arXiv subjects

Robert E. Kent

Publications and source records attributed to Robert E. Kent.

At least 19 recordsLinked to original sources

arXiv2024

The Architecture of Truth

The theory of institutions is framed as an indexed/fibered duality, where the indexed aspect specifies the fibered aspect. Tarski represented truth in terms of a satisfaction relation. The theory of institutions encodes satisfaction as its core architecture in the indexed aspect. Logical environments enrich this truth architecture by axiomatizing the truth adjunction in the fibered aspect. The truth architecture is preserved by morphisms of logical environments. (Although not every institution is a logical environment, each institution has an associated logical environment defined via the intent of the structures of the institution, and each institution is represented by an indexed functor into the structure category of the classification logical environment $\mathtt{Cls}$.)

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arXiv2024

Truth Factors

Truth refers to the satisfaction relation used to define the semantics of model-theoretic languages. The satisfaction relation for first order languages (truth classification), and the preservation of truth by first order interpretations (truth infomorphism), is a motivating example in the theory of Information Flow (IF) (Barwise and Seligman 1997). The abstract theory of satisfaction is the basis for the theory of institutions (Goguen and Burstall 1992). Factoring refers to categorical factorization systems. The concept lattice, which is the central structure studied by the theory of Formal Concept Analysis (FCA) (Ganter and Wille 1999), is constructed by a factorization. The study of classification structures (IF) and the study of conceptual structures (FCA) aim (at least is part) to provide a principled foundation for the logical theory of knowledge representation and organization. In an effort to unify these two areas, the paper "Distributed Conceptual Structures" (Kent 2002) abstracted the basic theorem of FCA in order to established three levels of categorical equivalence between classification structures and conceptual structures. In this paper we refine this approach by resolving the equivalence as the factorization of three isomorphic versions: relation, function and Galois connection. We develop the latter more algebraic version of the equivalence as the polar factorization of Galois connections. We advocate this abstract adjunctive representation of classification and conceptual structures.

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arXiv2024

The Characterization of Abstract Truth and its Factorization

Human knowledge is made up of the conceptual structures of many communities of interest. In order to establish coherence in human knowledge representation, it is important to enable communication between the conceptual structures of different communities The conceptual structures of any particular community is representable in an ontology. Such a ontology provides a formal linguistic standard for that community. However, a standard community ontology is established for various purposes, and makes choices that force a given interpretation, while excluding others that may be equally valid for other purposes. Hence, a given representation is relative to the purpose for that representation. Due to this relativity of representation, in the larger scope of all human knowledge it is more important to standardize methods and frameworks for relating ontologies than to standardize any particular choice of ontology. The standardization of methods and frameworks is called the semantic integration of ontologies.

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arXiv2023

FOLE Equivalence

The first-order logical environment FOLE [5] provides a rigorous and principled approach to distributed interoperable first-order information systems. FOLE has been developed in two forms: a classification form and an interpretation form. Two papers represent FOLE in a classification form corresponding to ideas of the Information Flow Framework [11],[12],[13]: the first paper [6] provides a foundation that connects elements of the ERA data model [2] with components of the first-order logical environment FOLE; the second paper [7] provides a superstructure that extends FOLE to the formalisms of first-order logic. The formalisms in the classification form of FOLE provide an appropriate framework for developing the relational calculus. Two other papers represent FOLE in an interpretation form: the first paper [8] develops the notion of the FOLE table following the relational model [3]; the second paper [9] discusses the notion of a FOLE relational database. All the operations of the relational algebra have been rigorously developed [10] using the interpretation form of FOLE. The present study demonstrates that the classification form of FOLE is informationally equivalent to the interpretation form of FOLE. In general, the FOLE representation uses a conceptual structures approach, that is completely compatible with formal concept analysis [4] and information flow [1].

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arXiv2023

The ERA of FOLE: Foundation

This paper discusses the representation of ontologies in the first-order logical environment {\ttfamily FOLE}. An ontology defines the primitives with which to model the knowledge resources for a community of discourse. These primitives consist of classes, relationships and properties. An ontology uses formal axioms to constrain the interpretation of these primitives. In short, an ontology specifies a logical theory. This paper continues the discussion of the representation and interpretation of ontologies in the first-order logical environment {\ttfamily FOLE}. The formalism and semantics of (many-sorted) first-order logic can be developed in both a \emph{classification form} and an \emph{interpretation form}. Two papers, the current paper, defining the concept of a structure, and ``The {\ttfamily ERA} of {\ttfamily FOLE}: Superstructure'', defining the concept of a sound logic, represent the \emph{classification form}, corresponding to ideas discussed in the ``Information Flow Framework''. Two papers, ``The {\ttfamily FOLE} Table'', defining the concept of a relational table, and ``The {\ttfamily FOLE} Database'', defining the concept of a relational database, represent the \emph{interpretation form}, expanding on material found in the paper ``Database Semantics''. Although the classification form follows the entity-relationship-attribute data model of Chen, the interpretation form incorporates the relational data model of Codd. A fifth paper ``{\ttfamily FOLE} Equivalence'' proves that the classification form is equivalent to the interpretation form. In general, the {\ttfamily FOLE} representation uses a conceptual structures approach, that is completely compatible with the theory of institutions, formal concept analysis and information flow.

