An Algebraic Theory of Information: An Introduction and Survey
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paper_title: A valuation-based language for expert systems

paper_content:

Abstract A new language based on valuations is proposed as an alternative to rule-based languages for constructing knowledge-based systems. Valuation-based languages are superior to rule-based languages for maintaining consistency in the knowledge base, for caching inferences, for managing uncertainty, and for nonmonotonic reasoning. An abstract description of a valuation-based language is given. Two specific instances of valuation-based languages are described. The first is designed to represent categorical knowledge. The ability of such a language to maintain consistency and cache inferences is demonstrated with an example. The second is an evidential language—a valuation-based language in which valuations are belief functions. The ability of evidential languages to perform nonmonotonic reasoning and manage uncertainty is demonstrated with an example.
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paper_title: Information Algebras: Generic Structures for Inference

paper_content:

1 Introduction.- 2 Valuation Algebras.- 2.1 The Framework.- 2.2 Axioms.- 2.3 Examples of Valuation Algebras.- 2.3.1 Indicator Functions.- 2.3.2 Relations.- 2.3.3 Probability Potentials.- 2.3.4 Possibility Potentials.- 2.3.5 Spohn Potentials.- 2.3.6 Set Potentials.- 2.3.7 Gaussian Potentials.- 2.4 Partial Marginalization.- 3 Algebraic Theory.- 3.1 Congruences.- 3.2 Domain-Free Valuation Algebras.- 3.3 Subalgebras, Homomorphisms.- 3.3.1 Subalgebras.- 3.3.2 Homomorphisms and Isomorphisms.- 3.3.3 Weak Subalgebras and Homomorphisms.- 3.4 Null Valuations.- 3.5 Regular Valuation Algebras.- 3.6 Separative Valuation Algebras.- 3.7 Scaled Valuation Algebras.- 4 Local Computation.- 4.1 Fusion Algorithm.- 4.2 Collect Algorithm.- 4.3 Computing Multiple Marginals.- 4.4 Architectures with Division.- 4.5 Computations in Valuation Algebras with Partial Marginalization.- 4.6 Scaling and Updating.- 5 Conditional Independence.- 5.1 Factorization and Graphical Models.- 5.2 Conditionals in Regular Algebras.- 5.3 Conditionals in Separative Algebras.- 6 Information Algebras.- 6.1 Idempotency.- 6.2 Partial Order of Information.- 6.3 File Systems.- 6.4 Information Systems.- 6.5 Examples.- 6.5.1 Propositional Logic.- 6.5.2 Boolean Information Algebras.- 6.5.3 Linear Equations.- 6.5.4 Linear Inequalities.- 6.6 Compact Systems.- 6.7 Mappings.- 7 Uncertain Information.- 7.1 Algebra of Random Variables.- 7.1.1 Random Variables.- 7.1.2 Allocation of Probability.- 7.1.3 Support Functions.- 7.2 Probabilistic Argumentation Systems.- 7.2.1 Assumption-Based Information.- 7.2.2 Probabilistic Argumentation Systems.- 7.2.3 Operations with Argumentation Systems.- 7.3 Allocations of Probability.- 7.3.1 Algebra of Allocations.- 7.3.2 Normalized Allocations.- 7.3.3 Random Variables and Allocations.- 7.3.4 Labeled Allocations and their Algebra.- 7.4 Independent Sources.- 7.4.1 Independent Random Variables.- 7.4.2 Algebra of Bpas.- References.
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paper_title: Local computations with probabilities on graphical structures and their application to expert systems

paper_content:

The invention comprises pigments of the general Formula I in which each X is hydrogen, chlorine or bromine, A is (a) an aliphatic hydrocarbon radical, (b) the phenylene group or (c) the naphthylene group and R is identical with the 1: 2-phthaloylcarbazole-aminocarbonyl radical shown in the above formula when A has the meaning defined under (a), and R is alkoxy, alkylamino, alkoylamino, trifluoroalkyl, alkylsulphonyl or fluoro when A has the meaning defined under (b) or (c), and in which general formula the group R-A-CO-NH- is in the 3- to 9-position. The invention further comprises a process for making pigments of the above Formula I in which each X is hydrogen, chloride, bromine or alkyl and (i) A is an aliphatic hydrocarbon radical and R is the 1: 2-phthaloylcarbazole-aminocarbonyl radical shown in the formula or (ii) A is an aromatic hydrocarbon radical and R is hydrogen, alkyl, alkoxy, alkylamino, alkoylamino, trifluoralkyl, alkylsulphonyl, halogen or the 1: 2-phthaloylcarbazoleaminocarbonyl radical, R-A-CO-NH- being linked in the 3- or 9-position, wherein an amino-1: 2-phthaloylcarbazole of the above Formula I in which the group R-A-CO- is replaced by hydrogen, is acylated with an appropriate aliphatic or aromatic dicarboxylic acid or an appropriate substituted or unsubstituted aromatic mono-carboxylic acid. The products are pigments for lacquers, printing pastes and plastics such as polyvinyl chloride.
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paper_title: Axioms for probability and belief-function propagation

paper_content:

In this paper, we describe an abstract framework and axioms under which exact local computation of marginals is possible. The primitive objects of the framework are variables and valuations. The primitive operators of the framework are combination and marginalization. These operate on valuations. We state three axioms for these operators and we derive the possibility of local computation from the axioms. Next, we describe a propagation scheme for computing marginals of a valuation when we have a factorization of the valuation on a hypertree. Finally we show how the problem of computing marginals of joint probability distributions and joint belief functions fits the general framework.
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paper_title: Computation in Valuation Algebras

paper_content:

The main goal of this chapter is to describe an abstract framework called valuation algebra for computing marginals using local computation. The valuation algebra framework is useful in many domains, and especially for managing uncertainty in expert systems using probability, Dempster-Shafer belief functions, Spohnian epistemic belief theory, and possibility theory.
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paper_title: General Lattice Theory

paper_content:

I First Concepts.- 1 Two Definitions of Lattices.- 2 How to Describe Lattices.- 3 Some Algebraic Concepts.- 4 Polynomials, Identities, and Inequalities.- 5 Free Lattices.- 6 Special Elements.- Further Topics and References.- Problems.- II Distributive Lattices.- 1 Characterization and Representation Theorems.- 2 Polynomials and Freeness.- 3 Congruence Relations.- 4 Boolean Algebras.- 5 Topological Representation.- 6 Pseudocomplementation.- Further Topics and References.- Problems.- III Congruences and Ideals.- 1 Weak Projectivity and Congruences.- 2 Distributive, Standard, and Neutral Elements.- 3 Distributive, Standard, and Neutral Ideals.- 4 Structure Theorems.- Further Topics and References.- Problems.- IV Modular and Semimodular Lattices.- 1 Modular Lattices.- 2 Semimodular Lattices.- 3 Geometric Lattices.- 4 Partition Lattices.- 5 Complemented Modular Lattices.- Further Topics and References.- Problems.- V Varieties of Lattices.- 1 Characterizations of Varieties.- 2 The Lattice of Varieties of Lattices.- 3 Finding Equational Bases.- 4 The Amalgamation Property.- Further Topics and References.- Problems.- VI Free Products.- 1 Free Products of Lattices.- 2 The Structure of Free Lattices.- 3 Reduced Free Products.- 4 Hopfian Lattices.- Further Topics and References.- Problems.- Concluding Remarks.- Table of Notation.- A Retrospective.- 1 Major Advances.- 2 Notes on Chapter I.- 3 Notes on Chapter II.- 4 Notes on Chapter III.- 5 Notes on Chapter IV.- 6 Notes on Chapter V.- 7 Notes on Chapter VI.- 8 Lattices and Universal Algebras.- B Distributive Lattices and Duality by B. Davey, II. Priestley.- 1 Introduction.- 2 Basic Duality.- 3 Distributive Lattices with Additional Operations.- 4 Distributive Lattices with V-preserving Operators, and Beyond.- 5 The Natural Perspective.- 6 Congruence Properties.- 7 Freeness, Coproducts, and Injectivity.- C Congruence Lattices by G. Gratzer, E. T. Schmidt.- 1 The Finite Case.- 2 The General Case.- 3 Complete Congruences.- D Continuous Geometry by F. Wehrung.- 1 The von Neumann Coordinatization Theorem.- 2 Continuous Geometries and Related Topics.- E Projective Lattice Geometries by M. Greferath, S. Schmidt.- 1 Background.- 2 A Unified Approach to Lattice Geometry.- 3 Residuated Maps.- F Varieties of Lattices by P. Jipsen, H. Rose.- 1 The Lattice A.- 2 Generating Sets of Varieties.- 3 Equational Bases.- 4 Amalgamation and Absolute Retracts.- 5 Congruence Varieties.- G Free Lattices by R. Frecse.- 1 Whitman's Solutions Basic Results.- 2 Classical Results.- 3 Covers in Free Lattices.- 4 Semisingular Elements and Tschantz's Theorem.- 5 Applications and Related Areas.- H Formal Concept Analysis by B. Cantor and R. Wille.- 1 Formal Contexts and Concept Lattices.- 2 Applications.- 3 Sublattices and Quotient Lattices.- 4 Subdirect Products and Tensor Products.- 5 Lattice Properties.- New Bibliography.
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paper_title: Information Algebras: Generic Structures for Inference

