Uses diagrammatic representations in place of formalisms to illustrate the notions of ontological modularity, namely conservative extensions, safety, modularity and locality. Discusses guarantees that must be achieved in modular ontological design. Discusses 'strongness' of modularity notions and plain, self-contained and depleting modules and of modularity. Modular ontologies are needed in life sciences and other sectors.

1. The Notions of Ontological Modularity
-Suri Chitti
In the pharmaceutical sector ontological reuse is fairly common. There is a bunch of
design challenges that is unique to ontological reuse. Ontology engineers aspiring to
contribute to this sector ought to be aware of these challenges. They are best
brought to the fore by a discussion on the notions of conservative extension, safety
with respect to a signature, modularity and locality. Discussions related to the topics
tend to involve imposing formalisms that make for arduous learning. We will use
diagrammatic representations to illustrate the notions. I hope they will provide some
visualization benefits and make the topics more convivial.
It is best to begin by reviewing a few basic elements. Ontology design starts with a
signature. A signature is a set of symbols that ontology engineers use to build
knowledge bases. The key construct of an ontology is the axiom. An axiom uses
symbols from a signature and the ontological language to form a logical statement.
A rudimentary understanding of ontological languages, knowledge bases,
signatures, symbols and logical statements will be beneficial to read this post.
Readers bereft of this basic attainment are also likely to find the topics captivating.
When building an ontology, it is common to require knowledge about a set of terms
(symbols) about which our own expertise is inadequate. We know they are related to
the terms we will be actively using in our ontology. We will call this set of symbols
the signature of interest. In this post we will call an ontology that can provide
knowledge an imparting ontology. We will call our ontology an importing ontology.
The signature of interest contributes symbols to both ontologies and plays a major
interfacing role. The imparting ontology will hopefully provide a bunch of logical
statements that express how the symbols in the signature of interest relate to each
other. This relationship structure (aka graph) is the knowledge we are seeking. It can
be helpful to model symbols in our ontology that do not belong to the signature of
interest. At this point it will be useful to delve a little more into how the logical
statements, signatures and knowledge are interrelated.
Let us denote the signature of interest by a set called S1. If we are blessed, the
imparting ontology will specify axioms that are exclusively derived from S1. The
structure, and thereby the knowledge we require, is more readily available. Many a
time, the ontology will specify axioms that contain symbols from S1 mixed with other
symbols. It will also contain axioms derived completely from non-S1 symbols. This
does not mean we are out of luck. There will be entailed axioms. The ‘mixed’ and
non- S1 symbol derived axioms could entail other axioms based on the logic that is
encoded in them. Entailments make developing ontologies a profitable venture.
Together with the specified axioms, the entailed axioms form the logical
consequences of an ontology. If the ontology we chose is rich in knowledge about
the symbols in S1, we can expect it to contain a sizeable bunch of entailed axioms

2. that exclusively derive from S1. The logical consequences over S1, for an ontology
or fragment in consideration, would then include any axioms specified and
entailments exclusively from symbols in S1. The logical consequences over S1 is a
(sub)-set or region of axioms (of the fragment under consideration) that will be
central to the discussion of all our notions.
Let there be a set of axioms denoted by A. A contains quite a few sorts of axioms –
specified and entailed, axioms that do not contain symbols from S1 and axioms that
mix symbols from S1 and symbols not from S1. More importantly A contains logical
consequences over signature S1. Let us bring another set of axioms denoted by B
into the picture. For the moment, let us not probe what B contains. Let us merge sets
A and B.
A ᴜ B is a conservative extension (CE) of A over the signature S1, if the logical
consequences over S1 in A ᴜ B remain identical to the ones in A.
If A is Pa, an imparting ontology and B is Po an importing ontology, we could merge
Pa with Po (or copy Pa into Po) to provide Po knowledge over S1. If Pa ᴜ Po is a CE
of Pa, then Po is safe w.r.t Pa because we have not modified the knowledge over
S1in Po. At this point it almost feels natural to ask ‘why would Po contain any axioms
that modify any knowledge over S1 anyways?'
1) Keep calm and drink your brew.
2) Keep calm and keep reading.
OntologydenotedbyA
Logical consequencesover
S1
(S1 region)
OntologydenotedbyB
A ᴜ B
Logical consequencesover
S1
(S1 region)

3. Now, it is time to probe into what Po (earlier B) contains. If Po did not use any
symbols from S1, the notion of CE could be relegated to Neverland. And with it goes
the idea of merging Pa with Po. It is not without reason that we wish to bring
knowledge over S1 into Po. We are building Po for purposes and we would like to
specify axioms that mix S1 symbols with symbols that are our own (neither S1, nor
the non-S1 symbols found in Pa). Besides, we may have already specified quite a
few (of our own) non-S1 axioms.
Different they may be, but the Po lot could provide entailments that could mean
logical consequences over S1, beyond what Pa has given us. This is very
undesirable because instead of using knowledge that we can repose our faith in, we
are tampering with it. Rather, we desire that entailments in our own non S1
containing axioms will be induced by the S1 logical consequences that Pa brings.
This would be in concurrence with our purpose. So there are two important tasks
laid out in front of us when building Po.
1) We need to use the symbols from S1 in the axioms we specify in Po
2) We need to make sure that our axioms will not entail additional logical
consequences that derive (exclusively) from S1.
When we achieve task 2, we have made a safe import and Po is safe w.r.t Pa. Pa ᴜ
Po is a CE of Pa.
It could quite be the case that Po does entail axioms derived from S1. They could be
identical with some logical consequences over S1 already laden in Pa. No problem,
we still have a CE. (A ᴜ B in our illustration in fact depicts that the S1 region is as
much a part of B as it is of A). This assertion will be valid for only as long as Pa
does not change! We are aware that it is not a good idea to tamper with an
accepted body of knowledge when we import or share it. This does not mean those
responsible to standardise the body of knowledge should not be permitted to modify
it. As with many other things in the world, Pa is likely to evolve over time. If Po is
laden with certain entailments over S1, it is very likely to produce a CE of only a
particular version of Pa upon merger. We do not want that outcome. We want Po to
form a CE of every version of Pa upon merger.
3) This means (in Po and its logical consequences) we want it to be devoid of any
entailments over S1.
Logical consequencesoverS1are
broughtintothe unionby Pa and
not Po.ThismeansPo can enter
intoa union with any versionof
Pa ina waythat will ensure the
unionisa CE overS1 of Pa
Ø

