Obvious inferences

The notion of obvious inference in predicate logic is discussed from the viewpoint of proof-checker applications in logic and mathematics education. A class of inferences in predicate logic is defined and it is proposed to identify it with the class of obvious logical inferences. The definition is compared with other approaches. The algorithm for implementing the obviousness decision procedure follows directly from the definition.     

Fuzzy measurement in the Mishnah and the Talmud

I discuss the attitude of Jewish law sources from the 2nd–:5th centuries to the imprecision of measurement. I review a problem that the Talmud refers to, somewhat obscurely, as impossible reduction. This problem arises when a legal rule specifies an object by referring to a maximized (or minimized) measurement function, e.g., when a rule applies to the largest part of a divided whole, or to the first incidence that occurs, etc. A problem that is often mentioned is whether there might be hypothetical situations involving more than one maximal (or minimal) value of the relevant measurement and, given such situations, what is the pertinent legal rule. Presumption of simultaneous occurrences or equally measured values are also a source of embarrassment to modern legal systems, in situations exemplified in the paper, where law determines a preference based on measured values. I contend that the Talmudic sources discussing the problem of impossible reduction were guided by primitive insights compatible with fuzzy logic presentation of the inevitable uncertainty involved in measurement. I maintain that fuzzy models of data are compatible with a positivistic epistemology, which refuses to assume any precision in the extra-conscious world that may not be captured by observation and measurement. I therefore propose this view as the preferred interpretation of the Talmudic notion of impossible reduction. Attributing a fuzzy world view to the Talmudic authorities is meant not only to increase our understanding of the Talmud but, in so doing, also to demonstrate that fuzzy notions are entrenched in our practical reasoning. If Talmudic sages did indeed conceive the results of measurements in terms of fuzzy numbers, then equality between the results of measurements had to be more complicated than crisp equations. The problem of impossible reduction could lie in fuzzy sets with an empty core or whose membership functions were only partly congruent. Reduction is impossible may thus be reconstructed as there is no core to the intersection of two measures. I describe Dirichlet maps for fuzzy measurements of distance as a rough partition of the universe, where for any region A there may be a non-empty set of - _A (upper approximation minus lower approximation), where the problem of impossible reduction applies. This model may easily be combined with probabilistic extention. The possibility of adopting practical decision standards based on -cuts (and therefore applying interval analysis to fuzzy equations) is discussed in this context. I propose to characterize the uncertainty that was presumably capped by the old sages as U-uncertainty, defined, for a non-empty fuzzy set A on the set of real numbers, whose -cuts are intervals of real numbers, as U(A) = 1/h(A) ₀ ^h(A) log [1+(A)]d, where h(A) is the largest membership value obtained by any element of A and (A) is the measure of the -cut of A defined by the Lebesge integral of its characteristic function.     

Two and three-quarter dimensional meshing facilitators

This paper presents generated enhancements for robust two and three-quarter dimensional meshing, including: (1) automated interval assignment by integer programming for submapped surfaces and volumes, (2) surface submapping, and (3) volume submapping. An introduction to the simplex method, an optimization technique of integer programming, is presented. Simplification of complex geometry is required for the formulation of the integer programming problem. A method of i-j unfolding is defined which explains how irregular geometry can be realigned into a simplified form that is suitable for submap interval assignment solutions. Also presented is the processes by which submapping eliminates the decomposition of surface geometry, through a pseudodecomposition process, producing suitable mapped meshes. The process of submapping involves the creation of interpolated virtual edges, user defined vertex types and i-j-k space traversals. The creation of interpolated virtual edges is the method by which submapping automatically subdivides surface geometry. The interpolated virtual edge is formulated according to an interpolation scheme using the node discretization of curves on the surface. User defined vertex types allow direct user control of surface decomposition and interval assignment by modifying i-j-k space traversals. Volume submapping takes the geometry decomposition to a higher level by using mapped virtual surfaces to eliminate decomposition of complex volumes.     

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 .     

