the problem of optimizing a piecewise-linear objective function over a non-convex domain. In particular we do not allow the solution to lie in the interior of a prespecified region R. We discuss the geometrical properties of this problems and present algorithms based on combinatorial arguments. In addition we show how we can construct quite complicated shaped sets R while maintaining the combinatorial properties.
In this paper we deal with the determination of the whole set of Pareto-solutions of location problems with respect to Q general criteria. These criteria include as particular instances median, center or cent-dian objective functions. The paper characterizes the set of Pareto-solutions of all these multicriteria problems. An efficient algorithm for the planar case is developed and its complexity is established. the proposed approach is more general than the previously published approaches to multicriteria location problems and includes almost all of them as particular instances.
The purpose of GPS-satellite-to-satellite tracking (GPS-SST) is to determine the gravitational potential at the earth's surface from measured ranges (geometrical distances) between a low-flying satellite and the high-flying satellites of the Global Posittioning System (GPS). In this paper GPS-satellite-to-satellite tracking is reformulated as the problem of determining the gravitational potential of the earth from given gradients at satellite altitude. Uniqueness and stability of the solution are investigated. The essential tool is to split the gradient field into a normal part (i.e. the first order radial derivative) and a tangential part (i.e. the surface gradient). Uniqueness is proved for polar, circular orbits corresponding to both types of data (first radial derivative and/or surface gradient). In both cases gravity recovery based on satellite-to-satellite tracking turns out to be an exponentially ill-posed problem. As an appropriate solution method regularization in terms of spherical wavelets is proposed based on the knowledge of the singular system. Finally, the extension of this method is generalized to a non-spherical earth and a non-spherical orbital surface based on combined terrestrial and satellite data material.
Here we consider the Kohonen algorithm with a constant learning rate as a Markov process evolving in a topological space. it is shown that the process is an irreducible and aperiodic T-chain, regardless of the dimension of both data space and network and the special shape of the neighborhood function. Moreover the validity of Deoblin's condition is proved. These imply the convergence in distribution of the process to a finite invariant measure with a uniform geometric rate. In addition we show the process is positive Harris recurrent, which enables us to use statistical devices to measure its centrality and variability as the time goes to infinity.
Comparison of kinetic theory and discrete element schemes for modelling granular Couette flows
(1999)
Discrete element based simulations of granular flow in a 2d velocity space are compared with a particle code that solves kinetic granular flow equations in two and three dimensions. The binary collisions of the latter are governed by the same forces as for the discrete elements. Both methods are applied to a granular shear flow of equally sized discs and spheres. The two dimensional implementation of the kinetic approach shows excellent agreement with the results of the discrete element simulations. When changing to a three dimensional velocity space, the qualitative features of the flow are maintained. However, some flow properties change quantitatively.
In the paper we discuss the transition from kinetic theory to macroscopic fluid equations, where the macroscopic equations are defined as aymptotic limits of a kinetic equation. This relation can be used to derive computationally efficient domain decomposition schemes for the simulaion of rarefied gas flows close to the continuum limit. Moreover, we present some basic ideas for the derivation of kinetic induced numerical schemes for macroscopic equations, namely kinetic schemes for general conservation laws as well as Lattice-Boltzmann methods for the incompressible Navier-Stokes equations.
The mathematical modelling of problems in science and engineering leads often to partial differential equations in time and space with boundary and initial conditions.The boundary value problems can be written as extremal problems(principle of minimal potential energy), as variational equations (principle of virtual power) or as classical boundary value problems.There are connections concerning existence and uniqueness results between these formulations, which will be investigated using the powerful tools of functional analysis.The first part of the lecture is devoted to the analysis of linear elliptic boundary value problems given in a variational form.The second part deals with the numerical approximation of the solutions of the variational problems.Galerkin methods as FEM and BEM are the main tools. The h-version will be discussed, and an error analysis will be done.Examples, especially from the elasticity theory, demonstrate the methods.