paper_content:

1 Introduction.- 2 Valuation Algebras.- 2.1 The Framework.- 2.2 Axioms.- 2.3 Examples of Valuation Algebras.- 2.3.1 Indicator Functions.- 2.3.2 Relations.- 2.3.3 Probability Potentials.- 2.3.4 Possibility Potentials.- 2.3.5 Spohn Potentials.- 2.3.6 Set Potentials.- 2.3.7 Gaussian Potentials.- 2.4 Partial Marginalization.- 3 Algebraic Theory.- 3.1 Congruences.- 3.2 Domain-Free Valuation Algebras.- 3.3 Subalgebras, Homomorphisms.- 3.3.1 Subalgebras.- 3.3.2 Homomorphisms and Isomorphisms.- 3.3.3 Weak Subalgebras and Homomorphisms.- 3.4 Null Valuations.- 3.5 Regular Valuation Algebras.- 3.6 Separative Valuation Algebras.- 3.7 Scaled Valuation Algebras.- 4 Local Computation.- 4.1 Fusion Algorithm.- 4.2 Collect Algorithm.- 4.3 Computing Multiple Marginals.- 4.4 Architectures with Division.- 4.5 Computations in Valuation Algebras with Partial Marginalization.- 4.6 Scaling and Updating.- 5 Conditional Independence.- 5.1 Factorization and Graphical Models.- 5.2 Conditionals in Regular Algebras.- 5.3 Conditionals in Separative Algebras.- 6 Information Algebras.- 6.1 Idempotency.- 6.2 Partial Order of Information.- 6.3 File Systems.- 6.4 Information Systems.- 6.5 Examples.- 6.5.1 Propositional Logic.- 6.5.2 Boolean Information Algebras.- 6.5.3 Linear Equations.- 6.5.4 Linear Inequalities.- 6.6 Compact Systems.- 6.7 Mappings.- 7 Uncertain Information.- 7.1 Algebra of Random Variables.- 7.1.1 Random Variables.- 7.1.2 Allocation of Probability.- 7.1.3 Support Functions.- 7.2 Probabilistic Argumentation Systems.- 7.2.1 Assumption-Based Information.- 7.2.2 Probabilistic Argumentation Systems.- 7.2.3 Operations with Argumentation Systems.- 7.3 Allocations of Probability.- 7.3.1 Algebra of Allocations.- 7.3.2 Normalized Allocations.- 7.3.3 Random Variables and Allocations.- 7.3.4 Labeled Allocations and their Algebra.- 7.4 Independent Sources.- 7.4.1 Independent Random Variables.- 7.4.2 Algebra of Bpas.- References.
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paper_title: Semiring induced valuation algebras: Exact and approximate local computation algorithms

paper_content:

Local computation in join trees or acyclic hypertrees has been shown to be linked to a particular algebraic structure, called valuation algebra. There are many models of this algebraic structure ranging from probability theory to numerical analysis, relational databases and various classical and non-classical logics. It turns out that many interesting models of valuation algebras may be derived from semiring valued mappings. In this paper we study how valuation algebras are induced by semirings and how the structure of the valuation algebra is related to the algebraic structure of the semiring. In particular, c-semirings with idempotent multiplication induce idempotent valuation algebras and therefore permit particularly efficient architectures for local computation. Also important are semirings whose multiplicative semigroup is embedded in a union of groups. They induce valuation algebras with a partially defined division. For these valuation algebras, the well-known architectures for Bayesian networks apply. We also extend the general computational framework to allow derivation of bounds and approximations, for when exact computation is not feasible.
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paper_title: Information Algebras: Generic Structures for Inference

paper_content:

1 Introduction.- 2 Valuation Algebras.- 2.1 The Framework.- 2.2 Axioms.- 2.3 Examples of Valuation Algebras.- 2.3.1 Indicator Functions.- 2.3.2 Relations.- 2.3.3 Probability Potentials.- 2.3.4 Possibility Potentials.- 2.3.5 Spohn Potentials.- 2.3.6 Set Potentials.- 2.3.7 Gaussian Potentials.- 2.4 Partial Marginalization.- 3 Algebraic Theory.- 3.1 Congruences.- 3.2 Domain-Free Valuation Algebras.- 3.3 Subalgebras, Homomorphisms.- 3.3.1 Subalgebras.- 3.3.2 Homomorphisms and Isomorphisms.- 3.3.3 Weak Subalgebras and Homomorphisms.- 3.4 Null Valuations.- 3.5 Regular Valuation Algebras.- 3.6 Separative Valuation Algebras.- 3.7 Scaled Valuation Algebras.- 4 Local Computation.- 4.1 Fusion Algorithm.- 4.2 Collect Algorithm.- 4.3 Computing Multiple Marginals.- 4.4 Architectures with Division.- 4.5 Computations in Valuation Algebras with Partial Marginalization.- 4.6 Scaling and Updating.- 5 Conditional Independence.- 5.1 Factorization and Graphical Models.- 5.2 Conditionals in Regular Algebras.- 5.3 Conditionals in Separative Algebras.- 6 Information Algebras.- 6.1 Idempotency.- 6.2 Partial Order of Information.- 6.3 File Systems.- 6.4 Information Systems.- 6.5 Examples.- 6.5.1 Propositional Logic.- 6.5.2 Boolean Information Algebras.- 6.5.3 Linear Equations.- 6.5.4 Linear Inequalities.- 6.6 Compact Systems.- 6.7 Mappings.- 7 Uncertain Information.- 7.1 Algebra of Random Variables.- 7.1.1 Random Variables.- 7.1.2 Allocation of Probability.- 7.1.3 Support Functions.- 7.2 Probabilistic Argumentation Systems.- 7.2.1 Assumption-Based Information.- 7.2.2 Probabilistic Argumentation Systems.- 7.2.3 Operations with Argumentation Systems.- 7.3 Allocations of Probability.- 7.3.1 Algebra of Allocations.- 7.3.2 Normalized Allocations.- 7.3.3 Random Variables and Allocations.- 7.3.4 Labeled Allocations and their Algebra.- 7.4 Independent Sources.- 7.4.1 Independent Random Variables.- 7.4.2 Algebra of Bpas.- References.
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paper_title: Quantifiers on distributive lattices

paper_content:

Abstract A Q -distributive lattice is an algebra 〈 L , ∧, ∨, ∇, 0, 1〉 of type (2, 2, 1, 0, 0) such that 〈 L , ∧, ∨, 0, 1〉 is a bounded distributive lattice and ∇ satisfies the equations: (1) ∇0 = 0, (2) x ∧ ∇ x = x , (3) ∇( x ∧ ∇ y ) = ∇ x ∧ ∇ y and (4) ∇( x ∨ y ) = ∇ x ∨ ∇ y . The opposite of the category of Q -distributive lattices is described in terms of Priestly spaces endowed with an equivalence relation. The simple and the sub-directly irreducible Q -distributive lattices are determined and it is shown that the lattices of equational classes of Q -distributive lattices is a chain of type ω + 1.
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paper_title: Information Algebras: Generic Structures for Inference

paper_content:

1 Introduction.- 2 Valuation Algebras.- 2.1 The Framework.- 2.2 Axioms.- 2.3 Examples of Valuation Algebras.- 2.3.1 Indicator Functions.- 2.3.2 Relations.- 2.3.3 Probability Potentials.- 2.3.4 Possibility Potentials.- 2.3.5 Spohn Potentials.- 2.3.6 Set Potentials.- 2.3.7 Gaussian Potentials.- 2.4 Partial Marginalization.- 3 Algebraic Theory.- 3.1 Congruences.- 3.2 Domain-Free Valuation Algebras.- 3.3 Subalgebras, Homomorphisms.- 3.3.1 Subalgebras.- 3.3.2 Homomorphisms and Isomorphisms.- 3.3.3 Weak Subalgebras and Homomorphisms.- 3.4 Null Valuations.- 3.5 Regular Valuation Algebras.- 3.6 Separative Valuation Algebras.- 3.7 Scaled Valuation Algebras.- 4 Local Computation.- 4.1 Fusion Algorithm.- 4.2 Collect Algorithm.- 4.3 Computing Multiple Marginals.- 4.4 Architectures with Division.- 4.5 Computations in Valuation Algebras with Partial Marginalization.- 4.6 Scaling and Updating.- 5 Conditional Independence.- 5.1 Factorization and Graphical Models.- 5.2 Conditionals in Regular Algebras.- 5.3 Conditionals in Separative Algebras.- 6 Information Algebras.- 6.1 Idempotency.- 6.2 Partial Order of Information.- 6.3 File Systems.- 6.4 Information Systems.- 6.5 Examples.- 6.5.1 Propositional Logic.- 6.5.2 Boolean Information Algebras.- 6.5.3 Linear Equations.- 6.5.4 Linear Inequalities.- 6.6 Compact Systems.- 6.7 Mappings.- 7 Uncertain Information.- 7.1 Algebra of Random Variables.- 7.1.1 Random Variables.- 7.1.2 Allocation of Probability.- 7.1.3 Support Functions.- 7.2 Probabilistic Argumentation Systems.- 7.2.1 Assumption-Based Information.- 7.2.2 Probabilistic Argumentation Systems.- 7.2.3 Operations with Argumentation Systems.- 7.3 Allocations of Probability.- 7.3.1 Algebra of Allocations.- 7.3.2 Normalized Allocations.- 7.3.3 Random Variables and Allocations.- 7.3.4 Labeled Allocations and their Algebra.- 7.4 Independent Sources.- 7.4.1 Independent Random Variables.- 7.4.2 Algebra of Bpas.- References.
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paper_title: Introduction to Lattices and Order

paper_content:

1. Ordered sets 2. Special types of ordered set 3. Lattices as algebraic structures 4. Boolean algebras 5. The structure of finite distributive lattices and finite Boolean algebras 6. Ideals, filters, and congruences 7. The Stone Representation Theorem for Boolean algebras 8. Lattices in algebra Appendix: outline of relevant basic topology.
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paper_title: Information Algebras: Generic Structures for Inference

paper_content:

1 Introduction.- 2 Valuation Algebras.- 2.1 The Framework.- 2.2 Axioms.- 2.3 Examples of Valuation Algebras.- 2.3.1 Indicator Functions.- 2.3.2 Relations.- 2.3.3 Probability Potentials.- 2.3.4 Possibility Potentials.- 2.3.5 Spohn Potentials.- 2.3.6 Set Potentials.- 2.3.7 Gaussian Potentials.- 2.4 Partial Marginalization.- 3 Algebraic Theory.- 3.1 Congruences.- 3.2 Domain-Free Valuation Algebras.- 3.3 Subalgebras, Homomorphisms.- 3.3.1 Subalgebras.- 3.3.2 Homomorphisms and Isomorphisms.- 3.3.3 Weak Subalgebras and Homomorphisms.- 3.4 Null Valuations.- 3.5 Regular Valuation Algebras.- 3.6 Separative Valuation Algebras.- 3.7 Scaled Valuation Algebras.- 4 Local Computation.- 4.1 Fusion Algorithm.- 4.2 Collect Algorithm.- 4.3 Computing Multiple Marginals.- 4.4 Architectures with Division.- 4.5 Computations in Valuation Algebras with Partial Marginalization.- 4.6 Scaling and Updating.- 5 Conditional Independence.- 5.1 Factorization and Graphical Models.- 5.2 Conditionals in Regular Algebras.- 5.3 Conditionals in Separative Algebras.- 6 Information Algebras.- 6.1 Idempotency.- 6.2 Partial Order of Information.- 6.3 File Systems.- 6.4 Information Systems.- 6.5 Examples.- 6.5.1 Propositional Logic.- 6.5.2 Boolean Information Algebras.- 6.5.3 Linear Equations.- 6.5.4 Linear Inequalities.- 6.6 Compact Systems.- 6.7 Mappings.- 7 Uncertain Information.- 7.1 Algebra of Random Variables.- 7.1.1 Random Variables.- 7.1.2 Allocation of Probability.- 7.1.3 Support Functions.- 7.2 Probabilistic Argumentation Systems.- 7.2.1 Assumption-Based Information.- 7.2.2 Probabilistic Argumentation Systems.- 7.2.3 Operations with Argumentation Systems.- 7.3 Allocations of Probability.- 7.3.1 Algebra of Allocations.- 7.3.2 Normalized Allocations.- 7.3.3 Random Variables and Allocations.- 7.3.4 Labeled Allocations and their Algebra.- 7.4 Independent Sources.- 7.4.1 Independent Random Variables.- 7.4.2 Algebra of Bpas.- References.
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paper_title: Information Algebras: Generic Structures for Inference

paper_content:

1 Introduction.- 2 Valuation Algebras.- 2.1 The Framework.- 2.2 Axioms.- 2.3 Examples of Valuation Algebras.- 2.3.1 Indicator Functions.- 2.3.2 Relations.- 2.3.3 Probability Potentials.- 2.3.4 Possibility Potentials.- 2.3.5 Spohn Potentials.- 2.3.6 Set Potentials.- 2.3.7 Gaussian Potentials.- 2.4 Partial Marginalization.- 3 Algebraic Theory.- 3.1 Congruences.- 3.2 Domain-Free Valuation Algebras.- 3.3 Subalgebras, Homomorphisms.- 3.3.1 Subalgebras.- 3.3.2 Homomorphisms and Isomorphisms.- 3.3.3 Weak Subalgebras and Homomorphisms.- 3.4 Null Valuations.- 3.5 Regular Valuation Algebras.- 3.6 Separative Valuation Algebras.- 3.7 Scaled Valuation Algebras.- 4 Local Computation.- 4.1 Fusion Algorithm.- 4.2 Collect Algorithm.- 4.3 Computing Multiple Marginals.- 4.4 Architectures with Division.- 4.5 Computations in Valuation Algebras with Partial Marginalization.- 4.6 Scaling and Updating.- 5 Conditional Independence.- 5.1 Factorization and Graphical Models.- 5.2 Conditionals in Regular Algebras.- 5.3 Conditionals in Separative Algebras.- 6 Information Algebras.- 6.1 Idempotency.- 6.2 Partial Order of Information.- 6.3 File Systems.- 6.4 Information Systems.- 6.5 Examples.- 6.5.1 Propositional Logic.- 6.5.2 Boolean Information Algebras.- 6.5.3 Linear Equations.- 6.5.4 Linear Inequalities.- 6.6 Compact Systems.- 6.7 Mappings.- 7 Uncertain Information.- 7.1 Algebra of Random Variables.- 7.1.1 Random Variables.- 7.1.2 Allocation of Probability.- 7.1.3 Support Functions.- 7.2 Probabilistic Argumentation Systems.- 7.2.1 Assumption-Based Information.- 7.2.2 Probabilistic Argumentation Systems.- 7.2.3 Operations with Argumentation Systems.- 7.3 Allocations of Probability.- 7.3.1 Algebra of Allocations.- 7.3.2 Normalized Allocations.- 7.3.3 Random Variables and Allocations.- 7.3.4 Labeled Allocations and their Algebra.- 7.4 Independent Sources.- 7.4.1 Independent Random Variables.- 7.4.2 Algebra of Bpas.- References.
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paper_title: Information Algebras: Generic Structures for Inference

paper_content:

1 Introduction.- 2 Valuation Algebras.- 2.1 The Framework.- 2.2 Axioms.- 2.3 Examples of Valuation Algebras.- 2.3.1 Indicator Functions.- 2.3.2 Relations.- 2.3.3 Probability Potentials.- 2.3.4 Possibility Potentials.- 2.3.5 Spohn Potentials.- 2.3.6 Set Potentials.- 2.3.7 Gaussian Potentials.- 2.4 Partial Marginalization.- 3 Algebraic Theory.- 3.1 Congruences.- 3.2 Domain-Free Valuation Algebras.- 3.3 Subalgebras, Homomorphisms.- 3.3.1 Subalgebras.- 3.3.2 Homomorphisms and Isomorphisms.- 3.3.3 Weak Subalgebras and Homomorphisms.- 3.4 Null Valuations.- 3.5 Regular Valuation Algebras.- 3.6 Separative Valuation Algebras.- 3.7 Scaled Valuation Algebras.- 4 Local Computation.- 4.1 Fusion Algorithm.- 4.2 Collect Algorithm.- 4.3 Computing Multiple Marginals.- 4.4 Architectures with Division.- 4.5 Computations in Valuation Algebras with Partial Marginalization.- 4.6 Scaling and Updating.- 5 Conditional Independence.- 5.1 Factorization and Graphical Models.- 5.2 Conditionals in Regular Algebras.- 5.3 Conditionals in Separative Algebras.- 6 Information Algebras.- 6.1 Idempotency.- 6.2 Partial Order of Information.- 6.3 File Systems.- 6.4 Information Systems.- 6.5 Examples.- 6.5.1 Propositional Logic.- 6.5.2 Boolean Information Algebras.- 6.5.3 Linear Equations.- 6.5.4 Linear Inequalities.- 6.6 Compact Systems.- 6.7 Mappings.- 7 Uncertain Information.- 7.1 Algebra of Random Variables.- 7.1.1 Random Variables.- 7.1.2 Allocation of Probability.- 7.1.3 Support Functions.- 7.2 Probabilistic Argumentation Systems.- 7.2.1 Assumption-Based Information.- 7.2.2 Probabilistic Argumentation Systems.- 7.2.3 Operations with Argumentation Systems.- 7.3 Allocations of Probability.- 7.3.1 Algebra of Allocations.- 7.3.2 Normalized Allocations.- 7.3.3 Random Variables and Allocations.- 7.3.4 Labeled Allocations and their Algebra.- 7.4 Independent Sources.- 7.4.1 Independent Random Variables.- 7.4.2 Algebra of Bpas.- References.
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paper_title: Information Algebras: Generic Structures for Inference

paper_content:

1 Introduction.- 2 Valuation Algebras.- 2.1 The Framework.- 2.2 Axioms.- 2.3 Examples of Valuation Algebras.- 2.3.1 Indicator Functions.- 2.3.2 Relations.- 2.3.3 Probability Potentials.- 2.3.4 Possibility Potentials.- 2.3.5 Spohn Potentials.- 2.3.6 Set Potentials.- 2.3.7 Gaussian Potentials.- 2.4 Partial Marginalization.- 3 Algebraic Theory.- 3.1 Congruences.- 3.2 Domain-Free Valuation Algebras.- 3.3 Subalgebras, Homomorphisms.- 3.3.1 Subalgebras.- 3.3.2 Homomorphisms and Isomorphisms.- 3.3.3 Weak Subalgebras and Homomorphisms.- 3.4 Null Valuations.- 3.5 Regular Valuation Algebras.- 3.6 Separative Valuation Algebras.- 3.7 Scaled Valuation Algebras.- 4 Local Computation.- 4.1 Fusion Algorithm.- 4.2 Collect Algorithm.- 4.3 Computing Multiple Marginals.- 4.4 Architectures with Division.- 4.5 Computations in Valuation Algebras with Partial Marginalization.- 4.6 Scaling and Updating.- 5 Conditional Independence.- 5.1 Factorization and Graphical Models.- 5.2 Conditionals in Regular Algebras.- 5.3 Conditionals in Separative Algebras.- 6 Information Algebras.- 6.1 Idempotency.- 6.2 Partial Order of Information.- 6.3 File Systems.- 6.4 Information Systems.- 6.5 Examples.- 6.5.1 Propositional Logic.- 6.5.2 Boolean Information Algebras.- 6.5.3 Linear Equations.- 6.5.4 Linear Inequalities.- 6.6 Compact Systems.- 6.7 Mappings.- 7 Uncertain Information.- 7.1 Algebra of Random Variables.- 7.1.1 Random Variables.- 7.1.2 Allocation of Probability.- 7.1.3 Support Functions.- 7.2 Probabilistic Argumentation Systems.- 7.2.1 Assumption-Based Information.- 7.2.2 Probabilistic Argumentation Systems.- 7.2.3 Operations with Argumentation Systems.- 7.3 Allocations of Probability.- 7.3.1 Algebra of Allocations.- 7.3.2 Normalized Allocations.- 7.3.3 Random Variables and Allocations.- 7.3.4 Labeled Allocations and their Algebra.- 7.4 Independent Sources.- 7.4.1 Independent Random Variables.- 7.4.2 Algebra of Bpas.- References.
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paper_title: Quantifiers on distributive lattices

paper_content:

Abstract A Q -distributive lattice is an algebra 〈 L , ∧, ∨, ∇, 0, 1〉 of type (2, 2, 1, 0, 0) such that 〈 L , ∧, ∨, 0, 1〉 is a bounded distributive lattice and ∇ satisfies the equations: (1) ∇0 = 0, (2) x ∧ ∇ x = x , (3) ∇( x ∧ ∇ y ) = ∇ x ∧ ∇ y and (4) ∇( x ∨ y ) = ∇ x ∨ ∇ y . The opposite of the category of Q -distributive lattices is described in terms of Priestly spaces endowed with an equivalence relation. The simple and the sub-directly irreducible Q -distributive lattices are determined and it is shown that the lattices of equational classes of Q -distributive lattices is a chain of type ω + 1.
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paper_title: Information Algebras: Generic Structures for Inference

paper_content:

1 Introduction.- 2 Valuation Algebras.- 2.1 The Framework.- 2.2 Axioms.- 2.3 Examples of Valuation Algebras.- 2.3.1 Indicator Functions.- 2.3.2 Relations.- 2.3.3 Probability Potentials.- 2.3.4 Possibility Potentials.- 2.3.5 Spohn Potentials.- 2.3.6 Set Potentials.- 2.3.7 Gaussian Potentials.- 2.4 Partial Marginalization.- 3 Algebraic Theory.- 3.1 Congruences.- 3.2 Domain-Free Valuation Algebras.- 3.3 Subalgebras, Homomorphisms.- 3.3.1 Subalgebras.- 3.3.2 Homomorphisms and Isomorphisms.- 3.3.3 Weak Subalgebras and Homomorphisms.- 3.4 Null Valuations.- 3.5 Regular Valuation Algebras.- 3.6 Separative Valuation Algebras.- 3.7 Scaled Valuation Algebras.- 4 Local Computation.- 4.1 Fusion Algorithm.- 4.2 Collect Algorithm.- 4.3 Computing Multiple Marginals.- 4.4 Architectures with Division.- 4.5 Computations in Valuation Algebras with Partial Marginalization.- 4.6 Scaling and Updating.- 5 Conditional Independence.- 5.1 Factorization and Graphical Models.- 5.2 Conditionals in Regular Algebras.- 5.3 Conditionals in Separative Algebras.- 6 Information Algebras.- 6.1 Idempotency.- 6.2 Partial Order of Information.- 6.3 File Systems.- 6.4 Information Systems.- 6.5 Examples.- 6.5.1 Propositional Logic.- 6.5.2 Boolean Information Algebras.- 6.5.3 Linear Equations.- 6.5.4 Linear Inequalities.- 6.6 Compact Systems.- 6.7 Mappings.- 7 Uncertain Information.- 7.1 Algebra of Random Variables.- 7.1.1 Random Variables.- 7.1.2 Allocation of Probability.- 7.1.3 Support Functions.- 7.2 Probabilistic Argumentation Systems.- 7.2.1 Assumption-Based Information.- 7.2.2 Probabilistic Argumentation Systems.- 7.2.3 Operations with Argumentation Systems.- 7.3 Allocations of Probability.- 7.3.1 Algebra of Allocations.- 7.3.2 Normalized Allocations.- 7.3.3 Random Variables and Allocations.- 7.3.4 Labeled Allocations and their Algebra.- 7.4 Independent Sources.- 7.4.1 Independent Random Variables.- 7.4.2 Algebra of Bpas.- References.
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paper_title: Logical Deduction using the Local Computation Framework

paper_content:

The Local Computation Framework has been used to improve the efficiency of computation in various uncertainty formalisms. This paper shows how it can be applied to logical deduction in first-order logic, modal and conditional logics, circumscription and possibilistic logic.
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paper_title: Information Algebras and Consequence Operators

paper_content:

We explore a connection between different ways of representing information in computer science. We show that relational databases, modules, algebraic specifications and constraint systems all satisfy the same ten axioms. A commutative semigroup together with a lattice satisfying these axioms is then called an “information algebra”. We show that any compact consequence operator satisfying the interpolation and the deduction property induces an information algebra. Conversely, each finitary information algebra can be obtained from a consequence operator in this way. Finally we show that arbitrary (not necessarily finitary) information algebras can be represented as some kind of abstract relational database called a tuple system.
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paper_title: Logical basis for information theory and probability theory

paper_content:

A new logical basis for information theory as well as probability theory is proposed, based on computing complexity.
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paper_title: Introduction to Lattices and Order

paper_content:

1. Ordered sets 2. Special types of ordered set 3. Lattices as algebraic structures 4. Boolean algebras 5. The structure of finite distributive lattices and finite Boolean algebras 6. Ideals, filters, and congruences 7. The Stone Representation Theorem for Boolean algebras 8. Lattices in algebra Appendix: outline of relevant basic topology.
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paper_title: Information Algebras: Generic Structures for Inference

paper_content:

1 Introduction.- 2 Valuation Algebras.- 2.1 The Framework.- 2.2 Axioms.- 2.3 Examples of Valuation Algebras.- 2.3.1 Indicator Functions.- 2.3.2 Relations.- 2.3.3 Probability Potentials.- 2.3.4 Possibility Potentials.- 2.3.5 Spohn Potentials.- 2.3.6 Set Potentials.- 2.3.7 Gaussian Potentials.- 2.4 Partial Marginalization.- 3 Algebraic Theory.- 3.1 Congruences.- 3.2 Domain-Free Valuation Algebras.- 3.3 Subalgebras, Homomorphisms.- 3.3.1 Subalgebras.- 3.3.2 Homomorphisms and Isomorphisms.- 3.3.3 Weak Subalgebras and Homomorphisms.- 3.4 Null Valuations.- 3.5 Regular Valuation Algebras.- 3.6 Separative Valuation Algebras.- 3.7 Scaled Valuation Algebras.- 4 Local Computation.- 4.1 Fusion Algorithm.- 4.2 Collect Algorithm.- 4.3 Computing Multiple Marginals.- 4.4 Architectures with Division.- 4.5 Computations in Valuation Algebras with Partial Marginalization.- 4.6 Scaling and Updating.- 5 Conditional Independence.- 5.1 Factorization and Graphical Models.- 5.2 Conditionals in Regular Algebras.- 5.3 Conditionals in Separative Algebras.- 6 Information Algebras.- 6.1 Idempotency.- 6.2 Partial Order of Information.- 6.3 File Systems.- 6.4 Information Systems.- 6.5 Examples.- 6.5.1 Propositional Logic.- 6.5.2 Boolean Information Algebras.- 6.5.3 Linear Equations.- 6.5.4 Linear Inequalities.- 6.6 Compact Systems.- 6.7 Mappings.- 7 Uncertain Information.- 7.1 Algebra of Random Variables.- 7.1.1 Random Variables.- 7.1.2 Allocation of Probability.- 7.1.3 Support Functions.- 7.2 Probabilistic Argumentation Systems.- 7.2.1 Assumption-Based Information.- 7.2.2 Probabilistic Argumentation Systems.- 7.2.3 Operations with Argumentation Systems.- 7.3 Allocations of Probability.- 7.3.1 Algebra of Allocations.- 7.3.2 Normalized Allocations.- 7.3.3 Random Variables and Allocations.- 7.3.4 Labeled Allocations and their Algebra.- 7.4 Independent Sources.- 7.4.1 Independent Random Variables.- 7.4.2 Algebra of Bpas.- References.
---
paper_title: On Semantic Information

paper_content:

In the last couple of decades, a logician or a philosopher has run a risk whenever he has put the term “information” into the title of one of his papers. In these days, the term “information” often creates an expectation that the paper has something to do with that impressive body of results in communication theory which was first known as theory of transmission of information but which now is elliptically called information theory (in the United States at least).1 For the purposes of this paper, I shall speak of it as statistical information theory. I want to begin by making it clear that I have nothing to contribute to this statistical information theory as it is usually developed.
---
paper_title: Logical Deduction using the Local Computation Framework

paper_content:

The Local Computation Framework has been used to improve the efficiency of computation in various uncertainty formalisms. This paper shows how it can be applied to logical deduction in first-order logic, modal and conditional logics, circumscription and possibilistic logic.
---
paper_title: Surface Information and Depth Information

paper_content:

In spite of its obvious importance, the notion of information has been strangely neglected by most contemporary epistemologists and philosophers of language. It seems to me more fruitful, however, to try to correct the situation than to lament it. In this paper, I shall point out some largely unexplored possibilities of using the concept of information in logic and especially in the philosophy of logic.
---
paper_title: Answers to Questions

paper_content:

In an earlier paper,1 have tried to answer the question of answerhood, that is, tried to specify the conditions on which a response to an English question counts as an answer to it. In this new paper, I shall strive to extend and to deepen that earlier analysis.
---
paper_title: Introduction to Lattices and Order

paper_content:

1. Ordered sets 2. Special types of ordered set 3. Lattices as algebraic structures 4. Boolean algebras 5. The structure of finite distributive lattices and finite Boolean algebras 6. Ideals, filters, and congruences 7. The Stone Representation Theorem for Boolean algebras 8. Lattices in algebra Appendix: outline of relevant basic topology.
---
paper_title: Mathematical Theory of Domains

paper_content:

Domain theory is the mathematical framework that is used to model the semantics of computer programs and the theory of computation. This is the first book on the subject that attempts to provide a rigorous introduction to the topic in a manner accessible to computer scientists by motivating the mathematics with computer science examples.
---
paper_title: Information Algebras: Generic Structures for Inference

paper_content:

