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]).     

Complexity and computability of solutions to linear programming systems

Through key examples and constructs, exact and approximate, complexity, computability, and solution of linear programming systems are reexamined in the light of Khachian's new notion of (approximate) solution. Algorithms, basic theorems, and alternate representations are reviewed. It is shown that the Klee-Minty example hasnever been exponential for (exact) adjacent extreme point algorithms and that the Balinski-Gomory (exact) algorithm continues to be polynomial in cases where (approximate) ellipsoidal centered-cutoff algorithms (Levin, Shor, Khachian, Gacs-Lovasz) are exponential. By model approximation, both the Klee-Minty and the new J. Clausen examples are shown to be trivial (explicitly solvable) interval programming problems. A new notion of computable (approximate) solution is proposed together with ana priori regularization for linear programming systems. New polyhedral constraint contraction algorithms are proposed for approximate solution and the relevance of interval programming for good starts or exact solution is brought forth. It is concluded from all this that the imposed problem ignorance of past complexity research is deleterious to research progress on computability or efficiency of computation.This research was partly supported by Project NR047-071, ONR Contract N00014-80-C-0242, and Project NR047-021, ONR Contract N00014-75-C-0569, with the Center for Cybernetic Studies, The University of Texas at Austin.     

Function-algebraic characterizations of log and polylog parallel time

The main results of this paper are recursion-theoretic characterizations of two parallel complexity classes: the functions computable by uniform bounded fan-in circuit families of log and polylog depth (or equivalently, the functions bitwise computable by alternating Turing machines in log and polylog time). The present characterizations avoid the complex base functions, function constructors, anda priori size or depth bounds typical of previous work on these classes. This simplicity is achieved by extending the tiered recursion techniques of Leivant and Bellantoni & Cook.     

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.     

Unextendible Product Bases and Locally Unconvertible Bound Entangled States

Mutual convertibility of bound entangled states under local quantum operations and classical communication (LOCC) is studied. We focus on states associated with unextendible product bases (UPB) in a system of three qubits. A complete classification of such UPBs is suggested. We prove that for any pair of UPBs S and T the associated bound entangled states _S and _T cannot be converted to each other by LOCC, unless S and T coincide up to local unitaries. More specifically, there exists a finite precision (S,T) > 0 such that for any LOCC protocol mapping _S into a probabilistic ensemble (p, ), the fidelity between _T and any possible final state satisfies F(_T, ) = 1 - (S,T).PACS: 03.65.Bz; 03.67.-a; 89.70+c.     

One Connection between Standard Invariance Conditions on Modal Formulas and Generalized Quantifiers

The language of standard propositional modal logic has one operator ( or ), that can be thought of as being determined by the quantifiers or , respectively: for example, a formula of the form is true at a point s just in case all the immediate successors of s verify .This paper uses a propositional modal language with one operator determined by a generalized quantifier to discuss a simple connection between standard invariance conditions on modal formulas and generalized quantifiers: the combined generalized quantifier conditions of conservativity and extension correspond to the modal condition of invariance under generated submodels, and the modal condition of invariance under bisimulations corresponds to the generalized quantifier being a Boolean combination of and .     

On Bounding Solutions of Underdetermined Systems

Sufficient conditions for the existence and uniqueness of a solution x* D (R ⁿ ) of Y(x) = 0 where : R ⁿ R ^m (m n) with C ²(D) where D R ⁿ is an open convex set and Y = (x)⁺ are given, and are compared with similar results due to Zhang, Li and Shen (Reliable Computing 5(1) (1999)). An algorithm for bounding zeros of f (·) is described, and numerical results for several examples are given.     

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.     

How to Build a Prototype for a Distributed Digital Mathematics Archive Library

The Electronic Mathematics Archiving Network Initiative (EMANI) recently announced by Springer Verlag, Tsinghua University Library (China), Göttingen State and University Library and Cornell University aims to insure the preservation and dissemination of mathematical information for future generations. In order to be able to use the resulting distributed digital mathematical archive, researchers have to be equipped with freely available readers which enable them to view digital downloaded articles and follow links to reviews in Zentralblatt für Mathematik or Mathematical Reviews of the papers they cite. Using then the permanent links between the review articles and their digital full texts on remote servers the mathematicians can also retrieve the digital full text of the cited articles. Such readers can only work if the scientific documents in the digital libraries have been provided with links between them and the review articles. In this article the different steps to solve these technical problems will be discussed. Methods will be presented which allow one to construct a prototype of a distributed searchable, linked mathematical digital archive library.     