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arXiv2023

The ERA of FOLE: Superstructure

This paper discusses the representation of ontologies in the first-order logical environment {\ttfamily FOLE}. An ontology defines the primitives with which to model the knowledge resources for a community of discourse. These primitives consist of classes, relationships and properties. An ontology uses formal axioms to constrain the interpretation of these primitives. In short, an ontology specifies a logical theory. This paper continues the discussion of the representation and interpretation of ontologies in the first-order logical environment {\ttfamily FOLE}. The formalism and semantics of (many-sorted) first-order logic can be developed in both a \emph{classification form} and an \emph{interpretation form}. Two papers, ``The {\ttfamily ERA} of {\ttfamily FOLE}: Foundation'', defining the concept of a structure, and the current paper, defining the concept of a sound logic, represent the \emph{classification form}, corresponding to ideas discussed in the ``Information Flow Framework''. Two papers, ``The {\ttfamily FOLE} Table'', defining the concept of a relational table, and ``The {\ttfamily FOLE} Database'', defining the concept of a relational database, represent the \emph{interpretation form}, expanding on material found in the paper ``Database Semantics''. Although the classification form follows the entity-relationship-attribute data model of Chen, the interpretation form incorporates the relational data model of Codd. A fifth paper ``{\ttfamily FOLE} Equivalence'' proves that the classification form is equivalent to the interpretation form. In general, the {\ttfamily FOLE} representation uses a conceptual structures approach, that is completely compatible with the theory of institutions, formal concept analysis and information flow.

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arXiv2023

The FOLE Table

This paper continues the discussion of the representation of ontologies in the first-order logical environment FOLE. According to Gruber, an ontology defines the primitives with which to model the knowledge resources for a community of discourse. These primitives, consisting of classes, relationships and properties, are represented by the entity-relationship-attribute ERA data model of Chen. An ontology uses formal axioms to constrain the interpretation of these primitives. In short, an ontology specifies a logical theory. A series of three papers by the author provide a rigorous mathematical representation for the ERA data model in particular, and ontologies in general, within FOLE. The first two papers, which provide a foundation and superstructure for FOLE, represent the formalism and semantics of (many-sorted) first-order logic in a classification form corresponding to ideas discussed in the Information Flow Framework (IFF). The third paper will define an interpretation of FOLE in terms of the transformational passage, first described in (Kent, 2013), from the classification form of first-order logic to an equivalent interpretation form, thereby defining the formalism and semantics of first-order logical/relational database systems. Two papers will provide a precise mathematical basis for FOLE interpretation: the current paper develops the notion of a FOLE relational table following the relational model of Codd, and a follow-up paper will develop the notion of a FOLE relational database. Both of these papers expand on material found in the paper (Kent, 2011). Although the classification form follows the entity-relationship-attribute data model of Chen, the interpretation form follows the relational data model of Codd. In general, the FOLE representation uses a conceptual structures approach, that is completely compatible with formal concept analysis and information flow.

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arXiv2023

The FOLE Database

This paper continues the discussion of the representation and interpretation of ontologies in the first-order logical environment {\ttfamily FOLE} (Kent). Ontologies are represented and interpreted in (many-sorted) first-order logic. Five papers provide a rigorous mathematical representation for the {\ttfamily ERA} (entity-relationship-attribute) data model (Chen) in particular, and ontologies in general, within the first-order logical environment {\ttfamily FOLE}. Two papers (Kent and another paper) represent the formalism and semantics of (many-sorted) first-order logic in a \emph{classification form} corresponding to ideas discussed in the Information Flow Framework (IFF). Two papers (Kent and the current paper) represent (many-sorted) first-order logic in an \emph{interpretation form} expanding on material found in the paper (Kent). A fifth paper (Kent) demonstrates that the classification form of {\ttfamily FOLE} is "informationally equivalent" to the interpretation form of {\ttfamily FOLE}, thereby defining the formalism and semantics of first-order logical/relational database systems. Although the classification form follows the entity-relationship-attribute data model of Chen, the interpretation form incorporates the relational data model of Codd. Two further papers discuss the "relational algebra" (Kent) and the "relational calculus". In general, the {\ttfamily FOLE} representation uses a conceptual structures approach, that is completely compatible with the theory of institutions (Goguen and Burstall), formal concept analysis (Ganter and Wille), and information flow (Barwise and Seligman).