Classical and Quantum Complexity of the Sturm–Liouville Eigenvalue Problem

We study the approximation of the smallest eigenvalue of a Sturm–Liouville problem in the classical and quantum settings. We consider a univariate Sturm–Liouville eigenvalue problem with a nonnegative function q from the class C² ([0,1]) and study the minimal number n() of function evaluations or queries that are necessary to compute an -approximation of the smallest eigenvalue. We prove that n()=(^–1/2) in the (deterministic) worst case setting, and n()=(^–2/5) in the randomized setting. The quantum setting offers a polynomial speedup with bit queries and an exponential speedup with power queries. Bit queries are similar to the oracle calls used in Grovers algorithm appropriately extended to real valued functions. Power queries are used for a number of problems including phase estimation. They are obtained by considering the propagator of the discretized system at a number of different time moments. They allow us to use powers of the unitary matrix exp((1/2) iM), where M is an n× n matrix obtained from the standard discretization of the Sturm–Liouville differential operator. The quantum implementation of power queries by a number of elementary quantum gates that is polylog in n is an open issue. In particular, we show how to compute an -approximation with probability (3/4) using n()=(^–1/3) bit queries. For power queries, we use the phase estimation algorithm as a basic tool and present the algorithm that solves the problem using n()=(log ^–1) power queries, log ^2–1 quantum operations, and (3/2) log ^–1 quantum bits. We also prove that the minimal number of qubits needed for this problem (regardless of the kind of queries used) is at least roughly (1/2) log ^–1. The lower bound on the number of quantum queries is proven in Bessen (in preparation). We derive a formula that relates the Sturm–Liouville eigenvalue problem to a weighted integration problem. Many computational problems may be recast as this weighted integration problem, which allows us to solve them with a polylog number of power queries. Examples include Grovers search, the approximation of the Boolean mean, NP-complete problems, and many multivariate integration problems. In this paper we only provide the relationship formula. The implications are covered in a forthcoming paper (in preparation).PACS: 03.67.Lx, 02.60.-x.     

Almost surely continuous solutions of a nonlinear stochastic integral equation

A nonlinear stochastic integral equation of the Hammerstein type in the formx(t; ) = h(t, x(t; )) + _s k(t, s; )f(s, x(s; ); )d(s) is studied wheret S, a measure space with certain properties, , the supporting set of a probability measure space (,A, P), and the integral is a Bochner integral. A random solution of the equation is defined to be an almost surely continuousm-dimensional vector-valued stochastic process onS which is bounded with probability one for eacht S and which satisfies the equation almost surely. Several theorems are proved which give conditions such that a unique random solution exists. AMS (MOS) subject classifications (1970): Primary; 60H20, 45G99. Secondary: 60G99.     

‘Use’ discourses in system development: Can communication be improved?

This paper aims to provide a basis for renewed talk about use in computing. Four current discourse arenas are described. Different intentions manifest in each arena are linked to failures in translation, different terminologies crossing disciplinary and national boundaries non-reflexively. Analysis of transnational use discourse dynamics shows much miscommunication. Conflicts like that between the Scandinavian System Development School and the usability approach have less current salience. Renewing our talk about use is essential to a participatory politics of information technology and will lead to clearer perception of the implications of letting new systems becoming primary media of social interaction.     

A geometric view of parametric linear programming

We present a new definition of optimality intervals for the parametric right-hand side linear programming (parametric RHS LP) Problem () = min{c ^t x¦Ax =b + ¯b,x 0}. We then show that an optimality interval consists either of a breakpoint or the open interval between two consecutive breakpoints of the continuous piecewise linear convex function (). As a consequence, the optimality intervals form a partition of the closed interval {; ¦()¦ < }. Based on these optimality intervals, we also introduce an algorithm for solving the parametric RHS LP problem which requires an LP solver as a subroutine. If a polynomial-time LP solver is used to implement this subroutine, we obtain a substantial improvement on the complexity of those parametric RHS LP instances which exhibit degeneracy. When the number of breakpoints of () is polynomial in terms of the size of the parametric problem, we show that the latter can be solved in polynomial time.This research was partially funded by the United States Navy-Office of Naval Research under Contract N00014-87-K-0202. Its financial support is gratefully acknowledged.     

