CONDITIONS FOR INTEROPERABILITY

Nick Rossiter

School of Informatics, Northumbria University, NE1 8ST, UK

Michael Heather

Northumbria University, NE1 8ST, UK

Keywords:

semantic interoperability, organisational interoperability, Godement calculus, commuting diagrams.

Abstract:

Interoperability for information systems remains a challenge both at the semantic and organisational levels.

The original three-level architecture for local databases needs to be replaced by a categorical four-level one

based on concepts, constructions, schema types and data together with the mappings between them. Such

an architecture provides natural closure as further levels are superﬂuous even in a global environment. The

architecture is traversed by means of the Godement calculus: arrows may be composed at any level as well as

across levles. The necessary and sufﬁcient conditions for interoperability are satisﬁed by composable (formal)

diagrams both for intension and extension in categories that are cartesian closed and locally cartesian closed.

Methods like partial categories and sketches in schema design can beneﬁt from Freyd’s punctured diagrams to

identify precisely type-forcing natural transformations. Closure is better achieved in standard full categories.

Global interoperability of extension can be achieved through semantic annotation but only if applied at run

time.

1 CLASSICAL DATA

STRUCTURES

Classical information systems employ some suitable

model to mediate between data and hardware. A data-

base model is a representation of policies in a struc-

tured form according to some perceived view of real-

ity.

In the ANSI/SPARC architecture (Tsichritzis,

1978) a conceptual schema or model is deﬁned as a

global logical deﬁnition of the data structure. This

schema relates to the internal (physical) deﬁnition by

a mapping from the logical level to the physical level.

The schema is protected from changes at the physical

level by adjusting this mapping. Each user has a par-

ticular view (external schema) of the database which

may be a restricted view. The architecture including

the series of mappings shown in Figure 1 provides as-

pects such as security and logical data independence.

The classical ANSI/SPARC architecture of Figure

1 has the disadvantage that the levels are not indepen-

dent of each other. This may be compared with the

natural architecture of Figure 2 (Heather and Rossiter,

Internal schema I

Conceptual schema C

External schema E

Figure 1: Classical ANSI/SPARC Architecture for Data-

bases

2002). The four levels (top-down) are categories for

concepts (real-world abstractions), constructs (facili-

ties available for schema design), schema (deﬁnition

of data types available in system) and data (the data

itself). In ANSI/SPARC the types of the three lev-

els are similar to the external schema but the internal

schema is composed of subcategories of the concep-

tual schema. The top level, the external schema, is

not a universal closure of types but a local closure

of the conceptual schema. The four-level architec-

ture in Figure 2 has orthogonal types with the rela-

Rossiter N. and Heather M. (2005).

CONDITIONS FOR INTEROPERABILITY.

In Proceedings of the Seventh International Conference on Enterprise Information Systems, pages 92-99

DOI: 10.5220/0002530200920099

 SciTePress

tionships between the levels expressed as categorical

adjunctions as already applied to structures in GRID

data processing (Heather and Rossiter, 2002). Cate-

gorical adjunctions relate one level to another. The

relationship between levels is measurable by the unit

of adjunction. For instance the adjunction P olicy ⊣

MetaMeta indicates that the free functor P olicy

is left adjoint to the underlying functor M etaM eta.

The unit of adjunction is given by η

cpt

: 1

cpt

−→

MetaMeta ◦ P olicy(cpt).

The ANSI/SPARC architecture was a useful way of

capturing abstractions of the relational model in the

1970s and 1980s. It has proved less suitable to facil-

itate the techniques needed today such as interoper-

ability where systems with different underlying mod-

els are required to work together. ANSI/SPARC can

be viewed as pseudo-natural. It was developed using

mathematical techniques and theories like sets. But

there is a gap between classical theory and real-world

performance and pragmatics. Triggers are an example

of an attempt to patch the weakness of the system by

providing some local strong anticipation using Event-

Conditions-Actions (ECA) (Date and Darwen, 2000).