We study a model for learning periodic signals in recurrent neural networks proposed by Doya and Yoshizawa [7] that can be considered as a model for temporal pattern memory in animal motoric systems. A network receives an external oscillatory input and adjusts its weights so that this signal can be reproduced approximately as the network output after some time. We use tools from adaptive control theory to derive an algorithm for weight matrices with special structure. If the input is generated by a network of the same structure the algorithm converges globally and does not exhibit the deficiencies of the back-propagation based approach of Doya and Yoshizawa under a persistency of excitation condition. This simple algorithm can also be used for open loop identification under quite restructive assumptions. The persistency of excitation condition cannot be proven even for the matrices with special structure but for a 3d system. For higher dimensional systems we give connections to the theory of linear time-varying systems where this condition is generically true (under assumption which are also needed in the time-invariant case). However, we cannot show that the linearized system related to the nonlinear neural network fulfills these generic assumptions.
Many interesting problems arise from the study of the behavior of fluids. From a theoretical point of view Fluid Dynamics works with a well defined set of equat ions for which it is expected to get a clear description of the solutions. Unfortunately, in ge neral this is not easy even if the many experiments performed in the field seem to indicate which path to follow. Some of the basic questions are still either partially or widely open. For example we would like to have a better understanding on : 1. Questions for both bounded and unbounded domains on regularity, uniqueness, long time behavior of the solutions. 2. How well do solutions to the fluid equations fit to the real flow. Depending on the type of data most of the answers to these questions are knonw, when we work in two dimensions. For solutions in three dimensions, in general, we have only partial answers.
We give a comparison of various differential cross-section models for a classical polyatomic gas for a homogeneous relaxation problem and the shock wave profiles at Mach numbers 1.71 and 12.9. Besides the standard Borgnakke-Larsen model and its generalizations to an energy dependent coefficient to control the amnount of rotationally elastic and completely inelastic collisions, we discuss some new models recently proposed by the same authors. Moreover, we present numerical algorithms to implement the models in a particle or Monte-Carlo code and compare the numerical shock wave profiles with existing experimental data.
The existence of maximum entropy solutions for a wide class of reduced moment problems on arbitrary open subsets of Rd is considered. In particular, new results for the case of unbounded domains are obtained. A precise condition is presented under which solvability of the moment problem implies existence of a maximum entropy solution.
The thermal equilibrium state of a bipolar, isothermal quantum fluid confined to a bounded domain \(\Omega\subset I\!\!R^d,d=1,2\) or \( d=3\) is the minimizer of the total energy \({\mathcal E}_{\epsilon\lambda}\); \({\mathcal E}_{\epsilon\lambda}\) involves the squares of the scaled Planck's constant \(\epsilon\) and the scaled minimal Debye length \(\lambda\). In applications one frequently has \(\lambda^2\ll 1\). In these cases the zero-space-charge approximation is rigorously justified. As \(\lambda \to 0 \), the particle densities converge to the minimizer of a limiting quantum zero-space-charge functional exactly in those cases where the doping profile satisfies some compatibility conditions. Under natural additional assumptions on the internal energies one gets an differential-algebraic system for the limiting \((\lambda=0)\) particle densities, namely the quantum zero-space-charge model. The analysis of the subsequent limit \(\epsilon \to 0\) exhibits the importance of quantum gaps. The semiclassical zero-space-charge model is, for small \(\epsilon\), a reasonable approximation of the quantum model if and only if the quantum gap vanishes. The simultaneous limit \(\epsilon =\lambda \to 0\) is analyzed.
A General Hilbert Space Approach to Wavelets and Its Application in Geopotential Determination
(1999)
A general approach to wavelets is presented within a framework of a separable functional Hilbert space H. Basic tool is the construction of H-product kernels by use of Fourier analysis with respect to an orthonormal basis in H. Scaling function and wavelet are defined in terms of H-product kernels. Wavelets are shown to be 'building blocks' that decorrelate the data. A pyramid scheme provides fast computation. Finally, the determination of the earth's gravitational potential from single and multipole expressions is organized as an example of wavelet approximation in Hilbert space structure.
