## Abstract

We present a characterization of the core entities and relationships of the Functional Requirements for Bibliographic Records (FRBR) in first order logic. Evidence from the text and diagrams in FRBR support the identification of axioms that capture the constraints and assumptions built into the model. The result is an axiomatic system that clarifies the meaning of the specification. This approach provides a more systematic way of clarifying ambiguities and confusions in the interpretation of FRBR than separate analyses of individual problems and allows scholars and system builders concerned with bibliographic theory to clarify their interpretation of the specification prior to the choice of an implementation language, such as RDF or OWL.

## Introduction

The Functional Requirements for Bibliographic Records (FRBR) presents a conceptual model of the objects and relationships relevant to the description and access of bibliographic materials (IFLA 1998). FRBR is a single document that contains natural language text and a series of entity-relationship diagrams describing the primary objects, relationships and attributes of the model. While the use of entity-relationship diagrams may give the impression of an unambiguous presentation, there are still aspects of the model that are unclear or easily misunderstood. The confusion over whether or not attributes within the model are inherited across entity types (Renear & Choi 2006) is one instance of this problem.

An exact picture of FRBR requires a systematic understanding of the objects that operate within the model in terms of their definitions, the possible relationships between them, and any restrictions that apply to those relationships. In what follows we work toward such a picture by encoding the core of the specification, the Group One entities and relationships, into a logical language. The result of this exercise is a first order theory of bibliographic objects as characterized by FRBR. We make no attempt to address whether or not FRBR is “correct” in its conceptions of the bibliographic domain. The focus is on giving a precise characterization of the model and its entailments.

Generally, a first order theory consists of a collection of predicates and axioms that conform to the rules of first order predicate logic, where theorems may be derived from axioms by the standard mechanisms available in the language. The choice of first order logic as a modeling language is due to its high degree of expressive power and well-studied semantic properties. Initial formalization in this general language will prevent the expressivity restrictions of a particular implementation languages, such as RDF or description logics, from prejudicing the analysis.

## FRBR in First Order Logic

We develop the theory of the FRBR Group One Entities by examining (1) the diagrams and (2) the text of FRBR for evidence of constraints on the relationships in the model. Using this evidence, we develop potential axioms in first order logic to add to the theory. The theory is then checked for redundancies or contradictions, and theorems may be derived. Relying on the text as the source for axioms provides confidence that the theory given here is an accurate reflection of FRBR and allows a clear separation between inherent restrictions and those that may be applied in addition to the model.

The entity types that appear in the Group One entity-relationship diagram are defined in the third chapter (pp.16–23) of the text:

“*work,*: a distinct intellectual or artistic creation

*expression:* the intellectual or artistic realization of a work

*manifestation:* the physical embodiment of an expression of a work

*item:* a single exemplar of a manifestation ”

The entity definitions take the notion of a work and the relationships of realization, embodiment, and exemplification to be primitive. The remaining entity types are defined in terms of the primitive constructs. The selection of primitive terms in this theory corresponds to how the definitions are presented in the text of FRBR, where the entities are all defined in terms of a work and the three primary bibliographic relationships. In the formalization below, entity sets are described with unary predicates and relationship sets with binary predicates.

The primitives are transcribed into first order logic as predicates and the definitions are built from those primitives. For ease of reading we use names for these predicates here that come from the language of the documentation. However, this choice is arbitrary and in terms of the features of the system that are of interest here, we could use any symbols. These primitives appear in the theory atomically, while the following definitions are built within the theory using the primitives constructs. In this way the concept of a work and the Group One relationships are taken as logically prior to the concepts of expression, manifestation, or item.

The definitions and axioms that follow are expressed using standard notations for first order logic. The universal and existential quantifiers, and the connectives of conjunction, disjunction, material implication, and equality are used with the familiar logical definitions.