Concepts

P olicy ⊣ MetaMeta

Constructs

O rg ⊣ Meta

Schema

Data

Instance ⊣ Classif y

Instance

mission

management

enterprise

information

execution

application

MetaMeta Policy

category

theory

levels

four levels

of Figure 2

1. data values objects

(identity

arrows)

dat

2. named values category DAT

3. classiﬁed val-

ues

functor I

DAT −→

SCH

4. contrasted rep-

resentation

natural

transforma-

tion

◦ β

Figure 4: Alternative Interpretation of Levels in the Archi-

tecture

It can be shown (Rossiter and Heather, 2003) that

the addition of further levels is possible but nothing

is gained by it type-wise. Thus addition of an ex-

tra level to the top of a four-level architecture sim-

ply results in the top level (comparison of mapping

from concepts to schema) being a composition of

three arrows rather than two. Thus consider the ad-

dition of a new top level PHI with the mappings

F : PHI −→ CPT, G : CPT −→ PHI and

op′

: F −→ F

′

where α

op′

compares the mappings

F and F

′

in two different approaches. The adjunction

is now I ◦ O ◦ P ◦ F ⊣ G ◦ A ◦ M ◦ C. The level

four of Figure 4 is now α

op′

◦ α

◦ β

and is still

a natural transformation through the rules of compo-

sition. The practical consequence is that a ﬁfth level

is equivalent to an alternative fourth level. The meta-

meta level gives ultimate closure of types.

2 NATURAL CALCULUS

We therefore have three types of mapping to con-

sider: within a category (for instance from a name to a

value), from one category to another (for instance the

functor P

′

from CPT to CST

′

) and from one func-

tor to another (for instance the natural transformation

α from P to P

′

Following the constructive principles of category

theory, the composition of these arrows is natural.

This consequently gives rise to a natural calculus

ﬁrst expounded by (Godement, 1958) and ((Barr and

Wells, 1999), 1st ed., pp 94-97) in the form of rules

governing composition. The composition of functors

and natural transformations is associative so that for

instance in Figure 3:

′

)α = I

′

α); γ(OP ) = (γO)P

Natural transformations may be composed with

each other:

γβ = (γO ) ◦ (I

′

β); βα = (βP ) ◦ (O

′

α)

Godement’s ﬁve rules are given by ((Barr and

Wells, 1999), 1st ed., p.96-97).

Consider as in Figure 5:

ﬁve categories A, B, C, D and E

the following eight functors:

E : A −→ B, F

, F

: B −→ C, G

, G

C −→ D, H : D −→ E

and the following four natural transformations:

α : F

−→ F

, β : F

−→ F

, γ : G

−→ G

δ : G

−→ G

A B C D E

E F

α γ

- - - -

- -

? ?

Figure 5: Godement in (Barr and Wells, 1999), 1st ed., p.96

Then the following rules hold:

(δ ◦ γ)(β ◦ α) = (δβ) ◦ (γα) (1)

(H ◦ G

)α = H(G

α) (2)

γ(F

◦ E) = (γF

)E (3)

(β ◦ α)E = (G

βE) ◦ (G

αE) (4)

γα = (γF

) ◦ (G

α) = (G

α) ◦ (γF

) (5)

Equation 5 is particularly interesting as it has dif-

ferent members on each side of the equation, permit-

ting solutions via simultaneous equations. The ﬁrst

four rules are concerned with interchange (commuta-

tivity), associativity and permutation.

Simmons ((Simmons, 1989) section 3.8) also deals

with Godement’s rules. For a simpliﬁed version of

Figure 5, omitting categories A, E and functors E, H,

as shown in Figure 6, he derives the commuting dia-

gram of Figure 7 to represent the composition of func-

tors and natural transformations.

ICEIS 2005 - DATABASES AND INFORMATION SYSTEMS INTEGRATION

B C D

α γ

- -

? ?

Figure 6: Godement in (Simmons, 1989) section 3.8

The commuting properties are shown in the dia-

gram in Figure 7. A small change in Simmons’ no-

tation has been made so that pairs beginning with a

natural transformation have the subsequent functor in

subscript form to indicate that the object of the natural

transformation is the functor ((Barr and Wells, 1999),

1st ed., p.94-95). Where the pairs begin with a functor

the notation is unchanged as it appropriately indicates

the application of a functor to the output from a nat-

ural transformation.

