- From: Pat Hayes <phayes@ihmc.us>
- Date: Sat, 1 Sep 2018 12:52:01 -0700
- To: semantic-web@w3.org
On 9/1/18 5:46 AM, Henry Story wrote: > A bit over 4 years have passed since this thread > ( https://lists.w3.org/Archives/Public/semantic-web/2014Apr/ ) > and I somehow ended up in University of Southampton studying > Category Theory and RDF. I am especially interested in bringing modal logic > to it, for reasons explained in the paper "Epistemology in the Cloud - on Fake news > and digital Sovereignty" where I argue from an epistemological point of view for a peer > to peer hyper web based on modal analysis of knowledge. > This has been accepted for the 2018 Decentralizing the Semantic Web Workshop > at ISWC. details here: https://medium.com/@bblfish/epistemology-in-the-cloud-472fad4c8282 > > That paper does not go into technical detail, but it did lead Pat Hayes to ping > me that he was not happy with having references to Possible Worlds in the original > RDF Semantics spec. (It would be nice to have precise reasons for why this was > later thought to be a mistake). OK, since you asked :-) The only place in the 2004 RDF specification where the phrase "possible world" is used is in a non-normative section which I wrote as an attempt to give a kind of intuitive outline of the basic ideas of Tarskian model theory, expressed in as intuitive a way as I could for readers who were new to the whole topic of formal semantics. And the phrase occurs even there only in an aside comment, inside scare quotes, to suggest one way to think intuitively about Tarskian interpretations. Nothing is said anywhere in that document or indeed anywhere else in the 2004 RDF specification documents to even remotely suggest that RDF has modal content or should be understood as a modal logic. (In fact, that idea is so far removed from the whole enterprise of RDF that the idea - and the concomitant possibility of misinterpreting that passing phrase in this way - did not even occur to me at the time.) However, this attempt to give an intuitive introduction in a technical specification document was not a success, as it was almost immediately misunderstood by a large number of readers (including, apparently, yourself) and in any case, in retrospect, it was not appropriate to give tutorial material in a technical specification. The revised RDF 1.1 semantics document, replacing the 2004 document which is now deprecated, therefore omits all such intuitive explanations and describes the (slightly modified) semantics directly and formally, without mentioning 'worlds' at all, possible or otherwise. Right at the end of the RDF 1.1 semantics document there is indeed a tiny mention of the possibility of some future extension of RDF using a modal interpretation of RDF datasets (basically quad stores). But any such interpretation would require a major change to the semantics (analogous to the extension of Tarskian model theory to the Kripke semantics for modal logics) and some kind of enrichment of the RDF syntax to provide some way to indicate the syntactic scope of any modal operator. RDF graph syntax has no scope marking, a fact that gave the RDF and OWL WGs many technical challenges. (For further discussion of this point and what could be done about it, see https://www.slideshare.net/PatHayes/blogic-iswc-2009-invited-talk starting from slide 15.) > > It is to me very clear that RDF has a modal aspect to it, which comes out very clearly > with Quad stores. That is totally unclear to me. Quad stores can be, and have been, used to represent all kinds of 'extra' content, including graphs with time-stamping or location-stamping or representing states of something or linking information about a person or topic to that person or topic. None of this is modal. But it looks like this may need proving - or perhaps someone has already > done so? Modal logic need not I suppose involve possible worlds, and the interesting thing > is that Category Theories believe to have proven that modal logic is to coalgebras what > equational reasoning is to algebras. See "Modal Logics are Coalgebraic" for a summary > https://academic.oup.com/comjnl/article-abstract/54/1/31/336864 > Coalgebras give us the mathematics of infinite streams, processes, a notion of co-induction, > and are to semantics what algebra is to syntax. > > All RDF semantics tells us is how to merge two graphs when one believes them both > to be true. Not quite "all", but your introduction of "believes" is gratuitous. The RDF (and OWL) semantics saying nothing about believing or beliefs. > But what if one believes that someone else believes them to be true? And how is that nested modality to be represented in a form that can be transmitted across the Web? You need to explain how RDF syntax can be extended to cover this kind of assertion. Then > by merging them one can find out what they think is true, and one can model that > in terms of possible worlds, or for those more syntactically oriented sets of all the > ways of completing those graphs in ways that are consistent (or sets of maximally complete > such graphs). There is a clear modal element to that, in so far as one cannot > merge graphs of what one believes to be true into someone else's belief store without getting > a wrong idea of what they believe. But one can say all of this without mentioning the modal notion of belief. You are here simply talking about truth, consistency and validity (or otherwise) of inference on RDF graphs, but adding 'believes' instead of 'true' throughout. > > So if this still needs to be proven What exactly "needs to be proven" ? it seems like Institution theory may help to do > so. In a very interesting paper from 2006 by Dorel Lucanu, Yuan Fang Li, and Jin Song Dong > entitled "Semantic Web Languages – Towards an Institutional Perspective" show how one can > use the theory of institutions to show how RDF, RDFS, OWL (light, DL,...,Full), ... that > seem to have very different semantics can in fact be seen to be consistent. The OWL specification documents show this already, in almost painful detail. (Well, insofar as it is correct. Some RDFS tautologies are not valid in any OWL dialect, for example.) > http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.119.5368&rep=rep1&type=pdf > So if someone tells you that these are incompatible semantics point them to that paper. Or read the specifications themselves :-) Pat > > It looks like work needs to be done to show that these are also compatible with > modal logics (with neighborhood semantics is my guess: ie coalgebras of the form > S -> S^2^2 > a.k.a > S -> 𝒫𝒫(S) > where 𝒫(S) is a predicate and 𝒫𝒫(S) is a set of predicates. Now if one thinks > of a graph as a predicate on possible worlds, one sees why this is similar to quad > stores. Those are known as a hyper-system as explained in "Universal Coalgebra: A Theory > of Systems" http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.159.2020&rep=rep1&type=pdf > > As for good introductions to CT, since that was part of the topic 4 years ago, > I think the best online intro (and more) for programmers are Bart Milewski's > ( https://bartoszmilewski.com/ ) videos on youtube > https://www.youtube.com/user/DrBartosz/playlists > I really recommend it. He is extremely clear without being boring. > > I also liked a lot "Category Theory for Computing Science" by Michael Barr and > Charles Wells (online http://www.tac.mta.ca/tac/reprints/articles/22/tr22.pdf ) > because they make the relation of categories to Graphs so clear. > > Indeed just because the relation is so striking I asked a question on Math > Stackexchange to illustrate how one could be (mis?)lead into a simple pattern > of thinking of the relationship > > https://math.stackexchange.com/questions/2896172/how-should-one-model-rdf-semantics-in-terms-of-category-theory > > Has anyone come across further developments in this space since then? > > Henry Story > > > >> On 17 Apr 2014, at 20:02, Obrst, Leo J. <lobrst@mitre.org> wrote: >> >> Back a few years, emerging from the old IEEE Standard Upper Ontology group’s work was Bob Kent’s Information Flow Framework, an ontology framework (a meta-level framework) based on Barwise & Seligman’s Information Flow Theory, itself an application of Category Theory. See, for example: http://arxiv.org/pdf/1109.0983v1. >> >> Mainly folks have used Information Flow Theory or Goguen’s notion of institutions as springboards from category theory to ontologies, especially for so-called “lattice of theories”, ontology mapping, and semantic interoperability applications. Work includes Mossakowski’s various papers: http://iws.cs.uni-magdeburg.de/~mossakow/. >> >> For a short “position” paper, see: >> Markus Kr¨otzsch, Pascal Hitzler, Marc Ehrig, York Sure. 2005. Category Theory in Ontology Research: Concrete Gain from an Abstract Approach. http://www.aifb.kit.edu/web/Techreport893. >> >> For RDF and category theory, the only paper I know of addresses graph transformations for RDF: >> Benjamin Braatz; Christoph Brandt. 2008. Graph Transformations for the Resource Description Framework. Proceedings of the Seventh International Workshop on Graph Transformation and Visual Modeling Techniques (GT-VMT 2008). http://journal.ub.tu-berlin.de/eceasst/article/view/158/142. >> >> Admittedly most of the above are applications beyond logic itself and RDF, but might shed some light on how category theory is being used for ontologies. >> >> Thanks, >> Leo >> >> From: henry.story@bblfish.net [mailto:henry.story@bblfish.net] >> Sent: Wednesday, April 16, 2014 6:09 PM >> To: Gregg Reynolds >> Cc: Antoine Zimmermann; SW-forum Web; public-philoweb@w3.org >> Subject: Re: rdf and category theory >> >> >> On 11 Apr 2014, at 16:32, Gregg Reynolds <dev@mobileink.com> wrote: >> >> >> On Fri, Apr 11, 2014 at 8:30 AM, Antoine Zimmermann <antoine.zimmermann@emse.fr> wrote: >> There're a lot of resources available online and for free about category theory. >> >> Some examples: >> - Jirà Adámek, Horst Herrlich, George E. Strecker. Abstract and Concrete Categories: The Joy of Cats (524 pages). http://katmat.math.uni-bremen.de/acc/acc.pdf >> - Maarten M. Fokkinga. A Gentle Introduction to Category Theory: the calculational approach.http://wwwhome.ewi.utwente.nl/~fokkinga/mmf92b.pdf (80 pages). >> - Jaap van Oosten. Basic Category Theory (88 pages). http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf >> >> >> One of the best is Robert Goldblatt's Topoi : The Categorial Analysis of Logic . He pays special attention to linking CT concepts to both classic math and ordinary intuition. >> >> I looked through Robert Goldblatt's Topoi quickly [1] and indeed it is the book that covers the subject probably most relevant to the semantic web community, since it aims to show how logic can be derived from Category Theory. In this area I found reading through the first part of Ralf Krömer's "Tool and Object: A History and Philosophy of Category Theory" to also be very interesting, as it gives an overview of the foundational debate in Mathematics started by CT. >> >> It's so odd that RDF is entirely about relations just as CT is ( except that RDF is one to many whereas CT arrows are functions). So I really look forward to understanding how these two domains fit together, and perhaps how they complement each other. >> >> >> Henry >> >> [1] Having read through half of "Conceptual Mathematics" by Willima Lawvere and done most of the exercises there, I am starting to be able to read a lot of these books much more easily. >> >> >> >> >> -Gregg >> >> Social Web Architect >> http://bblfish.net/ > > > > > -- ----------------------------------- call or text to 850 291 0667 www.ihmc.us/groups/phayes/ www.facebook.com/the.pat.hayes
Received on Saturday, 1 September 2018 19:52:29 UTC