This review article reports current activities and recent progress on constructive approximation and numerical analysis in physical geodesy. The paper focuses on two major topics of interest, namely trial systems for purposes of global and local approximation and methods for adequate geodetic application. A fundamental tool is an uncertainty principle, which gives appropriate bounds for the quantification of space and momentum localization of trial functions. The essential outcome is a better understanding of constructive approximation in terms of radial basis functions such as splines and wavelets.
The asymptotic behaviour of a singular-perturbed two-phase Stefan problem due to slow diffusion in one of the two phases is investigated. In the limit the model equations reduce to a one-phase Stefan problem. A boundary layer at the moving interface makes it necessary to use a corrected interface condition obtained from matched asymptotic expansions. The approach is validated by numerical experiments using a front-tracking method.
This paper is concerned with numerical algorithms for the bipolar quantum drift diffusion model. For the thermal equilibrium case a quasi-gradient method minimizing the energy functional is introduced and strong convergence is proven. The computation of current - voltage characteristics is performed by means of an extended emph{Gummel - iteration}. It is shown that the involved fixed point mapping is a contraction for small applied voltages. In this case the model equations are uniquely solvable and convergence of the proposed iteration scheme follows. Numerical simulations of a one dimensional resonant tunneling diode are presented. The computed current - voltage characteristics are in good qualitative agreement with experimental measurements. The appearance of negative differential resistances is verified for the first time in a Quantum Drift Diffusion model.
Compared to conventional techniques in computational fluid dynamics, the lattice Boltzmann method (LBM) seems to be a completely different approach to solve the incompressible Navier-Stokes equations. The aim of this article is to correct this impression by showing the close relation of LBM to two standard methods: relaxation schemes and explicit finite difference discretizations. As a side effect, new starting points for a discretization of the incompressible Navier-Stokes equations are obtained.
The problem of finding an optimal location X* minimizing the maximum Euclidean distance to existing facilities is well solved by e.g. the Elzinga-Hearn algorithm. In practical situations X* will however often not be feasible. We therefore suggest in this note a polynomial algorithm which will find an optimal location X^F in a feasible subset F of the plane R^2
In the following, we discuss a procedure for interpolating a spatial-temporal stochastic process. We stick to a particular, moderately general model but the approach can be easily transered to other similar problems. The original data, which motivated this work, are measurements of gas concentrations (SO2, NO, O2) and several meteorological parameters (temperature, sun radiation, procipitation, wind speed etc.). These date have been and are still recorded twice every hour at several irregularly located places in the forests of the state Rheinland-Pfalz as part of a program monitoring the air pollution in the forests.
We consider a scale discrete wavelet approach on the sphere based on spherical radial basis functions. If the generators of the wavelets have a compact support, the scale and detail spaces are finite-dimensional, so that the detail information of a function is determined by only finitely many wavelet coefficients for each scale. We describe a pyramid scheme for the recursive determination of the wavelet coefficients from level to level, starting from an initial approximation of a given function. Basic tools are integration formulas which are exact for functions up to a given polynomial degree and spherical convolutions.
Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems
(1999)
Some inequalities for the Boltzmann collision integral are proved. These inequalities can be considered as a generalization of the well-known Povzner inequality. The inequalities are used to obtain estimates of moments of solution to the spatially homogeneous Boltzmann equation for a wide class of intermolecular forces. We obtained simple necessary and sufficient conditions (on the potential) for the uniform boundedness of all moments. For potentials with compact support the following statement is proved. .....
Anhand des vom Gutachterausschuß der Stadt Kaiserlautern zur Verfügung gestellten Datenmaterials soll untersucht werden, welche Faktoren den Verkehrswert eines bebauten Grundstücks beeinflussen. Mit diesen Erkenntnissen soll eine möglichst einfache Formel ermittelt werden, die eine Schätzung für den Verkehrswert liefert, und die dabei die in der Vergangenheit erzielten Kaufpreise berücksichtigt. Für die Lösung dieser Aufgabe bietet sich das Verfahren der multiplen linearen Regression an. Auf die theoretischen Grundlagen soll hier nicht näher eingegangen werden, man findet sie in jedem Buch über mathematische Statistik, oder in [1]. Bei der Analyse der Daten wurde im großen und ganzen der Weg eingeschlagen, den Angelika Schwarz in [1] beschreibt. Ihre Ergebnisse lassen sich jedoch nicht direkt übertragen, da die dort betrachteten Grundstücke unbebaut waren. Da bei der statistischen Auswertung großer Datenmengen ein immenser Rechenaufwand anfällt, ist es unverzichtbar, professionelle statistische Software einzusetzen. Es stand das Programm S-Plus 2.0 (PC-Version für Windows) zur Verfügung. Sämtliche Berechnungen und alle Grafiken in diesem Bericht wurden in S-Plus erstellt.
