Re: rdf and category theory

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).

It is to me very clear that RDF has a modal aspect to it, which comes out very clearly
with Quad stores. 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. But what if one believes that someone else believes them to be true? 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.

So if this still 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.
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.

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 -> đť’«đť’«(S)
where đť’«(S) is a predicate and đť’«đť’«(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:
>  - Jirí 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/

Received on Saturday, 1 September 2018 12:46:59 UTC