The Relative Entropy of Entanglement of Schmidt Correlated States and Distillation

We show that a mixed state = _mna_mn|mn| can be realized by an ensemble of pure states {p_k, |_k} where . Employing this form, we discuss the relative entropy of entanglement of Schmidt correlated states. Also, we calculate the distillable entanglement of a class of mixed states. PACS: 03.67.-a; 03.65.Bz; 03.65.Ud     

Choosing reference classes and building provisional models

A well-known problem in default logic is the ability of naive reasoners to explain bothg and ¬g from a set of observations. This problem is treated in at least two different ways within that camp.One approach is examination of the various explanations and choosing among them on the basis of various explanation comparators. A typical comparator is choosing the explanation that depends on the most specific observation, similar to the notion of narrowest reference class.Others examine default extensions of the observations and choose whatever is true in any extension, or what is true in all extensions or what is true in preferred extensions. Default extensions are sometimes thought of as acceptable models of the world that are discarded as more knowledge becomes available.We argue that the notions of specificity and extension lack clear semantics. Furthermore, we show that the problems these ideas were supposed to solve can be handled easily within a probabilistic framework.     

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]).     

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.     

Low-Rank Approximation of Integral Operators by Interpolation

A central component of the analysis of panel clustering techniques for the approximation of integral operators is the so-called -admissibility condition min {diam(),diam()} 2dist(,) that ensures that the kernel function is approximated only on those parts of the domain that are far from the singularity. Typical techniques based on a Taylor expansion of the kernel function require a subdomain to be far enough from the singularity such that the parameter has to be smaller than a given constant depending on properties of the kernel function. In this paper, we demonstrate that any is sufficient if interpolation instead of Taylor expansionisused for the kernel approximation, which paves the way for grey-box panel clustering algorithms.     

Logic Based Abstractions of Real-Time Systems

When verifying concurrent systems described by transition systems, state explosion is one of the most serious problems. If quantitative temporal information (expressed by clock ticks) is considered, state explosion is even more serious. We present a notion of abstraction of transition systems, where the abstraction is driven by the formulae of a quantitative temporal logic, called qu-mu-calculus, defined in the paper. The abstraction is based on a notion of bisimulation equivalence, called , n-equivalence, where is a set of actions and n is a natural number. It is proved that two transition systems are , n-equivalent iff they give the same truth value to all qu-mu-calculus formulae such that the actions occurring in the modal operators are contained in , and with time constraints whose values are less than or equal to n. We present a non-standard (abstract) semantics for a timed process algebra able to produce reduced transition systems for checking formulae. The abstract semantics, parametric with respect to a set of actions and a natural number n, produces a reduced transition system , n-equivalent to the standard one. A transformational method is also defined, by means of which it is possible to syntactically transform a program into a smaller one, still preserving , n-equivalence.     

Super-pattern matching

Some recognition problems are either too complex or too ambiguous to be expressed as a simple pattern matching problem using a sequence or regular expression pattern. In these cases, a richer environment is needed to describe the patterns and recognition techniques used to perform the recognition. Some researchers have turned to artificial-intelligence techniques and multistep matching approaches for the problems of gene recognition [5], [7], [18], protein structure recognition [13], and on-line character recognition [6]. This paper presents a class of problems which involve finding matches to patterns of patterns, orsuper- patterns, given solutions to the lower-level patterns. The expressiveness of this problem class rivals that of traditional artificial-intelligence characterizations, and yet polynomial-time algorithms are described for each problem in the class.This work was supported in part by the National Institute of Health under Grant ROI LM04960 and by the Aspen Center for Physics.     

Context-free text grammars

A text is a triple=(, ₁, ₂) such that is a labeling function, and ₁ and ₂ are linear orders on the domain of ; hence may be seen as a word (, ₁) together with an additional linear order ₂ on the domain of . The order ₂ is used to give to the word (, ₁) itsindividual hierarchical representation (syntactic structure) which may be a tree but it may be also more general than a tree. In this paper we introducecontext-free grammars for texts and investigate their basic properties. Since each text has its own individual structure, the role of such a grammar should be that of a definition of a pattern common to all individual texts. This leads to the notion of ashapely context-free text grammar also investigated in this paper.     

Hierarchies of polynomially solvable satisfiability problems

In this paper, we introduce general techniques for extending classes of polynomially solvable SAT instances. We generalize the approach of Gallo and Scutellà, who defined the hierarchy { _i }, where _l corresponds to the Generalized Horn class. We propose a family of polynomial hierarchies, where a polynomial hierarchy { _i } is a sequence of polynomially solvable classes that cover the whole set of CNF formulas, and such that _{i i+1} fori0. Following a different approach, based on a new decomposition technique, we define the class of Split-Horn formulas, which is an extension of _l. We discuss and compare the basic properties of the proposed classes; polynomial time algorithms for recognition and solution are provided.     

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)     

Separating complexity classes related to bounded alternating ω-branching programs

We develop a theory of communication within branching programs that provides exponential lower bounds on the size of branching programs that are bounded alternating. Our theory is based on the algebraic concept of -branching programs, : , a semiring homomorphism, that generalizes ordinary branching programs, -branching programs [M2] andMOD _p-branching programs [DKMW].Due to certain exponential lower and polynomial upper bounds on the size of bounded alternating -branching programs we are able to separate the corresponding complexity classesN _ba ,co-N _{ba ba} , andMOD _p - _ba ,p prime, from each other, and from that classes corresponding to oblivious linear length-bounded branching programs investigated in the past.     