The composition, say λ, of (δ ∗ γ) ◦ (β ∗ α) fol-

lows the path from G

−→ G

−→

−→ G

; that, say ρ, of (δ ◦ β) ∗ (γ ◦ α) fol-

lows the path from G

−→ G

−→

−→ G

. Both routes start and end with the

same objects G

and G

respectively. Hence

they are equivalent and the interchange law is demon-

strated with λ = ρ.

α G

- -

? ? ?

Figure 7: Commuting Diagram in (Simmons, 1989) section

3.8

3 APPLICATION

The consequence of natural closure is that a categori-

cal approach ensures that the various arrows of differ-

ent types can be composed with each other, irrespec-

tive of their level in the system. Equations represent-

ing an equality of paths, can be solved for unknown

components that can be determined from an evalua-

tion of the known properties. For instance in com-

paring methods with the path IOP from CPT −→

CST −→ SCH −→ DAT deﬁning one approach,

then the path I

′

α from CPT −→ CST

′

−→

SCH

′

−→ DAT

′

might deﬁne an alternative ap-

proach if P

′

maps onto constructs in the category

CST

′

The diagram in Figure 8 shows the application of

the Godement calculus to handle semantic interop-

erability, deﬁned as the interoperation of one system

with another at the level of meaning of the data, that

is at the metadata level.

CPT CST SCH DAT

P O

′′

′

′′

- - -

- -

? ?

Figure 8: Semantic Interoperability in terms of Godement

The composition of the top line of functors I

◦O

◦

P gives the mapping from concepts to data for say

a relational system r. The composition of the mid-

dle line of functors I

◦ O

◦ P gives the mapping

from concepts to data for say an object-relational sys-

tem or. The composition of the bottom line of func-

tors I

◦ O

◦ P gives the mapping from concepts

to data for say an object-oriented system oo. Com-

paring these compositions gives a framework for in-

teroperability. For instance the natural transforma-

tion α

′

compares how the mapping is performed from

constructions to types in a relational system r with

that from constructions to types in an object-relational

system or. The natural transformation β

′′

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 through the various mappings. The diagram

in Figure 7 shows that a number of commuting equa-

tions can be derived, enabling solution of equations

for unknown values. For instance β

′′

◦ β

′

compares

the mapping from constructions to data in an object-

relational system or with that in an object-oriented

system oo.

To extend the categorical framework to handle or-

ganisational interoperability, deﬁned as the interoper-

ation 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 9.

The following canonical rules hold according to the

Godement calculus:

(β

′

◦ α

′

)(β ◦ α) = (β

′

β) ◦ (α

′

α) (6)

◦ O

)α = I

α) (7)

CONDITIONS FOR INTEROPERABILITY

CPT CST SCH DAT

′

′′

′

′′

- -

? ?

Figure 9: Organisational Interoperability in terms of Gode-

ment

′

◦ P

) = (α

′

(8)

(β

′

◦ α

′

= (I

′

) ◦ (I

′

) (9)

′′

′

= (α

′′

) ◦ (I

′

) = (I

′

) ◦ (α

′′

) (10)

A number of general principles in composition are

shown by the equations. Equation 6 indicates that of

commutativity (the interchange law); equations 7. . .8

indicate that of associativity; equation 9 indicates that

of permutation of paths. The last equation, 10, shows

the production of simultaneous equations represent-

ing different paths through the diagram. This is an im-

portant feature as it facilitates the solution for an un-

known mapping. For example, in equation 10 above,

if the values α

′

, α

′′

and I

are known, then O

is the

only unknown and a solution can be found for it. That

is if it is known how the mapping from constructions

to types and from types to data varies between a rela-

tional system r and an object-relational system or and

what the mapping is between types and data in an ob-

ject relational system or, then the mapping between

constructions and types in the relational system r can

be derived.

4 SEMANTIC

INTEROPERABILITY

The foregoing indicates that semantic interoperability

can be guaranteed therefore for a system that imple-

ments in full formal form the four-level categorical

diagram and approach as just described. In particular

all compositions of arrows (identity, function, func-

tor, natural transformation) must be natural, that is all

diagrams must commute. Semantic interoperability

depends on the horizontal and vertical composition

of both syntactical and semantic diagrams as well as

the interaction (contravariant) between the syntacti-

cal and the semantic. The semantics involve instan-

tiation everywhere, that is local extensionalities in-

terconnected one with another through global inten-

sionality. We have not explicitly mentioned the usual

point (because reality is equivalent to naturality in cat-

egory theory) that all the categories already referred to

in this paper are cartesian closed. In formal terms of

category theory, this further condition for global con-

nectivity in interoperability means that the categories

need to also be locally cartesian closed. This prop-

erty connects and integrates in a coherent way slice

categories

The method of semantic annotation as advanced at

present appears to be a local method carried out either

manually or by some automated agents. This will be

very reductionist but may be quite sufﬁcient if carried

out at run time.

