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Modularity notions

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Suri Chitti
Suri Chitti

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.

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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
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 ¨C 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 ? B is a conservative extension (CE) of A over the signature S1, if the logical consequences over S1 in A ? 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 ? 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 ? B Logical consequencesover S1 (S1 region)
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 ? 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 ? 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 ?
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 ? Po
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 ? 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
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 ¨C comments, questions, observations or whatever you wish to say. Referrals for opportunities will be awesome too. Happy holidays!!

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Modularity notions

  • 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 ¨C 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 ? B is a conservative extension (CE) of A over the signature S1, if the logical consequences over S1 in A ? 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 ? 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 ? 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 ? 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 ? 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 ?
  • 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 ? 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 ? 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 ¨C comments, questions, observations or whatever you wish to say. Referrals for opportunities will be awesome too. Happy holidays!!