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

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

How to Compute Interval Inclusions of Geodetic Coordinates from Interval Inclusions of Cartesian Coordinates

In satellite geodesy the position of a point P is usually determined by computing its coordinate vector x with respect to an earth-fixed Cartesian coordinate system S. S is chosen such that a rotational ellipsoid E, closely approximating the surface of the earth, has normal form with respect to S. Since the geodetic coordinates of P with respect to E (ellipsoidal latitude , ellipsoidal longitude , and ellipsoidal height h) describe the location of P with respect to the surface of the earth much better than x, a frequently appearing problem is to compute , , and h from x.In practice this problem is solved by iterative methods, the convergence properties of which are not analyzed in detail but for which fast (numerical) convergence is observed for points near to E.In the present paper a theoretically well founded new method is developed, working for all P having a unique nearest point in E.In addition it will be shown that the new method can be implemented such that interval inclusions for , , and h can be computed from interval inclusions of the components of x.     

Property preserving abstractions for the verification of concurrent systems

We study property preserving transformations for reactive systems. The main idea is the use of simulations parameterized by Galois connections (, ), relating the lattices of properties of two systems. We propose and study a notion of preservation of properties expressed by formulas of a logic, by a function mapping sets of states of a systemS into sets of states of a systemS'. We give results on the preservation of properties expressed in sublanguages of the branching time -calculus when two systemsS andS' are related via (, )-simulations. They can be used to verify a property for a system by verifying the same property on a simpler system which is an abstraction of it. We show also under which conditions abstraction of concurrent systems can be computed from the abstraction of their components. This allows a compositional application of the proposed verification method.This is a revised version of the papers [2] and [16]; the results are fully developed in [28].This work was partially supported by ESPRIT Basic Research Action REACT.Verimag is a joint laboratory of CNRS, Institut National Polytechnique de Grenoble, Université J. Fourier and Verilog SA associated with IMAG.     

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

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

Design sensitivity analysis of nonlinear dynamic response of structural and mechanical systems

This paper describes a unified variational theory for design sensitivity analysis of nonlinear dynamic response of structural and mechanical systems for shape, nonshape, material and mechanical properties selection, as well as control problems. The concept of an adjoint system, the principle of virtual work and a Lagrangian-Eulerian formulation to describe the deformations and the design variations are used to develop a unified view point. A general formula for design sensitivity analysis is derived and interpreted for usual performance functionals. Analytical examples are utilized to demonstrate the use of the theory and give insights for application to more complex problems that must be treated numerically.Derivatives The comma notation for partial derivatives is used, i.e. G,_u = G/u. An upper dot represents material time derivative, i.e. ü = ²u/t². A prime implies derivative with respect to the time measured in the reference time-domain, i.e. u = du/d.     

A Mathematical Analysis of Pānini’s <Emphasis Type="Italic">Śivasūtras</Emphasis>

In Pninis grammar of Sanskrit one finds the ivastras, a table which defines the natural classes of phonological segments in Sanskrit by intervals. We present a formal argument which shows that, using his representation method, Pninis way of ordering the phonological segments to represent the natural classes is optimal. The argument is based on a strictly set-theoretical point of view depending only on the set of natural classes and does not explicitly take into account the phonological features of the segments, which are, however, implicitly given in the way a language clusters its phonological inventory. The key idea is to link the graph of the Hasse-diagram of the set of natural classes closed under intersection to ivastra-style representations of the classes. Moreover, the argument is so general that it allows one to decide for each set of sets whether it can be represented with Pninis method. Actually, Pnini had to modify the set of natural classes to define it by the ivastras (the segment h plays a special role). We show that this modification was necessary and, in fact, the best possible modification. We discuss how every set of classes can be modified in such a way that it can be defined in a ivastra-style representation.¹     

Lattice Image Processing: A Unification of Morphological and Fuzzy Algebraic Systems