1 Introduction.- 2 Valuation Algebras.- 2.1 The Framework.- 2.2 Axioms.- 2.3 Examples of Valuation Algebras.- 2.3.1 Indicator Functions.- 2.3.2 Relations.- 2.3.3 Probability Potentials.- 2.3.4 Possibility Potentials.- 2.3.5 Spohn Potentials.- 2.3.6 Set Potentials.- 2.3.7 Gaussian Potentials.- 2.4 Partial Marginalization.- 3 Algebraic Theory.- 3.1 Congruences.- 3.2 Domain-Free Valuation Algebras.- 3.3 Subalgebras, Homomorphisms.- 3.3.1 Subalgebras.- 3.3.2 Homomorphisms and Isomorphisms.- 3.3.3 Weak Subalgebras and Homomorphisms.- 3.4 Null Valuations.- 3.5 Regular Valuation Algebras.- 3.6 Separative Valuation Algebras.- 3.7 Scaled Valuation Algebras.- 4 Local Computation.- 4.1 Fusion Algorithm.- 4.2 Collect Algorithm.- 4.3 Computing Multiple Marginals.- 4.4 Architectures with Division.- 4.5 Computations in Valuation Algebras with Partial Marginalization.- 4.6 Scaling and Updating.- 5 Conditional Independence.- 5.1 Factorization and Graphical Models.- 5.2 Conditionals in Regular Algebras.- 5.3 Conditionals in Separative Algebras.- 6 Information Algebras.- 6.1 Idempotency.- 6.2 Partial Order of Information.- 6.3 File Systems.- 6.4 Information Systems.- 6.5 Examples.- 6.5.1 Propositional Logic.- 6.5.2 Boolean Information Algebras.- 6.5.3 Linear Equations.- 6.5.4 Linear Inequalities.- 6.6 Compact Systems.- 6.7 Mappings.- 7 Uncertain Information.- 7.1 Algebra of Random Variables.- 7.1.1 Random Variables.- 7.1.2 Allocation of Probability.- 7.1.3 Support Functions.- 7.2 Probabilistic Argumentation Systems.- 7.2.1 Assumption-Based Information.- 7.2.2 Probabilistic Argumentation Systems.- 7.2.3 Operations with Argumentation Systems.- 7.3 Allocations of Probability.- 7.3.1 Algebra of Allocations.- 7.3.2 Normalized Allocations.- 7.3.3 Random Variables and Allocations.- 7.3.4 Labeled Allocations and their Algebra.- 7.4 Independent Sources.- 7.4.1 Independent Random Variables.- 7.4.2 Algebra of Bpas.- References.
---
paper_title: Logical basis for information theory and probability theory

paper_content:

A new logical basis for information theory as well as probability theory is proposed, based on computing complexity.
---
paper_title: ON TWO TYPES OF CONTINUOUS INFORMATION ALGEBRAS

paper_content:

In this paper (strong) continuity of domain-free information algebras and labeled information algebras are introduced. Relationships between domain-free information algebras and labeled information algebras are mainly explored. It is shown that continuity and compactness can be preserved under certain canonical correspondences between domain-free information algebras and labeled information algebras. Some equivalent characterizations and examples for continuous information algebras are also given.
---
paper_title: Introduction to Lattices and Order

paper_content:

1. Ordered sets 2. Special types of ordered set 3. Lattices as algebraic structures 4. Boolean algebras 5. The structure of finite distributive lattices and finite Boolean algebras 6. Ideals, filters, and congruences 7. The Stone Representation Theorem for Boolean algebras 8. Lattices in algebra Appendix: outline of relevant basic topology.
---
paper_title: Information Algebras: Generic Structures for Inference

paper_content:

1 Introduction.- 2 Valuation Algebras.- 2.1 The Framework.- 2.2 Axioms.- 2.3 Examples of Valuation Algebras.- 2.3.1 Indicator Functions.- 2.3.2 Relations.- 2.3.3 Probability Potentials.- 2.3.4 Possibility Potentials.- 2.3.5 Spohn Potentials.- 2.3.6 Set Potentials.- 2.3.7 Gaussian Potentials.- 2.4 Partial Marginalization.- 3 Algebraic Theory.- 3.1 Congruences.- 3.2 Domain-Free Valuation Algebras.- 3.3 Subalgebras, Homomorphisms.- 3.3.1 Subalgebras.- 3.3.2 Homomorphisms and Isomorphisms.- 3.3.3 Weak Subalgebras and Homomorphisms.- 3.4 Null Valuations.- 3.5 Regular Valuation Algebras.- 3.6 Separative Valuation Algebras.- 3.7 Scaled Valuation Algebras.- 4 Local Computation.- 4.1 Fusion Algorithm.- 4.2 Collect Algorithm.- 4.3 Computing Multiple Marginals.- 4.4 Architectures with Division.- 4.5 Computations in Valuation Algebras with Partial Marginalization.- 4.6 Scaling and Updating.- 5 Conditional Independence.- 5.1 Factorization and Graphical Models.- 5.2 Conditionals in Regular Algebras.- 5.3 Conditionals in Separative Algebras.- 6 Information Algebras.- 6.1 Idempotency.- 6.2 Partial Order of Information.- 6.3 File Systems.- 6.4 Information Systems.- 6.5 Examples.- 6.5.1 Propositional Logic.- 6.5.2 Boolean Information Algebras.- 6.5.3 Linear Equations.- 6.5.4 Linear Inequalities.- 6.6 Compact Systems.- 6.7 Mappings.- 7 Uncertain Information.- 7.1 Algebra of Random Variables.- 7.1.1 Random Variables.- 7.1.2 Allocation of Probability.- 7.1.3 Support Functions.- 7.2 Probabilistic Argumentation Systems.- 7.2.1 Assumption-Based Information.- 7.2.2 Probabilistic Argumentation Systems.- 7.2.3 Operations with Argumentation Systems.- 7.3 Allocations of Probability.- 7.3.1 Algebra of Allocations.- 7.3.2 Normalized Allocations.- 7.3.3 Random Variables and Allocations.- 7.3.4 Labeled Allocations and their Algebra.- 7.4 Independent Sources.- 7.4.1 Independent Random Variables.- 7.4.2 Algebra of Bpas.- References.
---
paper_title: Information Algebras: Generic Structures for Inference

paper_content:

1 Introduction.- 2 Valuation Algebras.- 2.1 The Framework.- 2.2 Axioms.- 2.3 Examples of Valuation Algebras.- 2.3.1 Indicator Functions.- 2.3.2 Relations.- 2.3.3 Probability Potentials.- 2.3.4 Possibility Potentials.- 2.3.5 Spohn Potentials.- 2.3.6 Set Potentials.- 2.3.7 Gaussian Potentials.- 2.4 Partial Marginalization.- 3 Algebraic Theory.- 3.1 Congruences.- 3.2 Domain-Free Valuation Algebras.- 3.3 Subalgebras, Homomorphisms.- 3.3.1 Subalgebras.- 3.3.2 Homomorphisms and Isomorphisms.- 3.3.3 Weak Subalgebras and Homomorphisms.- 3.4 Null Valuations.- 3.5 Regular Valuation Algebras.- 3.6 Separative Valuation Algebras.- 3.7 Scaled Valuation Algebras.- 4 Local Computation.- 4.1 Fusion Algorithm.- 4.2 Collect Algorithm.- 4.3 Computing Multiple Marginals.- 4.4 Architectures with Division.- 4.5 Computations in Valuation Algebras with Partial Marginalization.- 4.6 Scaling and Updating.- 5 Conditional Independence.- 5.1 Factorization and Graphical Models.- 5.2 Conditionals in Regular Algebras.- 5.3 Conditionals in Separative Algebras.- 6 Information Algebras.- 6.1 Idempotency.- 6.2 Partial Order of Information.- 6.3 File Systems.- 6.4 Information Systems.- 6.5 Examples.- 6.5.1 Propositional Logic.- 6.5.2 Boolean Information Algebras.- 6.5.3 Linear Equations.- 6.5.4 Linear Inequalities.- 6.6 Compact Systems.- 6.7 Mappings.- 7 Uncertain Information.- 7.1 Algebra of Random Variables.- 7.1.1 Random Variables.- 7.1.2 Allocation of Probability.- 7.1.3 Support Functions.- 7.2 Probabilistic Argumentation Systems.- 7.2.1 Assumption-Based Information.- 7.2.2 Probabilistic Argumentation Systems.- 7.2.3 Operations with Argumentation Systems.- 7.3 Allocations of Probability.- 7.3.1 Algebra of Allocations.- 7.3.2 Normalized Allocations.- 7.3.3 Random Variables and Allocations.- 7.3.4 Labeled Allocations and their Algebra.- 7.4 Independent Sources.- 7.4.1 Independent Random Variables.- 7.4.2 Algebra of Bpas.- References.
---
paper_title: Information Flow: The Logic of Distributed Systems

paper_content:

Information is a central topic in computer science, cognitive science, and philosophy. In spite of its importance in the "information age," there is no consensus on what information is, what makes it possible, and what it means for one medium to carry information about another. Drawing on ideas from mathematics, computer science, and philosophy, this book addresses the definition and place of information in society. The authors, observing that information flow is possible only within a connected distribution system, provide a mathematically rigorous, philosophically sound foundation for a science of information. They illustrate their theory by applying it to a wide range of phenomena, from file transfer to DNA, from quantum mechanics to speech act theory.
---
paper_title: Continuous Lattices and Domains

paper_content:

Preface Acknowledgements Foreword Introduction 1. A primer on ordered sets and lattices 2. Order theory of domains 3. The Scott topology 4. The Lawson Topology 5. Morphisms and functors 6. Spectral theory of continuous lattices 7. Compact posets and semilattices 8. Topological algebra and lattice theory: Applications Bibliography List of symbols List of categories Index.
---
paper_title: On Semantic Information

paper_content:

In the last couple of decades, a logician or a philosopher has run a risk whenever he has put the term “information” into the title of one of his papers. In these days, the term “information” often creates an expectation that the paper has something to do with that impressive body of results in communication theory which was first known as theory of transmission of information but which now is elliptically called information theory (in the United States at least).1 For the purposes of this paper, I shall speak of it as statistical information theory. I want to begin by making it clear that I have nothing to contribute to this statistical information theory as it is usually developed.
---
paper_title: Probabilistic argumentation systems

paper_content:

Different formalisms for solving problems of inference under uncertainty have been developed so far. The most popular numerical approach is the theory of Bayesian inference [Lauritzen and Spiegelhalter, 1988]. More general approaches are the Dempster-Shafer theory of evidence [Shafer, 1976], and possibility theory [Dubois and Prade, 1990], which is closely related to fuzzy systems. For these systems computer implementations are available. In competition with these numerical methods are different symbolic approaches. Many of them are based on different types of non-monotonic logic.
---
paper_title: Generalized information theory for hints

paper_content:

This paper develops a new uncertainty measure for the theory of hints that complies with the established semantics of statistical information theory and further satisfies all classical requirements for such a measure imposed in the literature. The proposed functional decomposes into conversant uncertainty measures and therefore discloses a new interpretation of the latters as well. By abstracting to equivalence classes of hints we transport the new measure to mass functions in Dempster-Shafer theory and analyse its relationship with the aggregate uncertainty, which currently is the only known functional for the Dempster-Shafer theory of evidence that satisfies the same set of properties. Moreover, the perspective of hints reveals that the standard independence notion in Dempster-Shafer theory called non-interactivity corresponds to an amalgamation of probabilistic independence and qualitative independence between frames of discernment. All results in this paper are developed for arbitrary families of compatible frames generalizing the very specialized multi-variate systems that are usually studied in information theory.
---
paper_title: Surface Information and Depth Information

paper_content:

In spite of its obvious importance, the notion of information has been strangely neglected by most contemporary epistemologists and philosophers of language. It seems to me more fruitful, however, to try to correct the situation than to lament it. In this paper, I shall point out some largely unexplored possibilities of using the concept of information in logic and especially in the philosophy of logic.
---
paper_title: Allocations of Probability

paper_content:

This paper studies belief functions, set functions which are normalized and monotone of order 8. The concepts of continuity and condensability are defined for belief functions, and it is shown how to extend continuous or condensable belief functions from an algebra of subsets to the corresponding power set. The main tool used in this extension is the theorem that every belief function can be represented by an allocation of probability i.e., by a n-homomorphism into a positive and completely additive probability algebra. This representation can be deduced either from an integral representation due to Choquet or from more elementary work by Revuz and Honeycutt.
---
paper_title: Comparing Questions and Answers: A bit of Logic, a bit of Language, and some bits of Information.

paper_content:

Notions like ‘entropy’ and ‘(expected) value of observations’ are widely used in science to determine which experiment to conduct to make a better informed choice between a set of scientific theories that are all consistent with the data, and in military campaigns to determine the coordinates of the enemy’s target. But these notions are, or so I will argue, almost equally important for our use of language in daily life as they are for scientific inquiry’s and nuclear warfare, although the stakes are, perhaps, a bit lower. I will make use of these notions to measure how ‘good’ particular questions and answers are in particular circumstances. In doing so, I will extend and/or refine the qualitative approach towards such measurements proposed by Groenendijk & Stokhof (1984). The refinements are due to the fact that I also take into account (i) probabilities, (ii) utilities, and (iii) the idea (cf. van Rooy, 1999b) that we ask questions to resolve decision problems. In this paper I will first explain Groenendijk & Stokhof’s partition based analysis of questions, and then discuss their qualitative method of measurement. Next, I will take also probabilities into account, and show how a natural quantitative measure of informativity can be defined in terms of it. Following the lead of Communication Theory and Inductive Logic, I will then show that we can also describe a natural measure of the informative value of questions and answers in terms of conditional entropy’s, when we take into account that questions are asked to resolve decision problems. Finally, I will argue that to measure the value of questions and answers we should in general also take utilities seriously, and following standard practice in Statistical Decision Theory, I show how some intuitively appealing utility values can be specified.
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paper_title: Questions and answers: Semantics and logic

paper_content:

I sketch a semantic interpretation of questions in first-order logic. Questions are interpreted as partitions of sets of possible worlds. The semantics I use is an update semantics. I discuss several notions of entailment and answerhood, and give the bare outlines of Balder ten Cate & Chung-chieh Shan’s syntactic characterization of these notions.
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paper_title: SUPPORT- AND PLAUSIBILITY FUNCTIONS INDUCED BY FILTER-VALUED MAPPINGS

paper_content:

Abstract This paper introduces a mathematical model of a hint as a body of imprecise and uncertain information. Hints are used to judge hypotheses: the degree to which a hint supports a hypothesis and the degree to which a hypothesis appears as plausible in the light of a hint are defined. This leads in turn to support-and plausibility functions. Those functions are characterized as set functions which are normalized and monotone or alternating of order ∞. This relates the present work to G. Shafer's mathematical theory of evidence. However, whereas Shafer starts out with an axiomatic definition of belief functions, the notion of a hint is considered here as the basic element of the theory. It is shown that a hint contains more information than is conveyed by its support function alone. Also hints allow for a straightforward and logical derivation of Dempster's rule for combining independent and dependent bodies of information. Thus, this paper presents the mathematical theory of evidence for general, inf...
---
paper_title: Axioms for probability and belief-function propagation

paper_content:

In this paper, we describe an abstract framework and axioms under which exact local computation of marginals is possible. The primitive objects of the framework are variables and valuations. The primitive operators of the framework are combination and marginalization. These operate on valuations. We state three axioms for these operators and we derive the possibility of local computation from the axioms. Next, we describe a propagation scheme for computing marginals of a valuation when we have a factorization of the valuation on a hypertree. Finally we show how the problem of computing marginals of joint probability distributions and joint belief functions fits the general framework.
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paper_title: Formal Properties of Conditional Independence in Different Calculi of AI

paper_content:

In this paper formal properties of CI in different frameworks are studied. The first part is devoted to the comparison of three different frameworks for study CI: probability theory, theory of relational databases and Spohn's theory of ordinal conditional functions. Although CI-models arising in these frameworks are very similar (they satisfy semigraphoid axioms) we give examples showing that their formal properties still differ (each other). On the other hand, we find that (within each of these frameworks) there exists no finite complete axiomatic characterization of CI-models by finding an infinite set of sound inference rules (the same in all three frameworks). In the second part further frameworks for CI are discussed: Dempster-Shafer theory, possibility theory and (general) Shenoy's theory of valuation-based systems.
---
paper_title: Logical basis for information theory and probability theory

paper_content:

A new logical basis for information theory as well as probability theory is proposed, based on computing complexity.
---
paper_title: Upper and Lower Probabilities Induced by a Multivalued Mapping.

paper_content:

A multivalued mapping from a space X to a space S carries a probability measure defined over subsets of X into a system of upper and lower probabilities over subsets of S. Some basic properties of such systems are explored in Sects. 1 and 2. Other approaches to upper and lower probabilities are possible and some of these are related to the present approach in Sect. 3. A distinctive feature of the present approach is a rule for conditioning, or more generally, a rule for combining sources of information, as discussed in Sects. 4 and 5. Finally, the context in statistical inference from which the present theory arose is sketched briefly in Sect. 6.
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paper_title: Theory of information : fundamentality, diversity and unification

paper_content:

How Do We Know What Information Is? Information in Society, Information in Nature, Technological Aspects of Information General Theory of Information: Information Ontology, Information Axiology, Information Typology Information, Data, and Knowledge Emotions and Information Statistical Information Theory: Information and Entropy, Information and Communication in Technical Systems, Quantum Information, Information and Problem Solving, Axiomatization of Information Measurement, Information in Physics Semantic Information Theory: The Three Dimensions of Information, Logic in Information, Knowledge from Information Algorithmic Information Theory: Information and Complexity, Recursive Algorithmic Information Theory, Inductive Algorithmic Information Theory, Relative Information Measures, Axiomatic Information Measures Pragmatic Information Theory: Economic Approach, Mission-Oriented Approach and Value of Information, Transformational Approach, Dynamics of Information: Information Flow, Operator Approach, Information Algebra.
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paper_title: On the Desirability of Acyclic Database Schemes

paper_content:

A class of database schemes, called acychc, was recently introduced. It is shown that this class has a number of desirable properties. In particular, several desirable properties that have been studied by other researchers m very different terms are all shown to be eqmvalent to acydicity. In addition, several equivalent charactenzauons of the class m terms of graphs and hypergraphs are given, and a smaple algorithm for determining acychclty is presented. Also given are several eqmvalent characterizations of those sets M of multivalued dependencies such that M is the set of muRlvalued dependencies that are the consequences of a given join dependency. Several characterizations for a conflict-free (in the sense of Lien) set of muluvalued dependencies are provided.
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paper_title: Separoids: A Mathematical Framework for Conditional Independence and Irrelevance

paper_content:

We introduce an axiomatic definition of a mathematical structure that we term a separoid. We develop some general mathematical properties of separoids and related axiom systems, as well as connections with other mathematical structures, such as distributive lattices, Hilbert spaces, and graphs. And we show, by means of a detailed account of a number of models of the separoid axioms, how the concept of separoid unifies a variety of notions of ‘irrelevance’ arising out of different formalisms for representing uncertainty in Probability, Statistics, Artificial Intelligence, and other fields.
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paper_title: Computation in Valuation Algebras

paper_content:

The main goal of this chapter is to describe an abstract framework called valuation algebra for computing marginals using local computation. The valuation algebra framework is useful in many domains, and especially for managing uncertainty in expert systems using probability, Dempster-Shafer belief functions, Spohnian epistemic belief theory, and possibility theory.
---
paper_title: Answers to Questions

paper_content:

In an earlier paper,1 have tried to answer the question of answerhood, that is, tried to specify the conditions on which a response to an English question counts as an answer to it. In this new paper, I shall strive to extend and to deepen that earlier analysis.
---
paper_title: Algorithmic Information Theory

paper_content:

This paper reviews algorithmic information theory, which is an attempt to apply information-theoretic and probabilistic ideas to recursive function theory. Typical concerns in this approach are, for example, the number of bits of information required to specify an algorithm, or the probability that a program whose bits are chosen by coin flipping produces a given output. During the past few years the definitions of algorithmic information theory have been reformulated. The basic features of the new formalism are presented here and certain results of R. M. Solovay are reported.
---
paper_title: Probabilistic Networks and Expert Systems: Exact Computational Methods for Bayesian Networks

paper_content:

WINNER OF THE 2001 DEGROOT PRIZE! Probabilistic expert systems are graphical networks that support the modelling of uncertainty and decisions in large complex domains, while retaining ease of calculation. Building on original research by the authors over a number of years, this book gives a thorough and rigorous mathematical treatment of the underlying ideas, structures, and algorithms, emphasizing those cases in which exact answers are obtainable. It covers both the updating of probabilistic uncertainty in the light of new evidence, and statistical inference, about unknown probabilities or unknown model structure, in the light of new data. The careful attention to detail will make this work an important reference source for all those involved in the theory and applications of probabilistic expert systems. This book was awarded the first DeGroot Prize by the International Society for Bayesian Analysis for a book making an important, timely, thorough, and notably original contribution to the statistics literature.
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paper_title: Three approaches to the quantitative definition of information

paper_content:

A method and apparatus for cutting resilient foamed synthetic plastics filter material by pressing a die against a block of such material to provide cuts which are interleaved extending from opposite faces of the block the cuts at each such face being joined by a curved cut conforming to the desired exterior surface shape of the cut block when it is stretched to provide a corrugated configuration.
---
paper_title: Introduction to Lattices and Order

paper_content:

1. Ordered sets 2. Special types of ordered set 3. Lattices as algebraic structures 4. Boolean algebras 5. The structure of finite distributive lattices and finite Boolean algebras 6. Ideals, filters, and congruences 7. The Stone Representation Theorem for Boolean algebras 8. Lattices in algebra Appendix: outline of relevant basic topology.
---

Title: An Algebraic Theory of Information: An Introduction and Survey

Section 1: Introduction: Modeling Information
Description 1: Explain the proposed approach to the concept of information, including the motivation for and basic principles behind the algebraic structures used to model information.

Section 2: Modeling Questions
Description 2: Discuss the concept of questions within the algebraic framework, including the granularity and ordering of questions.

Section 3: Labeled Information
Description 3: Describe the concept of labeled pieces of information and the algebraic operations for combination and projection within the context of a lattice of questions.

Section 4: Transport of Information
Description 4: Elaborate on extending the projection operator to a total transport map and discuss the properties and implications of this transport operation.

Section 5: Another View
Description 5: Present an alternative perspective on information algebras focusing on domain-free information and the algebraic structure it induces.

Section 6: Order
Description 6: Introduce and explore the concept of order within information algebras, including how pieces of information can be compared based on their information content.

Section 7: Ideal Completion
Description 7: Discuss the concept of ideal completion and how ideals form an information algebra by themselves.

Section 8: Support
Description 8: Investigate the notion of support in domain-free information algebras and its implications for extracting information related to specific questions.

Section 9: Atoms
Description 9: Examine the role of atomic elements in labeled information algebras and their use in constructing generalized relational information algebras.

Section 10: Generic Constructions
Description 10: Present general methods of constructing information algebras, including generalized relational algebras, quantifier algebras, and consequence operators.

Section 11: Morphisms
Description 11: Discuss the concept of morphisms between information algebras and how maps between information algebras form an algebra themselves.

Section 12: Finiteness
Description 12: Consider the problem of finiteness in information algebras and the conditions under which finite information elements can approximate infinite information.

Section 13: Continuous Maps
Description 13: Define and elaborate on continuous maps between compact information algebras and how these maps form a continuous information algebra.

Section 14: Categories of Information Algebras
Description 14: Introduce the categories of information algebras and compact information algebras, and discuss their properties such as Cartesian closure.

Section 15: Outlook
Description 15: Provide an outlook on additional topics and open questions in the theory of information algebras, such as its relation to other information theories and potential areas for further research.