A Typology of Translation Problems for Eurotra Translation Machines

This paper presents a detailed study of Eurotra Machine Translation engines, namely the mainstream Eurotra software known as the E-Framework, and two unofficial spin-offs – the C,A,T and Relaxed Compositionality translator notations – with regard to how these systems handle hard cases, and in particular their ability to handle combinations of such problems. In the C,A,T translator notation, some cases of complex transfer are wild, meaning roughly that they interact badly when presented with other complex cases in the same sentence. The effect of this is that each combination of a wild case and another complex case needs ad hoc treatment. The E-Framework is the same as the C,A,T notation in this respect. In general, the E-Framework is equivalent to the C,A,T notation for the task of transfer. The Relaxed Compositionality translator notation is able to handle each wild case (bar one exception) with a single rule even where it appears in the same sentence as other complex cases.     

Pushing Convertible Constraints in Frequent Itemset Mining

Recent work has highlighted the importance of the constraint-based mining paradigm in the context of frequent itemsets, associations, correlations, sequential patterns, and many other interesting patterns in large databases. Constraint pushing techniques have been developed for mining frequent patterns and associations with antimonotonic, monotonic, and succinct constraints. In this paper, we study constraints which cannot be handled with existing theory and techniques in frequent pattern mining. For example, avg(S)v, median(S)v, sum(S)v (S can contain items of arbitrary values, {<, <, , } and v is a real number.) are customarily regarded as tough constraints in that they cannot be pushed inside an algorithm such as Apriori. We develop a notion of convertible constraints and systematically analyze, classify, and characterize this class. We also develop techniques which enable them to be readily pushed deep inside the recently developed FP-growth algorithm for frequent itemset mining. Results from our detailed experiments show the effectiveness of the techniques developed.     

The surface of language and humanities computing

Recent articles have noted that humanities computing techniques and methodologies remain marginal to mainstream literary scholarship. Mark Olsen's paper discusses this phenomenon and argues for large scale analyses of text databases that would incorporate a shift in theoretical orientation to include greater stress on intertextuality and sign theory. Part of Olsen's argument revolves on the need to move away from the syntactic and overt grammatical elements of textual language to more subtle semantics and meaning systems. While provocative and important, Olsen's stance remains rooted in literary theoretical constructs. Another level of language, the cognitive, offers equally interesting challenges for humanities computing, though the paradigms for this type of computer-based exploration are derived from disciplines traditionally removed from the humanities. The riddle, a nearly universal genre, offers a window onto some of the cognitive processes involved in deep level language function. By analyzing the riddling process, different methods of computational modelling can be inferred, suggesting new avenues for computing in the humanities.Charles Henry has a Ph.D. in Comparative Literature and is presently the Director of Libraries at Vassar College. His research includes non-literal aspects of language, and the cognitive processes involved in understanding symbolic language. Recent articles include The Image of Word,in Humanities and the Computer: New Directions, ed. D. Miall (Oxford, 1990), and Non-Literal Aspects of Language and Knowledge Structuring, inCybernetics and Systems Research '92, ed. R. Trappl (World Scientific Press, 1992).     

Unification in partially commutative semigroups

Unification algorithms have been constructed for semigroups and commutative semigroups. This paper considers the intermediate case of partially commutative semigroups. We introduce classesN and of such semigroups and justify their use. We present an equation-solving algorithm for any member of the classN. This algorithm is relative to having an algorithm to determine all non-negative solutions of a certain class of diophantine equations of degree 2 which we call -equations. The difficulties arising when attempting to solve equations in members of the class are discussed, and we present arguments that strongly suggest that unification in these semigroups is undecidable.     