4. This is the most significant task we need to accomplish. This means Po is safe w.r.t
S1 and we've made that possible. We have accomplished safety w.r.t a signature of
interest, S1. We have achieved an important guarantee in ontology design
We desired knowledge over S1 in Po. We imported Pa into Po. Good idea but not
the best. Instead, we could have identified a subset of Pa that contained all the
logical consequences over S1 that Pa contains. It is a sensible assumption that all
the axioms in Pa are not required to churn the logical consequences over S1. It is
now time to talk about a module.
A module is a subset of an ontology that contains all the logical consequences over
the signature of our interest that the ontology contains. Our notion of CE is handy to
illustrate a module. If an ontology Pa is a CE of a subset of itself denoted by Pa'
over a signature of interest (S1 being the one in this case), we can call Pa' a plain
(S1-) module of Pa. The smallest Pa' we can identify is called the minimal plain
module (not illustrated) of Pa over S1.
We have covered some ground now. Let us take a minute and discuss how the
notions we have illustrated relate to the guarantees we wish to put in place in our
design. . Safety w.r.t import is a major guarantee desired in ontological modularity
and our discussions of CE have explained it for us. Coverage is another guarantee
desired in modular design. Coverage ensures that no part of the knowledge
contained by an ontology over a signature is omitted during import. Clearly,
modularity, which in turn is based on CE, establishes coverage. Minimal modules
guarantee economy. There is another guarantee called independence that comes
into picture when we are importing from multiple ontologies and are interested in
multiple signature sets. But we will talk about independence after the holidays
Pa
S1 region
S1-module
Pa ᴜ Po

5. I realize that this post has become longer than I had wished for. Please take heart.
It won’t take much longer to cover the rest. Should a caffeine or nicotine fix be
needed take a break now, but please come back.
Much like love, the notions we have discussed are quite 'undecidable'. (When there
does not exist a terminating procedure for a computer to determine an outcome, a
problem is called undecidable.) . The problem of deciding whether one ontology is a
conservative extension of a fragment w.r.t a given signature and a language is
undecidable for most but rather 'light' ontological languages. The problem of
determining whether a subset of an ontology is a module for a given signature is
undecidable even for rather 'light' languages.
Fortunately, approximations are available that will help us realise our notions and
guarantees, Approximations can be expected to be larger than minimal. One
approximation is based on a condition called locality. Locality has received much
attention and has been intensely studied in recent years. Before we understand
locality, we need to understand what a depleting module means
Once we have identified the subset of axioms that make up our module, we are
bound to have a few symbols other than those that belong to S1 in our subset. In
case we do not want miss entailments that include these additional symbols, we will
be interested in making sure our module is self-contained. This means we desire to
add the 'extra' symbols to our signature of interest.
Our signature of interest is now S1’ = S1 ᴜ Sig(Pa’).
A self-contained module contains all the logical consequences over signature S1'. It
is considered to be a 'stronger' notion of modularity than plain modularity that we
have discussed so far. Perhaps the strongest notion of modularity is that of a
depleting module.
Our module Pa’ would be called a depleting module on the signature S1’ if we could
make sure that Pa-Pa’ (Pa minus Pa’) contains absolutely no entailments over S1'. It
is much like our safe ontology Po in that it does not contain any logical
consequences over S1. (In contrast, a plain module will allow Pa-Pa' to retain
entailments, albeit ones not different from Pa'). A clever way to check for depletion
would be to replace every non S1’ symbol in Pa-Pa’ with an 'empty' representation
and see if the ontology reduces to an empty one. Locality approaches employ this
tactic and provide guidelines for 'trivially interpreting' non S1’ symbols. Using these
approaches, if all the axioms in Pa-Pa’ are identified as local, Pa’ can be assumed to
be a depleting model CE S1’-module. Pa’ thus obtained would not be a minimal
module but would satisfy our notions and guarantees. In the last couple of
sentences we brought two new words - interpretation and model. Keep away from
the panic button. For the moment, It will suffice to mention that the notions we
discussed in this post can also be described in terms of interpretations or models of
ontologies. Of consequence to us is the fact that we end up with stronger notions

6. than the ones we described using axioms in this post, when we use models. Locality
conditions are attractive because locality checks present very decidable problems.
Syntactic locality checks particularly, can be achieved on an axiom in polynomial
time.
I hope you enjoyed reading this. If you did, please provide me with your feedback –
comments, questions, observations or whatever you wish to say. Referrals for
opportunities will be awesome too.
Happy holidays!!