Elementary Closure Research Articles

A quantitative approach to logical inference

Logical inference is of central importance in the information and decision sciences but presents a very hard computational problem. Since the traditional symbolic inference methods have had limited success on large knowledge bases, this papers investigates a quantitative approach. It surveys the application of integer programming methods to inference problems in propositional logic. It displays a number of remarkable parallels between logic and mathematics and shows that these can lead to fast inference methods, both quantitative and symbolic. In particular it explains why the logical concepts of resolution, extended resolution, input and unit refutation, the Davis-Putnam procedure, and drawing of inferences pertinent to a given topic are closely related to the mathematical concepts of cutting planes, Chvátal's method, elementary closure, branch and bound, and projection of a polytope, respectively. Much of the paper should be intelligible to persons with limited background in logic and mathematical programming, but recent mathematical results are stated precisely.

Flows of Energy and Materials in Marine Ecosystems: Theory and Practice.

One - Invited Papers.- The Importance of Measuring Fluxes in Marine Ecosystems.- Community Measures of Marine Food Networks and their Possible Applications.- Thermodynamics of the Pelagic Ecosystem: Elementary Closure Conditions for Biological Production in the Open Ocean.- The Structure of Aquatic Ecosystems and its Dependency on Environmental Conditions.- Eutrophication of a Coastal Marine Ecosystem - An Experimental Study Using the MERL Microcosms.- Oceanic Nutrient Cycles.- State-of-the-Art in the Measurement of Primary Production.- Functional Types of Marine Planktonic Primary Producers and their Relative Significance in the Food Web.- Measuring the Metabolism of the Benthic Ecosystem.- The Measurement of the Enthalpy of Metabolism of Marine Organisms.- Bacterial Production in the Marine Food Chain: The Emperor's New Suit of Clothes?.- Suspended Marine Bacteria as a Food Source.- The Biological Role of Detritus in the Marine Environment.- The Cycling of Organic Matter by Bacterioplankton in Pelagic Marine Ecosystems: Microenvironmental Considerations.- An Overview of Secondary Production in Pelagic Ecosystems.- The Quantitative Significance of Gelatinous Zooplankton as Pelagic Consumers.- Fish Production in Open Ocean Ecosystems.- Ecological Efficiency and Activity Metabolism.- Detrital Organic Fluxes Through Pelagic Ecosystems.- The Supply of Food to the Benthos.- Coupling the Sub-Systems - the Baltic Sea as a Case Study.- On the Relation of Primary Production to Grazing during the Fladen Ground Experiment 1976.- Two - Working Group Reports.- Studies on Marine Autotrophs: Recommendations for the 1980s.- Respiration.- Nutrient Cycling in Estuarine and Coastal Marine Ecosystems.- Excretion and Mineralisation Processes in the Open Sea.- Herbivory.- Detritivory.- Carnivory.- The Role of Free Bacteria and Bactivory.- Participants.

Some elementary closure properties of $n$-cylinders.

(i) A set is a cylinder if and only if it is an ^-cylinder for each n > 1. (ii) The class of all (n+l)-cylinders is a proper subset of the class of all π-cylinders for each n > 1. (iii) For each n > 1, the class of all ^-cylinders contains a simple set and hence a nonsplinter. (iv) For each n > 1, the class of all ^-cylinders contains a nonsimple nonsplinter. (v) A set P is an ^-cylinder (n > 1) if and only if there is a recursive function g such that for all x,

Functionals defined by transfinite recursion

This paper deals mainly with quantifier-free second order systems (i.e., with free variables for numbers and functions, and constants for numbers, functions, and functionals) whose basic rules are those of primitive recursive arithmetic together with definition of functionals by primitive recursion and explicit definition. Precise descriptions are given in §2. The additional rules have the form of definition by transfinite recursion up to some ordinal ξ (where ξ is represented by a primitive recursive (p.r.) ordering). In §3 we discuss some elementary closure properties (under rules of inference and definition) of systems with recursion up to ξ. Let Rξ denote (temporarily) the system with recursion up to ξ. The main results of this paper are of two sorts:Sections 5–7 are concerned with less elementary closure properties of the systems Rξ. Namely, we show that certain classes of functional equations in Rη can be solved in Rη for some explicitly determined η < ε(η) (the least ε-number > ξ). The classes of functional equations considered all have roughly the form of definition by recursion on the partial ordering of unsecured sequences of a given functional F, or on some ordering which is obtained from this by simple ordinal operations. The key lemma (Theorem 1) needed for the reduction of these equations to transfinite recursion is simply a sharpening of the Brouwer-Kleene idea.