Linked with the typing problems exhibited in punc-

tured diagrams, there is the whole question of the cap-

ture of type information. Semantic interoperability

depends on as complete a picture as possible being

obtained of types in the different systems. Semantic

annotation is employed in the semantic web (Hendler

et al., 2002) and in other techniques such as metadata

creation (Soo et al., 2003) where agents are used to

explore the data structures for type information.

For semantic annotation we are investigating the

use of natural database techniques (Rossiter and

Heather, 2004) to see how much of such informa-

tion can be collected automatically through analysis

of the data in a categorical framework. Collection

of metameta data is essentially an open architecture

task and we intend to employ the categorial topos and

its internal intuitionistic logic i.e. Heyting (Lane and

Moerdijk, 1991; Johnstone, 2002) for this purpose.

5 COMPOSITION FAILURE

Composition is only certain for categories with ar-

rows as properly deﬁned (Simmons, 1989). Figure

1 is not a formal diagram and the arrows in that ﬁg-

ure for the classical ANSI/SPARC architecture often

do not satisfy such requirements. Implementations of

database schemas typically make use of partial func-

tions in a reductionist view of real-world naturality.

For instance the relational data model of Codd has

had to be compromised for the various SQL standards

(Date and Darwen, 2000). Partial functions are in

such common use in mathematical modelling that var-

ious attempts have been made to carry them over into

the use of category theory. Peter Freyd, an early cate-

gorist pioneer, has proposed that composition failure

be acknowledged in formal diagrams by a puncture

mark.

Barr & Wells provide a comprehensive deﬁnition of lo-

cally cartesian closed categories ((Barr and Wells, 1999),

3rd ed.) and the signiﬁcance of slice categories in computer

science is extensively dealt with by (Goguen and Burstall,

1984).

ICEIS 2005 - DATABASES AND INFORMATION SYSTEMS INTEGRATION

5.1 Punctured Diagrams

A B

? ?

Figure 10: Punctured Commuting Diagram

A commuting diagram is itself a proof of some nat-

urality. A weaker version of the commuting diagram

may be represented by the punctured diagram pro-

moted by ((Freyd and Scedrov, 1990) section 1.251)

as in Figure 10 where the puncture mark

removes

the commutation of the right-hand triangle although it

retains the commutation of the left-hand triangle and

the commutation of the rectangle as a whole. That is

the puncture sign

removes the equivalent of one

equation. A weakness of punctured diagrams is that

which diagram is punctured is not always explicitly

shown. What about the outer square in Figure 10?

Does it need a separate puncture mark? In databases

there are examples where the puncture might be used,

for instance, with the problem of representing partial

functions. If in Figure 11, ST K is a library stock,

ISS is the category of books issued on loan at the

current time, CAT is the catalogue, ACC is the ac-

cession numbers, then t is a total function from CAT

to ACC, u is a total function from ST K to CAT , x

is a total function from ISS to ACC and z is a total

function from ST K to ACC. y is, however, a partial

function from ST K to ISS as not all books will be

out on loan at any one time. If all the functions were

total, then z = tu = xy so we have three commut-

ing equations: z = tu, z = xy and tu = xy. With

the partial function y, then we lose the commuting

equation z = xy but retain z = tu. The outer dia-

gram will still commute if ACC is derived indepen-

dently of ISS so we have lost one commuting equa-

tion, hence the one puncture mark.

Punctured diagrams represent a type failure in a

category: the type of an input is unexpected and

an appropriate output cannot be generated. One of

the causes shown here of partial functions could be

avoided by making all functions total as is usual in

category theory. Source objects that are unassigned

ST K ISS

ACCCAT

? ?

