The co-invariant generator: An aid in deriving loop bodies

Given a loop invariant,I, and an assignment,, which decreases the variant, we define a constructive function, cg, called the co-invariant generator, which has the property thatI cg (, I) wp (, I), where wp (, I) is the weakest precondition for to establish I. Several results about the co-invariant generator are proved, important special cases are considered, and a non-trivial example of its use in deriving the body of a loop is given. We also define a function which performs a related constructive action on terms formed from binary operations. The coinvariant generator makes a useful contribution to formalising and automating a key step in program derivation.     

Proving total correctness of nondeterministic programs in infinitary logic

Summary It is shown how the weakest precondition approach to proving total correctness of nondeterministic programs can be formalized in infinitary logic. The weakest precondition technique is extended to hierarchically structured programs by adding a new primitive statement for operational abstraction, the nondeterministic assignment statement, to the guarded commands of Dijkstra. The infinitary logic L ₁ is shown to be strong enough to express the weakest preconditions for Dijkstra's guarded commands, but too weak for the extended guarded commands. Two possible solutions are considered: going to the essentially stronger infinitary logic L ₁₁ and restricting the power of the nondeterministic assignment statement in a way which allows the weakest preconditions to be expressed in L ₁.     

The Objective Conception of Context and Its Logic

In this paper, an objective conception of contexts based loosely upon situation theory is developed and formalized. Unlike subjective conceptions, which take contexts to be something like sets of beliefs, contexts on the objective conception are taken to be complex, structured pieces of the world that (in general) contain individuals, other contexts, and propositions about them. An extended first-order language for this account is developed. The language contains complex terms for propositions, and the standard predicate ist that expresses the relation that holds between a context and a proposition just in case the latter is true in the former. The logic for the objective conception features a global classical predicate calculus, a local logic for reasoning within contexts, and axioms for propositions. The specter of paradox is banished from the logic by allowing ist to be nonbivalent in problematic cases: it is not in general the case, for any context c and proposition p, that either ist(c,p) or ist(c, ¬ p). An important representational capability of the logic is illustrated by proving an appropriately modified version of an illustrative theorem from McCarthy's classic Blocks World example.     

States and the free orthogonality monoid

Let (X, #) be an orthogonality space such that the lattice C(X, #) of closed subsets of (X, #) is orthomodular and let (, ) denote the free orthogonality monoid over (X, #). Let C₀(, ) be the subset of C(, ), consisting of all closures of bounded orthogonal sets. We show that C₀(, ) is a suborthomodular lattice of C(, ) and we provide a necessary and sufficient condition for C₀(, ) to carry a full set of dispersion free states.The work of the second author on this paper was supported by National Science Foundation Grant GP-9005.     

Designing broadcasting algorithms in the postal model for message-passing systems

In many distributed-memory parallel computers and high-speed communication networks, the exact structure of the underlying communication network may be ignored. These systems assume that the network creates a complete communication graph between the processors, in which passing messages is associated with communication latencies. In this paper we explore the impact of communication latencies on the design of broadcasting algorithms for fully connected message-passing systems. For this purpose, we introduce thepostal model that incorporates a communication latency parameter 1. This parameter measures the inverse of the ratio between the time it takes an originator of a message to send the message and the time that passes until the recipient of the message receives it. We present an optimal algorithm for broadcasting one message in systems withn processors and communication latency , the running time of which is (( logn)/log( + 1)). For broadcastingm 1 messages, we first examine several generalizations of the algorithm for broadcasting one message and then analyze a family of broadcasting algorithms based on degree-d trees. All the algorithms described in this paper are practical event-driven algorithms that preserve the order of messages.     

Tight bounds for oblivious routing in the hypercube

We prove that in anyN-node communication network with maximum degreed, any deterministic oblivious algorithm for routing an arbitrary permutation requires (N/d) parallel communication steps in the worst case. This is an improvement upon the (N/d ^3/2) bound obtained by Borodin and Hopcroft. For theN-node hypercube, in particular, we show a matching upper bound by exhibiting a deterministic oblivious algorithm that routes any permutation in (N/logN) steps. The best previously known upper bound was (N). Our algorithm may be practical for smallN (up to about 2¹⁴ nodes).C. Kaklamanis was supported in part by NSF Grant NSF-CCR-87-04513. T. Tsantilas was supported in part by NSF Grants NSF-DCR-86-00379 and NSF-CCR-89-02500.     

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.     