A semidynamic construction of higher-order voronoi diagrams and its randomized analysis

Thek-Delaunay tree extends the Delaunay tree introduced in [1], and [2]. It is a hierarchical data structure that allows the semidynamic construction of the higher-order Voronoi diagrams of a finite set ofn points in any dimension. In this paper we prove that a randomized construction of thek-Delaunay tree, and thus of all the orderk Voronoi diagrams, can be done inO(n logn+k ³n) expected time and O(k²n) expected storage in the plane, which is asymptotically optimal for fixedk. Our algorithm extends tod-dimensional space with expected time complexityO(k ^(d+1)/2+1 n ^(d+1)/2) and space complexityO(k ^(d+1)/2 n ^(d+1)/2). The algorithm is simple and experimental results are given.This work has been supported in part by the ESPRIT Basic Research Action No. 3075 (ALCOM).     

A short note on simultaneous splitting

We show that for any alphabet there is a setL ^* such that ifC is any infinite co-infinite context-free language over , thenL splitsC (i.e., each ofL C,L , C, and is infinite).Preparation of this paper was supported in part by the National Science Foundation under Grant No. MCS77-11360.     

From Conditional Events to Conditional Measures: A New Axiomatic Approach

Our starting point is a definition of conditional event EH which differs from many seemingly similar ones adopted in the relevant literature since 1935, starting with de Finetti. In fact, if we do not assign the same third value u (undetermined) to all conditional events, but make it depend on EH, it turns out that this function t(EH) can be taken as a general conditional uncertainty measure, and we get (through a suitable – in a sense, compulsory – choice of the relevant operations among conditional events) the natural axioms for many different (besides probability) conditional measures.     

Maintaining the visibility map of spheres while moving the viewpoint on a circle at infinity

We investigate three-dimensional visibility problems for scenes that consist ofn non-intersecting spheres. The viewing point moves on a flightpath that is part of a circle at infinity given by a planeP and a range of angles {(t)¦t[01]} [02]. At timet, the lines of sight are parallel to the ray inP, which starts in the origin ofP and represents the angle(t) (orthographic views of the scene). We give an algorithm that computes the visibility graph at the start of the flight, all time parameters at which the topology of the scene changes, and the corresponding topology changes. The algorithm has running time0(n + k + p) logn), wheren is the number of spheres in the scene;p is the number of transparent topology changes (the number of different scene topologies visible along the flight path, assuming that all spheres are transparent); andk denotes the number of vertices (conflicts) which are in the (transparent) visibility graph at the start and do not disappear during the flight.The second author was supported by the ESPRIT II Basic Research Actions Program, under Contract No. 3075 (project ALCOM).     

Weaving patterns of lines and line segments in space

Aweaving W is a simple arrangement of lines (or line segments) in the plane together with a binary relation specifying which line is above the other. A system of lines (or line segments) in 3-space is called arealization ofW, if its projection into the plane isW and the above-below relations between the lines respect the specifications. Two weavings are equivalent if the underlying arrangements of lines are combinatorially equivalent and the above-below relations are the same. An equivalence class of weavings is said to be aweaving pattern. A weaving pattern isrealizable if at least one element of the equivalence class has a three-dimensional realization. A weaving (pattern)W is calledperfect if, along each line (line segment) ofW, the lines intersecting it are alternately above and below. We prove that (i) a perfect weaving pattern ofn lines is realizable if and only ifn 3, (ii) a perfect m byn weaving pattern of line segments (in a grid-like fashion) is realizable if and only if min(m, n) 3, (iii) ifn is sufficiently large, then almost all weaving patterns ofn lines are nonrealizable.Jànos Pach has been supported in part by Hungarian NFSR Grant 1812, NSF Grant CCR-8901484, and the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), a National Science Foundation Science and Technology Center, under NSF Grant STC88-09648. Richard Pollack has been supported in part by NSA Grant MDA904-89-H-2030, NSF Grants DMS-85-01947 and CCR-8901484, and DIMACS. Emo Welzl has been supported in part by the ESPRIT II Basic Research Actions Program of the EC under Contract No. 3075 (project ALCOM) and DIMACS.     

Deterministic sequential functions

The simple rational partial functions accepted by generalized sequential machines are shown to coincide with the compositions _P ^–1 , where _P consists of the prefix codings. The rational functions accepted by generalized sequential machines are proved to coincide with the compositions _P ^–1 , where is the family of endmarkers and is the family of removals of endmarkers. (The compositions are read from left to right). We also show that _P ^–1 is the family of the subsequential functions.This work was partially supported by the Esprit Basic Research Action Working Group No. 3166 ASMICS, the CNRS and the Academy of Finland     

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.     

Types with intersection: An introduction

This paper is an informal introduction to the theory of types which use a connective for the intersection of two types and a constant for a universal type, besides the usual connective for function-types. This theory was first devised in about 1977 by Coppo, Dezani and Sallé in the context of-calculus and its main development has been by Coppo and Dezani and their collaborators in Turin. With suitable axioms and rules to assign types to-calculus terms, they obtained a system in which (i) the set of types given to a term does not change under-conversion, (ii) some interesting sets of terms, for example the solvable terms and the terms with normal form, can be characterised exactly by the types of their members, and (iii) the type-apparatus is not so complex as polymorphic systems with quantifier-containing types and therefore probably not so expensive to implement mechanically as these systems.There are in fact several variant systems with different detailed properties. This paper defines and motivates the simplest one from which the others are derived, and describes its most basic properties. No proofs are given but the motivation is shown by examples. A comprehensive bibliography is included.