Figure 11: Punctured Commuting Diagram for Library Ex-

ample

ACC = accessions, ST K = stock, ISS = issues, CAT =

catalogue

by the function in its normal operation may then be

assigned to the initial object (⊥) so that a complete

assignment is made of objects in the source category.

In categorical terms, type forcing is necessary to avoid

punctured diagrams and maintain interoperability.

5.2 Lifted Categories and Sketches

There are alternative approaches to composition fail-

ure in category theory. Two of these have been the

focus of database workers.

(Lellahi and Spyratos, 1990), in the FIDE project,

attempted to adapt category theory to partial functions

by creating a new categorical type of lifted (or partial)

functions. This technique has not been further devel-

oped, perhaps because of its inherent complexity and

its conﬂict with much of the established theory of cat-

egories.

Sketches have had more advocates. The reduction-

ism of set theoretic methods to represent real-world

activities means that corresponding categories may

not commute because of the departure from natural-

ity. The purpose of sketches was to identify, as in

punctured diagrams, the departures from naturality in

the internal components of a diagram like the cones

and cocones. For instance (Johnson et al., 2002)

applied sketches to entity-relationship and relational

modelling and (Diskin and Cadish, 1995) to object

databases.

Sketches are strictly outside category theory as they

permit diagrams that do not commute but they may

be mapped onto categories by a model functor. Many

types of sketches have been developed in the theory

itself. For instance (Johnstone, 2002) deﬁnes eight at

D2.1.3 2 p.863-864. They separate out four compo-

nents of a diagram in order to ﬂag the parts for which

composition fails. These four components may be de-

ﬁned as a 4-tuple < E, L, R, S > where E is a ﬁnite

graph for the data structure, L is a set of diagrams in

E giving the constraints as commuting diagrams, R

CONDITIONS FOR INTEROPERABILITY

is a ﬁnite set of discrete cones in E giving the rela-

tionships and S a ﬁnite set of discrete cocones in E

specifying the attributes. For example the omission

of a diagram from L means that it is not required to

commute, so this diagram is effectively punctured.

Sketches lack ﬂexibility as all structures and con-

straints have to be pre-speciﬁed. In difﬁcult ar-

eas such as interoperability, sketches are inadequate

as they do not offer natural closure. (Johnson and

Rosebrugh, 2000) attempt to adapt their sketches to

achieve interoperability but the aim is to achieve only

logical independence, as in the three-level architec-

ture of Figure 1, not semantic interoperability, as in

the four-level architecture of Figure 2. The difference

between a natural structure and a sketch is like that be-

tween typing and labelling. A graph is richer than an

entity-relationship model as its arrows are typed with

identity functors. Labelling in the entity-relationship

model is an informal typing whereas the identity ar-

row is a formal typing.

6 NATURAL COMPOSITION

Some problems with partial functions can be avoided

by altering the data design so that the partial functions

only operate in the assignment to the end of the chain

(the terminal object). For instance an alternative de-

sign can be considered for Figure 11. Here the natural

order would be to consider ﬁrst accessions, which are

then put into the stack and can be issued later. For this

schema the composition diagram would be as in Fig-

ure 12. These are full categories without composition

failure and the puncture sign can be removed. There

are no punctured diagrams if ISS is the codomain of

each of x

′

, t

′

and z

′

. This is because these are all par-

tial functions, mapping onto a category which is last

in the sequence, the terminal object. There is a type

change but it occurs just once, in the ﬁnal step. It

is when partial functions map onto intermediate cat-

egories in a chain that typing problems are likely to

occur, because of the ﬂuctuations of the types.

7 CONCLUSIONS

The use of a formal four-level architecture, based on

category theory, provides an encouraging framework

for tackling both semantic and organisational inter-

operability. The use of the Godement calculus, in

particular, enables many different paths at a number

of level to be compared and analysed. A number of

problems remain. Failure of composition, particularly

due to the existence of partial functions, needs to be

ACC ST K

ISSCAT

′

? ?

Figure 12: Non-punctured Commuting Diagram for Library

Example

ACC = accessions, ST K = stock, ISS = issues, CAT =

catalogue

identiﬁed. Punctured categorical diagrams are used

for this purpose in preference to lifted categories or

sketches. Semantic annotation remains a challenging

area where the open Heyting logic may be of assis-

tance.

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CONDITIONS FOR INTEROPERABILITY