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arXiv2021

Relational Operations in FOLE

This paper discusses relational operations in the first-order logical environment {FOLE}. Here we demonstrate how FOLE expresses the relational operations of database theory in a clear and implementable representation. An analysis of the representation of database tables/relations in FOLE reveals a principled way to express the relational operations. This representation is expressed in terms of a distinction between basic components versus composite relational operations. The 9 basic components fall into three categories: reflection (2), Booleans or basic operations (3), and adjoint flow (4). Adjoint flow is given for signatures (2) and for type domains (2), which are then combined into full adjoint flow. The basic components are used to express various composite operations, where we illustrate each of these with a flowchart. Implementation of the composite operations is then expressed in an input/output table containing four parts: constraint, construction, input, and output. We explain how limits and colimits are constructed from diagrams of tables, and then classify composite relational operations into three categories: limit-like, colimit-like and unorthodox.

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arXiv2018

Conceptua: Institutions in a Topos

Tarski's semantic definition of truth is the composition of its extensional and intensional aspects. Abstract satisfaction, the core of the semantic definition of truth, is the basis for the theory of institutions (Goguen and Burstall). The satisfaction relation for first order languages (the truth classification), and the preservation of truth by first order interpretations (the truth infomorphism), form a key motivating example in the theory of Information Flow (IF) (Barwise and Seligman). The concept lattice notion, which is the central structure studied by the theory of Formal Concept Analysis (FCA) (Ganter and Wille), is constructed by the polar factorization of derivation. The study of classification structures (IF) and the study of conceptual structures (FCA) provide a principled foundation for the logical theory of knowledge representation and organization. In an effort to unify these two areas, the paper "Distributed Conceptual Structures" (Kent arXiv:1810.04774) abstracted the basic theorem of FCA in order to established three levels of categorical equivalence between classification structures and conceptual structures. In this paper, we refine this approach by resolving the equivalence as the category-theoretic factorization of the Galois connection of derivation. The equivalence between classification and conceptual structures is mediated by the opposite motions of factorization and composition. Abstract truth factors through the concept lattice of theories in terms of its extensional and intensional aspects.

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arXiv2018

The IFF Approach to the Lattice of Theories

The IFF approach for the notion of "lattice of theories" uses the idea of a concept lattice from Formal Concept Analysis (Ganter and Wille) and the idea of the truth classification from Information Flow (Barwise and Seligman). The IFF approach is concentrated in the joining of these two important ideas. The result is called the truth concept lattice, the concept lattice of the truth classification. The IFF provides a principled (versus ad hoc) approach for John Sowa's "lattice of theories" framework. The "lattice of theories" is represented by the truth concept lattice, each theory in the lattice is represented by a formal concept in the truth concept lattice.

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arXiv2018

Conceptual Analysis of Resource Meta-information

It's ease of use and the availability of browsers for various platforms have paved the way for the enormous popularity that the World Wide Web currently enjoys. In the near future, by providing not only easy access to information, but also means for conducting business transactions, the Web could form the base technology for the information superhighway. In such a large distributed information system, resource discovery becomes a critical problem. Recent developments in resource discovery systems, such as Harvest and Whois++, provide scalable mechanisms for the identification, location and characterization of networked information resources based upon resource meta-information. However, the Web's vast information space can only be handled effectively, when resources are meaningfully classified into coherent conceptual structures. The automatic classification of resource meta-information is at the heart of the WAVE system, which employs methods from the mathematical theory of concept analysis to analyze and interactively explore the vast information space defined by wide area resource discovery services. In this paper we discuss these methods by interpreting various synoptic and summary interchange formats for resource meta-information, such as the Harvest SOIF and the Whois++ urc, in terms of basic ideas from concept analysis. In so doing, we advocate concept analysis as a principled approach to effective resource discovery.

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arXiv2018

Soft Concept Analysis

In this chapter we discuss soft concept analysis, a study which identifies an enriched notion of "conceptual scale" as developed in formal concept analysis with an enriched notion of "linguistic variable" as discussed in fuzzy logic. The identification "enriched conceptual scale" = "enriched linguistic variable" was made in a previous paper (Enriched interpretation, Robert E. Kent). In this chapter we offer further arguments for the importance of this identification by discussing the philosophy, spirit, and practical application of conceptual scaling to the discovery, conceptual analysis, interpretation, and categorization of networked information resources. We argue that a linguistic variable, which has been defined at just the right generalization of valuated categories, provides a natural definition for the process of soft conceptual scaling. This enrichment using valuated categories models the relation of indiscernability, a notion of central importance in rough set theory. At a more fundamental level for soft concept analysis, it also models the derivation of formal concepts, a process of central importance in formal concept analysis. Soft concept analysis is synonymous with enriched concept analysis. From one viewpoint, the study of soft concept analysis that is initiated here extends formal concept analysis to soft computational structures. From another viewpoint, soft concept analysis provides a natural foundation for soft computation by unifying and explaining notions from soft computation in terms of suitably generalized notions from formal concept analysis, rough set theory and fuzzy set theory.