Where does the end begin? Problems in musico-cognitive modeling

Research with computer systems and musical grammars into improvisation as found in the tabla drumming system of North India has indicated that certain musical sentences comprise (a) variable prefixes, and (b) fixed suffixes (or cadences) identical with those of their original rhythmic themes. It was assumed that the cadence functioned as a kind of target in linear musical space, and yet experiments showed that defining what exactly constituted the cadence was problematic. This paper addresses the problem of the status of cadential patterns, and demonstrates the need for a better understanding and formalization of ambiguity in musico-cognitive processing. It would appear from the discussion that the cadence is not a discrete unit in itself, but just part of an ever-present underlying framework comprising the entire original rhythmic theme. Improvisations (variations), it is suggested, merely break away from and rejoin this framework at important structural points. This endorses the theory of simultaneity. However, the general cognitive implications are still unclear, and further research is required to explore musical ambiguity and the interaction of musical, linguistic, and spatio-motor grammars.     

How to Pass a Turing Test

I advocate a theory of syntactic semantics as a way of understanding how computers can think (and how the Chinese-Room-Argument objection to the Turing Test can be overcome): (1) Semantics, considered as the study of relations between symbols and meanings, can be turned into syntax – a study of relations among symbols (including meanings) – and hence syntax (i.e., symbol manipulation) can suffice for the semantical enterprise (contra Searle). (2) Semantics, considered as the process of understanding one domain (by modeling it) in terms of another, can be viewed recursively: The base case of semantic understanding –understanding a domain in terms of itself – is syntactic understanding. (3) An internal (or narrow), first-person point of view makes an external (or wide), third-person point of view otiose for purposes of understanding cognition.     

Theoretical comparisons of search strategies in branch-and-bound algorithms

Four known search strategies used in branch-and-bound algorithms-heuristic search, depth-first search, best-bound search, and breadth-first search-are theoretically compared from the viewpoint of the performance of the resulting algorithms. Heuristic search includes the other three as special cases. Since heuristic search is determined by a heuristic functionh, we first investigate how the performance of the resulting algorithms depends onh. In particular, we show that heuristic search is stable in the sense that a slight change inh causes only a slight change in its performance. The best and the worst heurstic functions are clarified, and also discussed is how the heuristic functionh should be modified to obtain a branch-and-bound algorithm with an improved performance. Finally, properties and limitations of depth-first search, best-bound search, and breadth-first search viewed as special cases of heuristic search are considered. In particular, it is shown that the stability observed for heuristic search no longer holds for depth-first search.     

Linear scale-space

The formulation of afront-end orearly vision system is addressed, and its connection with scale-space is shown. A front-end vision system is designed to establish a convenient format of some sampled scalar field, which is suited for postprocessing by various dedicated routines. The emphasis is on the motivations and implications of symmetries of the environment; they pose natural, a priori constraints on the design of a front-end.The focus is on static images, defined on a multidimensional spatial domain, for which it is assumed that there are no a priori preferred points, directions, or scales. In addition, the front-end is required to be linear. These requirements are independent of any particular image geometry and express the front-end's pure syntactical, bottom up nature.It is shown that these symmetries suffice to establish the functionality properties of a front-end. For each location in the visual field and each inner scale it comprises a hierarchical family of tensorial apertures, known as the Gaussian family, the lowest order of which is the normalised Gaussian. The family can be truncated at any given order in a consistent way. The resulting set constitutes a basis for alocal jet bundle. Note that scale-space theory shows up here without any call upon the prohibition of spurious detail, which, in some way or another, usually forms the basic starting point for diffusion-like scale-space theories.     

Petri net based verification of distributed algorithms: An example

A technique to model and to verify distributed algorithms is suggested. This technique (based on Petri nets) reduces the modelling and analysis effort to a reasonable level. The paper outlines the technique using the example of a typical network algorithm, theecho algorithm.Supported by the DFG-projects Verteilte Algorithmen and Konsensalgorithmen     

On a Stroboscopic Approach to Quantum Tomography of Qudits Governed by Gaussian Semigroups

In this paper, we discuss the minimal number of observables Q ₁, ..., Q , where expectation values at some time instants t ₁, ..., t _r determine the trajectory of a d-level quantum system (qudit) governed by the Gaussian semigroup . We assume that the macroscopic information about the system in question is given by the mean values E _j(Q _i) = tr(Q _i(t _j)) of n selfadjoint operators Q ₁, ..., Q _n at some time instants t ₁ < t ₂ < ... < t _r, where n < d ²– 1 and r deg (, ). Here (, ) stands for the minimal polynomial of the generator of the Gaussian flow (t).