Constructing sets of functions which have a givenF-cardinality

The termF-cardinality of (=F-card()) is introduced whereF: ^{n n} is a partial function and is a set of partial functionsf: ^{n n} . TheF-cardinality yields a lower bound for the worst-case complexity of computingF if only functionsf can be evaluated by the underlying abstract automaton without conditional jumps. This complexity bound isindependent from the oracles available for the abstract machine. Thus it is shown that any automaton which can only apply the four basic arithmetic operations needs (n logn) worst-case time to sortn numbers; this result is even true if conditional jumps witharbitrary conditions are possible. The main result of this paper is the following: Given a total functionF: ^{n n} and a natural numberk, it is almost always possible to construct a set such that itsF-cardinality has the valuek; in addition, can be required to be closed under composition of functionsf,g . Moreover, ifF is continuous, then consists of continuous functions.     

Modular Control and Coordination of Discrete-Event Systems

In the supervisory control of discrete-event systems based on controllable languages, a standard way to handle state explosion in large systems is by modular supervision: either horizontal (decentralized) or vertical (hierarchical). However, unless all the relevant languages are prefix-closed, a well-known potential hazard with modularity is that of conflict. In decentralized control, modular supervisors that are individually nonblocking for the plant may nevertheless produce blocking, or even deadlock, when operating on-line concurrently. Similarly, a high-level hierarchical supervisor that predicts nonblocking at its aggregated level of abstraction may inadvertently admit blocking in a low-level implementation. In two previous papers, the authors showed that nonblocking hierarchical control can be guaranteed provided high-level aggregation is sufficiently fine; the appropriate conditions were formalized in terms of control structures and observers. In this paper we apply the same technique to decentralized control, when specifications are imposed on local models of the global process; in this way we remove the restriction in some earlier work that the plant and specification (marked) languages be prefix-closed. We then solve a more general problem of coordination: namely how to determine a high level coordinator that forestalls conflict in a decentralized architecture when it potentially arises, but is otherwise minimally intrusive on low-level control action. Coordination thus combines both vertical and horizontal modularity. The example of a simple production process is provided as a practical illustration. We conclude with an appraisal of the computational effort involved.     

Thiele Rational Interpolation for the Numerical Computation of the Reversible Randles–Sevcik Function in Electrochemistry

This paper uses Thiele rational interpolation to derive a simple method for computing the Randles–Sevcik function ^1/2(x), with relative error at most 1.9 × 10^–5 for – < x < . We develop a piecewise approximation method for the numerical computation of ^1/2(x) on the union (–, –10) [–10, 10] (10, ). This approximation is particularly convenient to employ in electrochemical applications where four significant digits of accuracy are usually sufficient. Although this paper is primarily concerned with the approximation of the Randles–Sevcik function, some examples are included that illustrate how Thiele rational interpolation can be employed to generate useful approximations to other functions of interest in scientific work.     

Parallel geometric algorithms on a mesh-connected computer

We show that a number of geometric problems can be solved on a n × n mesh-connected computer (MCC) inO(n) time, which is optimal to within a constant factor, since a nontrivial data movement on an MCC requires (n) time. The problems studied here include multipoint location, planar point location, trapezoidal decomposition, intersection detection, intersection of two convex polygons, Voronoi diagram, the largest empty circle, the smallest enclosing circle, etc. TheO(n) algorithms for all of the above problems are based on the classical divide-and-conquer problem-solving strategy.This work was supported in part by the National Science Foundation under Grant DCR 8420814. A preliminary version was presented in the 1987 FJCC, Dallas, TX.     

Toward Cultural Oncology: The Evolutionary Information Dynamics of Cancer

'Racial' disparities among cancers, particularly of the breast and prostate, are something of a mystery. For the US, in the face of slavery and its sequelae, centuries of interbreeding has greatly leavened genetic differences between Blacks and Whites, but marked contrasts in disease prevalence and progression persist. Adjustment for socioeconomic status and lifestyle, while statistically accounting for much of the variance in breast cancer, only begs the question of ultimate causality. Here we propose a more basic biological explanation that extends the theory of immune cognition to include an elaborate tumor control mechanism constituting the principal selection pressure acting on pathologically mutating cell clones. The interplay between them occurs in the context of an embedding, highly structured, system of culturally-specific psychosocial stress. A rate distortion argument finds that larger system able to literally write an image of itself onto the disease process, in terms of enhanced risk behaviour, accelerated mutation rate, and depressed mutation control. The dynamics are analogous to punctuated equilibrium in simple evolutionary systems, accounting for the staged nature of disease progression. We conclude that 'social exposures' are, for human populations, far more than incidental cofactors in cancer etiology. Rather, they are part of the basic biology of the disorder. The aphorism that culture is as much a part of human biology as the enamel on our teeth appears literally true at a fundamental cellular level.     