cs.AI↗

arXiv2018

Enriched Interpretation

The theory introduced, presented and developed in this paper, is concerned with an enriched extension of the theory of Rough Sets pioneered by Zdzislaw Pawlak. The enrichment discussed here is in the sense of valuated categories as developed by F.W. Lawvere. This paper relates Rough Sets to an abstraction of the theory of Fuzzy Sets pioneered by Lotfi Zadeh, and provides a natural foundation for "soft computation". To paraphrase Lotfi Zadeh, the impetus for the transition from a hard theory to a soft theory derives from the fact that both the generality of a theory and its applicability to real-world problems are substantially enhanced by replacing various hard concepts with their soft counterparts. Here we discuss the corresponding enriched notions for indiscernibility, subsets, upper/lower approximations, and rough sets. Throughout, we indicate linkages with the theory of Formal Concept Analysis pioneered by Rudolf Wille. We pay particular attention to the all-important notion of a "linguistic variable" - developing its enriched extension, comparing it with the notion of conceptual scale from Formal Concept Analysis, and discussing the pragmatic issues of its creation and use in the interpretation of data. These pragmatic issues are exemplified by the discovery, conceptual analysis, interpretation, and categorization of networked information resources in WAVE, the Web Analysis and Visualization Environment currently being developed for the management and interpretation of the universe of resource information distributed over the World-Wide Web.

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arXiv2018

The Information Flow Foundation for Conceptual Knowledge Organization

The sharing of ontologies between diverse communities of discourse allows them to compare their own information structures with that of other communities that share a common terminology and semantics - ontology sharing facilitates interoperability between online knowledge organizations. This paper demonstrates how ontology sharing is formalizable within the conceptual knowledge model of Information Flow (IF). Information Flow indirectly represents sharing through a specifiable, ontology extension hierarchy augmented with synonymic type equivalencing - two ontologies share terminology and meaning through a common generic ontology that each extends. Using the paradigm of participant community ontologies formalized as IF logics, a common shared extensible ontology formalized as an IF theory, participant community specification links from the common ontology to the participating community ontology formalizable as IF theory interpretations, this paper argues that ontology sharing is concentrated in a virtual ontology of community connections, and demonstrates how this virtual ontology is computable as the fusion of the participant ontologies - the quotient of the sum of the participant ontologies modulo the ontological sharing structure.

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arXiv2018

Semantic Integration in the Information Flow Framework

The Information Flow Framework (IFF) is a descriptive category metatheory currently under development, which is being offered as the structural aspect of the Standard Upper Ontology (SUO). The architecture of the IFF is composed of metalevels, namespaces and meta-ontologies. The main application of the IFF is institutional: the notion of institutions and their morphisms are being axiomatized in the upper metalevels of the IFF, and the lower metalevel of the IFF has axiomatized various institutions in which semantic integration has a natural expression as the colimit of theories.

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arXiv2018

Creating a Web Analysis and Visualization Environment

Due to the rapid growth of the World Wide Web, resource discovery becomes an increasing problem. As an answer to the demand for information management, a third generation of World-Wide Web tools will evolve: information gathering and processing agents. This paper describes WAVE (Web Analysis and Visualization Environment), a 3D interface for World-Wide Web information visualization and browsing. It uses the mathematical theory of concept analysis to conceptually cluster objects, and to create a three-dimensional layout of information nodes. So-called "conceptual scales" for attributes, such as location, title, keywords, topic, size, or modification time, provide a formal mechanism that automatically classifies and categorizes documents, creating a conceptual information space. A visualization shell serves as an ergonomically sound user interface for exploring this information space.

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arXiv2018

The Institutional Approach

This chapter discusses the institutional approach for organizing and maintaining ontologies. The theory of institutions was named and initially developed by Joseph Goguen and Rod Burstall. This theory, a metatheory based on category theory, regards ontologies as logical theories or local logics. The theory of institutions uses the category-theoretic ideas of fibrations and indexed categories to develop logical theories. Institutions unite the lattice approach of Formal Concept Analysis of Ganter and Wille with the distributed logic of Information Flow of Barwise and Seligman. The institutional approach incorporates locally the lattice of theories idea of Sowa from the theory of knowledge representation. The Information Flow Framework, which was initiated within the IEEE Standard Upper Ontology project, uses the institutional approach in its applied aspect for the comparison, semantic integration and maintenance of ontologies. This chapter explains the central ideas of the institutional approach to ontologies in a careful and detailed manner.

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