### Participation

The textual definitions of the entity types reveal participation constraints. Every item must exemplify a manifestation, every manifestation must embody an expression, and every expression must realize some work. In the terminology of entity-relationship modeling, these are total participation constraints which induce existence dependencies between the entity types.

Specifically, there is an existence dependency between each defined entity type and the entity type that appears in its definition. Thus, in order for any items to exist, some manifestations must exist. Similarly, the existence of manifestations requires the existence of some expressions, and the existence of expressions requires the existence of works. However, as has been noted by Renear et al. (2006), there is no existence dependency in the other direction. That is, within the model, it is possible to have un-realized works, un-embodied expressions, and un-exemplified manifestations.

While the total participation constraints are a direct consequence of the definitions given in the text, they are not explicitly indicated in the ER diagram shown in Figure 1 (group1ER.png). This is despite the fact that there are standard techniques for indicating this constraint (usually with the use of double lines) in ER modeling and participation constraints are considered important structural information about a model.

### Cardinality

The Group One ER diagram does indicate cardinality constraints through the use of double and single arrow heads on the relationship arcs. From this notation we are informed that realization and exemplification are one-to-many relationships, and embodiment is a many-to-many relationship.

In the case of a one-to-many relationship, there is a constraint placed on the relationship. For realization, this means that while a single work may be realized through many expressions, a single expression can only realize a single work. Similarly, a single item can only realize a single manifestation.

These constraints can be expressed within the theory with the following cardinality axioms.

Since a many-to-many relationship may be view as unconstrained in terms of its cardinality, there is no corresponding cardinality axiom for the embodiment relationship.

### Domain/Range

Another important constraint is asserted in the fifth chapter (p.58) of the text:

“each of the three primary relationships… is unique and operates between only one pair of entities in the model.”

This text asserts the following domain and range restrictions.

Given the domain and range axioms, the final clause in each definition is redundant and may be removed.

### Disjointness

Next we ask whether the Group One entity types are disjoint. That is, is it possible for some individual object within FRBR to be a member of more than one entity set?

Because nothing can be both abstract and concrete, the following passage implies that the work entity set and the item entity set are disjoint:

Beyond this, there seems to be no direct textual warrant for a more general statement of disjointness amongst the entity sets. However, disjointness seems reasonable and seems to be assumed in most discussions of the FRBR Group One entities (Tillet 2004). In our system, we make disjointness explicit in order to better examine its consequences and determine whether it is in fact consistent with bibliographic practice.

Although this set may appear incomplete, the fact that items may not be members of any other entity set is entailed by the three stated axioms. The redundant formula was not included in order to maintain the independence of the axiom set. The choice of which formula to exclude is arbitrary, and any selection of three from the four possible formulas will be equivalent.

### Relational Properties

Taking the disjointness and domain/range axioms together, we can prove as threorems that each of the relationships described in the Group One ER diagram are irreflexive and asymmetric.

Consider realization as an example. Because works are the only things that can be realized and expressions the only things that can realize (formula 6), and nothing can be both a work and an expression (formula 12), no object within the model may realize itself. A similar argument shows that each of the relationships is asymmetric. Returning to the example of realization, this means that, for some x and some y, if realizes(x,y) is true then realizes(y,x) is false.

## Discussion

The axioms presented above are a partial characterization of the FRBR Group One entities and relationships in first order logic. This approach provides a more systematic way of clarifying ambiguities and confusions in the interpretation of FRBR than separate analyses of individual problems. In addition, such a characterization allows scholars and system builders concerned with bibliographic theory to clarify their interpretation of the specification prior to the choice of an implementation language, such as RDF or OWL. So far, the formal characteristics of these axioms align them closely with the constructors and semantics found in description logics, which serve as the basis of OWL. In addition, this treatment may serve as an initial step in the “refactoring” that has been argued for by Renear and Dubin (2007), supporting the application of FRBR within a general ontology of cultural objects.

## Acknowledgements

We thank Michael Sperberg-McQueen for his insightful comments.