this paper we briefly review some of these issues and then concentrate on the problem of generalizing the formal framework of the relational data model to include null values. A basic problem with null values is that they have many plausible interpretations. The ANSI/SPARC interim report, for instance, cites 14 different manifestations of nulls. Most authors, however, agree that the various manifestations of nulls can be reduced to two basic interpretations. These are: (a) the unknown interpretation: a value exists but it is not known; and lb) the nonexistent interpretation: a value does not exist

Uncertainty is a fundamental and irreducible aspect of our knowledge about the world. Probability is the most well-understood and widely applied logic for computational scientific reasoning under uncertainty. As theory and practice advance, general-purpose languages are beginning to emerge for which the fundamental logical basis is probability. However, such languages have lacked a logical foundation that fully integrates classical first-order logic with probability theory. This paper presents such an integrated logical foundation. A formal specification is presented for multi-entity Bayesian networks (MEBN), a knowledge representation language based on directed graphical probability models. A proof is given that a probability distribution over interpretations of any consistent, finitely axiomatizable first-order theory can be defined using MEBN. A semantics based on random variables provides a logically coherent foundation for open world reasoning and a means of analyzing tradeoffs between accuracy and computation cost. Furthermore, the underlying Bayesian logic is inherently open, having the ability to absorb new facts about the world, incorporate them into existing theories, and/or modify theories in the light of evidence. Bayesian inference provides both a proof theory for combining prior knowledge with observations, and a learning theory for refining a representation as evidence accrues. The results of this paper provide a logical foundation for the rapidly evolving literature on first-order Bayesian knowledge representation, and point the way toward Bayesian languages suitable for general-purpose knowledge representation and computing. Because first-order Bayesian logic contains classical first-order logic as a deterministic subset, it is a natural candidate as a universal representation for integrating domain ontologies expressed in languages based on classical first-order logic or subsets thereof.

We characterize the real line by properties similar to the so-called Peano axioms for natural numbers. These properties include an induction principle and a corresponding recursion scheme. The recursion scheme allows us to define functions such as addition, multiplication, exponential, logarithm, sine, arc sine, etc. from simpler ones. In order to obtain such a characterization, we introduce a notion of infinitely iterated composition of morphisms in categories, and we state a fixed point theorem and an infinite composition theorem for uniform spaces. 1 Introduction We characterize the real line by properties similar to the so-called Peano axioms for natural numbers [10, 11, 19]. These properties include an induction principle and a corresponding recursion scheme. The recursion scheme allows us to define functions such as addition, multiplication, exponential, logarithm, sine, arc sine, etc. from simpler ones. 1.1 Programme We begin by characterizing the unit interval I = [0; 1]. F...

This paper describes a construction of the real numbers in the HOL theorem-prover by strictly definitional means using a version of Dedekind's method. It also outlines the theory of mathematical analysis that has been built on top of it and discusses current and potential applications in verification and computer algebra. Keywords: Mathematical Logic; Deduction and Theorem Proving 1 The real numbers For some mathematical tasks, the natural numbers N = f0; 1; 2; : : :g are sufficient. However for many purposes it is convenient to use a more extensive system, such as the integers (Z) or the rational (Q ), real (R) or complex (C ) numbers. In particular the real numbers are normally used for the measurement of physical quantities which (at least in abstract models) are continuously variable, and are therefore ubiquitous in scientific applications. 1.1 Properties of the real numbers We can characterize the reals as the unique `complete ordered field'. More precisely, the reals are a set ...

Uncertainty is a fundamental and irreducible aspect of our knowledge about the world. Until recently, classical first-order logic has reigned as the de facto standard logical foundation for artificial intelligence. The lack of a built-in, semantically grounded capability for reasoning under uncertainty renders classical first-order logic inadequate for many important classes of problems. General-purpose languages are beginning to emerge for which the fundamental logical basis is probability. Increasingly expressive probabilistic languages demand a theoretical foundation that fully integrates classical first-order logic and probability. In first-order Bayesian logic (FOBL), probability distributions are defined over interpretations of classical first-order axiom systems. Predicates and functions of a classical first-order theory correspond to a random variables in the corresponding first-order Bayesian theory. This is a natural correspondence, given that random variables are formalized in mathematical statistics as measurable functions on a probability space. A formal system called Multi-Entity Bayesian Networks (MEBN) is presented for composing distributions on interpretations by instantiating and combining parameterized fragments of directed graphical models. A construction is given of a MEBN theory that assigns a non-zero

The partial real line is an extension of the Euclidean real line with partial real numbers, which has been used to model exact real number computation in the programming language Real PCF. We introduce induction principles and recursion schemes for the partial unit interval, which allow us to verify that Real PCF programs meet their specification. They resemble the so-called Peano axioms for natural numbers. The theory is based on a domain-equation-like presentation of the partial unit interval. The principles are applied to show that Real PCF is universal in the sense that all computable elements of its universe of discourse are definable. These elements include higher-order functions such as integration operators. Keywords: Induction, coinduction, exact real number computation, domain theory, Real PCF, universality. Introduction The partial real line is the domain of compact real intervals ordered by reverse inclusion [28,21]. The idea is that singleton intervals represent total rea...

The accompanying thesis is part of a long-term project to enable computers to do mathematics in the same way that humans do. I will sketch something of the nature of mathematics and the project, and then turn to role of the thesis. Mathematics Mathematics arises from the interaction of two dissimilar modes of reasoning: a ‘soft ’ side, dealing with ideas and analogies, and a ‘hard ’ side, dealing with verification. The ‘hard ’ side is easier to pin down. It consists primarily of formal ‘proofs’, each consisting of a series of assertions. A mathematician can verify that a proof is correct by following it, step by step, checking that each step follows from previous ones via facts already proved to be correct. The ‘soft ’ side is less easily described. It consists of intuitions about the formal objects constructed in mathematical proofs; ideas that one piece of mathematics may analogically correspond to another piece of mathematics; or even analogies between mathematics and objects in the physical world.

We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. We test the notion in categories of interest. In the category of sets, any closed and bounded interval of real numbers is an interval object. In the category of topological spaces, the interval objects are closed and bounded intervals with the Euclidean topology. We also prove that an interval object exists in any elementary topos with natural numbers object. The universal property of an interval object provides a mechanism for defining functions on the interval. We use this to define basic arithmetic operations, and to verify equations between them. It also allows us to develop an analogue of the primitive recursive functions, yielding a natural class of computable functions on the interval. Contents 1

Fuzzy logic is based on the theory of fuzzy sets, where an object’s membership of a set is gradual rather than just member or not a member. Fuzzy logic uses the whole interval of real numbers between zero (False) and one (True) to develop a logic as a basis for rules of inference. Particularly the fuzzified version of the modus ponens rule of inference enables

of bifree algebras