This paper explores some aspects of the algebraic theory of mathematical morphology from the viewpoints of minimax algebra and translation-invariant systems and extends them to a more general algebraic structure that includes generalized Minkowski operators and lattice fuzzy image operators. This algebraic structure is based on signal spaces that combine the sup-inf lattice structure with a scalar semi-ring arithmetic that possesses generalized additions and -multiplications. A unified analysis is developed for: (i) representations of translation-invariant operators compatible with these generalized algebraic structures as nonlinear sup- convolutions, and (ii) kernel representations of increasing translation-invariant operators as suprema of erosion-like nonlinear convolutions by kernel elements. The theoretical results of this paper develop foundations for unifying large classes of nonlinear translation-invariant image and signal processing systems of the max or min type. The envisioned applications lie in the broad intersection of mathematical morphology, minimax signal algebra and fuzzy logic.Petros Maragos received the Diploma degree in electrical engineering from the National Technical University of Athens in 1980, and the M.Sc.E.E. and Ph.D. degrees from Georgia Tech, Atlanta, USA, in 1982 and 1985.In 1985 he joined the faculty of the Division of Applied Sciences at Harvard University, Cambridge, Massachusetts, where heworked for 8 years as professor of electrical engineering, affiliated with the interdisciplinary Harvard Robotics Lab. He has also been a consultant to several industry research groups including Xeroxs research on document image analysis. In 1993, he joined the faculty of the School of Electrical and Computer Engineering at Georgia Tech. During parts of 1996-98 he was on academic leave working as a senior researcher at the Institute for Language and Speech Processing in Athens. In 1998, he joined the faculty of the National Technical University of Athens where he is currently working as professor of electrical and computer engineering. His current research and teaching interests include the general areas of signal processing, systems theory, pattern recognition, and their applications to image processing and computer vision, and computer speech processing and recognition.He has served as associate editor for the IEEE Trans. on Acoustics, Speech, and Signal Processing, editorial board member for the Journal of Visual Communications and Image Representation, and guest editor for the IEEE Trans. on Image Processing; general chairman for the 1992 SPIE Conference on Visual Communications and Image Processing, and co-chairman for the 1996 International Symposium on Mathematical Morphology; member of two IEEE DSP committees; and president of the International Society for Mathematical Morphology.Dr. Maragos research work has received several awards, including: a 1987 US National Science Foundation Presidential Young Investigator Award; the 1988 IEEE Signal Processing Societys Paper Award for the paper Morphological Filters; the 1994 IEEE Signal Processing Societys Senior Award and the 1995 IEEE Baker Award for the paper Energy Separation in Signal Modulations with Application to Speech Analysis; and the 1996 Pattern Recognition Societys Honorable Mention Award for the paper Min-Max Classifiers. In 1995, he was elected Fellow of IEEE for his contributions to the theory and applications of nonlinear signal processing systems.     

Context calibration

The basic starting point of this paper is that context constitutes most of the user interface when doing VR-related experiments, but even so one bases performance measures on only a few active tasks. Thus, in order to meaningfully compare results obtained in vastly different experiments one needs to somehow subtract the contribution to observables that are due to the context. For the case where one is investigating whether changes in one observable causes changes in another, a method, context calibration, is proposed that does just that. This method is expected to, to a large extent, factor out the part of one's results that are due to factors that are not explicitly considered when evaluating the experiment, factors that the experimenter might not even suspect influences the experiment. A procedure for systematically investigating the theoretical assumptions underlying context calibration is also discussed as is an initial experiment adhering to the proposed methodology.     

On Computer-Assisted Proofs in Ordinal Number Theory

Some basic theorems about ordinal numbers were proved using McCunes computer program OTTER, building on Quaifes modification of Gödels class theory. Our theorems are based on Isbells elegant definition of ordinals. Neither the axiom of regularity nor the axiom of choice is used.     

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.     

Single axioms for groups

We study equations of the form (=x) which are single axioms for group theory. Earlier examples of such were found by Neumann and McCune. We prove some lower bounds on the complexity of these , showing that McCune's examples are the shortest possible. We also show that no such can have only two distinct variables. We do produce an with only three distinct variables, refuting a conjecture of Neumann. Automated reasoning techniques are used both positively (searching for and verifying single axioms) and negatively (proving that various candidate (=x) hold in some nongroup and are, hence, not single axioms).     

On a random Volterra integral equation

Summary Tsokos [12] showed the existence of a unique random solution of the random Volterra integral equation (*)x(t; ) = h(t; ) + _o ^t k(t, ; )f(, x(; )) d, where , the supporting set of a probability measure space (,A, P). It was required thatf must satisfy a Lipschitz condition in a certain subset of a Banach space. By using an extension of Banach's contraction-mapping principle, it is shown here that a unique random solution of (*) exists whenf is (, )-uniformly locally Lipschitz in the same subset of the Banach space considered in [12].