We consider a multiple objective linear program (MOLP) max{Cx|Ax = b,x in N_{0}^{n}} where C = (c_ij) is the p x n - matrix of p different objective functions z_i(x) = c_{i1}x_1 + ... + c_{in}x_n , i = 1,...,p and A is the m x n - matrix of a system of m linear equations a_{k1}x_1 + ... + a_{kn}x_n = b_k , k=1,...,m which form the set of constraints of the problem. All coefficients are assumed to be natural numbers or zero. The set M of admissable solutions {hat x} is an admissible solution such that there exists no other admissable solution x' with C{hat x} Cx'. The efficient solutions play the role of optimal solutions for the MOLP and it is our aim to determine the set of all efficient solutions
In this paper we give the definition of a solution concept in multicriteria combinatorial optimization. We show how Pareto, max-ordering and lexicographically optimal solutions can be incorporated in this framework. Furthermore we state some properties of lexicographic max-ordering solutions, which combine features of these three kinds of optimal solutions. Two of these properties, which are desirable from a decision maker" s point of view, are satisfied if and only of the solution concept is that of lexicographic max-ordering.
The Weber Problem for a given finite set of existing facilities {cal E}x = {Ex_1,Ex_2, ... ,Ex_M} subset R^2 with positive weights w_m (m = 1, ... ,M) is to find a new fcility X* such that sum_{m=1}^{M} w_{m}d(X,Ex_m) is minimized for some distance function d. A variation of this problem is obtained of the existing facilities are situated on two sides of a linear barrier. Such barriers like rivers, highways, borders or mountain ranges are frequently encountered in practice. Structural results as well as algorithms for this non-convex optimization problem depending on the distance function and on the number and location of passages through the barrier are presented. A reduction to convex optimization problems is used to derive efficient algorithms.
In this paper we introduce a new type of single facility location problems on networks which includes as special cases most of the classical criteria in the literature. Structural results as well as a finite dominationg set for the optimal locations are developed. Also the extension to the multi-facility case is discussed.
In this paper network location problems with several objectives are discussed, where every single objective is a classical median objective function. We will lock at the problem of finding Pareto optimal locations and lexicographically optimal locations. It is shown that for Pareto optimal locations in undirected networks no node dominance result can be shown. Structural results as well as efficient algorithms for these multi-criteria problems are developed. In the special case of a tree network a generalization of Goldman's dominance algorithm for finding Pareto locations is presented.
An approach to generating all efficient solutions of multiple objective programs with piecewise linear objective functions and linear constraints is presented. The approach is based on the decomposition of the feasible set into subsets, referred to as cells, so that the original problem reduces to a series of lenear multiple objective programs over the cells. The concepts of cell-efficiency and complex-efficiency are introduced and their relationship with efficiency is examined. A generic algorithm for finding efficent solutions is proposed. Applications in location theory as well as in worst case analysis are highlighted.
Facility location problems in the plane play an important role in mathematical programming. When looking for new locations in modeling real-word problems, we are often confronted with forbidden regions, that are not feasible for the placement of new locations. Furthermore these forbidden regions may habe complicated shapes. It may be more useful or even necessary to use approcimations of such forbidden regions when trying to solve location problems. In this paper we develop error bounds for the approximative solution of restricted planar location problems using the so called sandwich algorithm. The number of approximation steps required to achieve a specified error bound is analyzed. As examples of these approximation schemes, we discuss round norms and polyhedral norms. Also computational tests are included.