An Estimation Problem for Quasilinear Stochastic Partial Differential Equations

A problem of estimating a functional parameter (x) and functionals () based on observation of a solution u (t, x) of the stochastic partial differential equation is considered. The asymptotic problem setting, as the noise intensity 0, is investigated.     

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     

On improving the accuracy of the Hough transform

The subject of this paper is very high precision parameter estimation using the Hough transform. We identify various problems that adversely affect the accuracy of the Hough transform and propose a new, high accuracy method that consists of smoothing the Hough arrayH(, ) prior to finding its peak location and interpolating about this peak to find a final sub-bucket peak. We also investigate the effect of the quantizations and ofH(, ) on the final accuracy. We consider in detail the case of finding the parameters of a straight line. Using extensive simulation and a number of experiments on calibrated targets, we compare the accuracy of the method with results from the standard Hough transform method of taking the quantized peak coordinates, with results from taking the centroid about the peak, and with results from least squares fitting. The largest set of simulations cover a range of line lengths and Gaussian zero-mean noise distributions. This noise model is ideally suited to the least squares method, and yet the results from the method compare favorably. Compared to the centroid or to standard Hough estimates, the results are significantly better—for the standard Hough estimates by a factor of 3 to 10. In addition, the simulations show that as and are increased (i.e., made coarser), the sub-bucket interpolation maintains a high level of accuracy. Experiments using real images are also described, and in these the new method has errors smaller by a factor of 3 or more compared to the standard Hough estimates.     

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     

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].     

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.     

Viscous Lattices

Let E be an arbitrary space, and an extensive dilation of P(E) into itself, with an adjoint erosion . Then, the image [P(E)] of P(E) by is a complete lattice P where the sup is the union and the inf the opening of the intersection according to . The lattice L, named viscous, is not distributive, nor complemented. Any dilation on P(E) admits the same expression in L. However, the erosion in L is the opening according to of the erosion in P(E). Given a connection C on P(E) the image of C under turns out to be a connection C on L as soon as (C)eq C. Moreover, the elementary connected openings _x of C and _(x) are linked by the relation _(x) = _x. A comprehensive class of connection preverving closings is constructed. Two examples, binary and numerical (the latter comes from the heart imaging), prove the relevance of viscous lattices in interpolation and in segmentation problems.Jean Serra obtained the degree of Mining Engineer, in 1962 in Nancy, France, and in 1967 his Ph.D. for a work dealing with the estimation of the iron ore body of Lorraine by geostatistics. In cooperation with Georges Matheron, he laid the foundations of a new method, that he called Mathematical Morphology (1964). Its purpose was to describe quantitatively shapes and textures of natural phenomena, at micro and macro scales. In 1967, he founded with G. Matheron, the Centre de Morphologie Mathematique, at School of Mines of Paris, on the campus of Fontainebleau. Since this time, he has been working in this framework as a Directeur de Recherches. His main book is a two-volume treatise entitled Image Analysis and Mathematical Morphology (Ac. Press, 1982, 1988). He has been Vice President for Europe of the International Society for Stereology from 1979 to 1983. He founded the International Society for Mathematical Morphology in 1993, and was elected his first president. His achievements include several patents of devices for image processing, various awards and titles, such as the first AFCET award, in 1988, or Doctor Honoris Causa of the University of Barcelona (Spain) in 1993. He recently developed a new theory of segmentation, which is based on set connections (2001–2004), and currently works on colour image processing.     

A New Implementation of Automath

This paper presents aut, a modern Automath checker. It is a straightforward re-implementation of the Zandleven Automath checker from the seventies. It was implemented about five years ago, in the programming language C. It accepts both the AUT-68 and AUT-QE dialects of Automath. This program was written to restore a damaged version of Jutting's translation of Landau's Grundlagen. Some notable features: It is fast. On a 1 GHz machine it will check the full Jutting formalization (736 K of nonwhitespace Automath source) in 0.6 seconds. Its implementation of -terms does not use named variables or de Bruijn indices (the two common approaches) but instead uses a graph representation. In this representation variables are represented by pointers to a binder. The program can compile an Automath text into one big Automath single line-style -term. It outputs such a term using de Bruijn indices. (These -terms cannot be checked by modern systems like Coq or Agda, because the -typed -calculi of de Bruijn are different from the -typed -calculi of modern type theory.)The source of aut is freely available on the Web at the address .