A Natural Basis for Interoperability

Rossiter, Nick; Heather, Michael; Nelson, David

doi:10.1007/978-1-84628-714-5_39

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A Natural Basis for Interoperability

Michael Heather

2007, Enterprise Interoperability

https://doi.org/10.1007/978-1-84628-714-5_39

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12 pages

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Abstract

Successful interoperability of systems requires a sound basis for activity across levels and up to the highest global level. The interoperability is non-local and subject to the conditions of naturality found in reality. The axiomatic models over the last two centuries can guarantee no reliability at higher levels. Category theory is free from these twin problems and can therefore offer a theoretical basis on which to base standards for interoperability. Four levels are used to give closure for policy, organisation, instantiation, naming, classification and meta and metameta relationships. Such constructions provide facilities for relating arrows in general, both descriptive and manipulative, including the specification of constraints and a calculus. The implications for standards are discussed. c c c

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Fig. 3. Comparison of Mappings in two Systems
As mentioned earlier, there are three areas of interoperability that our architec-
ture must satisfy: data structures, constraints and data manipulation. Each is
covered in turn. — Fig. 3. Comparison of Mappings in two Systems As mentioned earlier, there are three areas of interoperability that our architec- ture must satisfy: data structures, constraints and data manipulation. Each is covered in turn.

Fig. 2. Interpretation of Levels as Natural Schema in General Terms

Fig. 4. Defining the Four Levels with Contravariant Functors and Intension-Extension
(I-E) Pairs
The four levels of Figure 3 can now be viewed as the two intension-extension
pairs in Figure 4. The pairs are for CPT/CST (concepts/constructs) and
SCH/DAT (schema/data). For interoperability purposes, it has been shown
by the fundamental nature of category theory that four levels are sufficient for
all purposes [Rossiter & Heather 2005]. Further levels are possible but unneces-
sary. To maintain the coherence of the present approach it would be necessary to
go up to six levels as the next step to maintain the intension- extension pairings. — Fig. 4. Defining the Four Levels with Contravariant Functors and Intension-Extension (I-E) Pairs The four levels of Figure 3 can now be viewed as the two intension-extension pairs in Figure 4. The pairs are for CPT/CST (concepts/constructs) and SCH/DAT (schema/data). For interoperability purposes, it has been shown by the fundamental nature of category theory that four levels are sufficient for all purposes [Rossiter & Heather 2005]. Further levels are possible but unneces- sary. To maintain the coherence of the present approach it would be necessary to go up to six levels as the next step to maintain the intension- extension pairings.

Fig. 6. Composition of Adjoints is Natural
Fig. 5. Examples of Levels in the Four-Level Architecture — Fig. 6. Composition of Adjoints is Natural Fig. 5. Examples of Levels in the Four-Level Architecture

Fig. 7. Semantic Interoperability in terms of Godement
The diagram in Figure 7 shows the application of the Godement calculus
to handle semantic interoperability, defined as the interoperation of one system
with another at the level of meaning of the data, that is at the metadata level. — Fig. 7. Semantic Interoperability in terms of Godement The diagram in Figure 7 shows the application of the Godement calculus to handle semantic interoperability, defined as the interoperation of one system with another at the level of meaning of the data, that is at the metadata level.

Fig. 8. Organisational Interoperability in terms of Godement
To extend the categorical framework to handle organisational interoperabil-
ity, defined as the interoperation of systems at the business process level, we
need to vary the functor P for each environment so that the metameta level is
variable. The required diagram is shown in Figure 8.
The following canonical rules hold according to the Godement calculus:
performed from constructions to types in a relational system r with that from
constructions to types in an object-relational system or. The natural transfor-
mation 3” compares how the mapping is performed from types to data in an
object-relational system or with that from types to data in an object-oriented
system oo. The advantage of the Godement approach is that arrows of any type
can be composed with each other so that any route can be taken. — Fig. 8. Organisational Interoperability in terms of Godement To extend the categorical framework to handle organisational interoperabil- ity, defined as the interoperation of systems at the business process level, we need to vary the functor P for each environment so that the metameta level is variable. The required diagram is shown in Figure 8. The following canonical rules hold according to the Godement calculus: performed from constructions to types in a relational system r with that from constructions to types in an object-relational system or. The natural transfor- mation 3” compares how the mapping is performed from types to data in an object-relational system or with that from types to data in an object-oriented system oo. The advantage of the Godement approach is that arrows of any type can be composed with each other so that any route can be taken.

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A Natural Basis for Interoperability

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