Re: [External] My summary of discussion with Pat was: modal logic, rdf and category theory

I’ve been noodling about Category Theory and RDF too. It seems like “RDF Graph” is a category. Related categories would be “Table” (e.g. HBase) and “Tree” (e.g. XML/JSON). The objects, morphisms, composition, identity and associativity manifested in each category represent different perspectives, with complementary features (advantages and disadvantages).

Given categories from this POV, the functors would be operations like SPARQL SELECT<https://www.w3.org/TR/sparql11-query/#select> (graph to table), R2RML<https://www.w3.org/TR/r2rml/> (table to graph), XSLT (tree to graph), SPARQL CONSTRUCT<https://www.w3.org/TR/sparql11-query/#construct> (graph to graph), etc. These functors allow applications to move seamlessly between the categories so the advantages of each can be leveraged efficiently.

I haven’t thought about if/how this way of thinking relates to modal logic, though.

Jeff

From: Henry Story <henry.story@bblfish.net>
Date: Wednesday, September 19, 2018 at 6:56 AM
To: SW-forum Web <semantic-web@w3.org>
Cc: "public-philoweb@w3.o" <public-philoweb@w3.org>
Subject: [External] My summary of discussion with Pat was: modal logic, rdf and category theory
Resent-From: <semantic-web@w3.org>
Resent-Date: Wednesday, September 19, 2018 at 6:51 AM

Here is my (subjective) summary of the long thread on modal logic, rdf and CT.
I have very likely missed a lot of details, but I hope it can help see the wood
for the trees.


(A) CT Structure of RDF
-------------------

For a while I thought Pat Hayes did not want one to talk about sets of possible models,
as I think one can use that to then build a definition of the meaning of a graph as the
set of models in which it is true, and that then seems to me very close to modal logic.
Since the RDF1.1 spec mixes model and interpretation I wanted to get a clear view on this.
So I asked on MathExchange a simple question on RDF

"What kind of Categorical object is an RDF Model?"
It got what looks to me like an excellent answer
https://math.stackexchange.com/questions/2905226/what-kind-of-categorical-object-is-an-rdf-model/2905287#2905287<https://math.stackexchange.com/questions/2905226/what-kind-of-categorical-object-is-an-rdf-model/2905287#2905287>
It would be interesting to see how to extend this into the notion of RDF Datasets
found in the RDF1.1 spec, and that comes from the SPARQL query language

This is useful as it allows me then to speak of a category of RDF models.


(B) The Misunderstanding on "Context"
--------------------------------

There was a bit of a misunderstanding on the word "context". I had a minimalist
view of contexts as a quotation mechanism, whereas Pat Hayes was referring to
a well developed notion of context that has a long history in AI. It turns out that
Pat actually has written a lot on this topic.

1) I was thinking in terms of simple quotation where one can have an object of
a relation be a graph. In N3 one writes it with {... } so one can write
S believes { a r b }
The structure that allows one to write that is known as RDF Datasets, it has
a 4 th element for the graph name, so that one can specify for each triple the graph
in which it can be found. (There is a default graph )

2) I have been basing myself (and in my paper on foaf+ssl) on Tim Berners Lee's work on N3
which he wrote up
Berners-Lee, T., Connolly, D., Kagal, L., Scharf, Y., & Hendler, J. (2008).
N3logic: A logical framework for the world wide web.
Theory and Practice of Logic Programming, 8(3), 249-269.
http://dig.csail.mit.edu/2006/Papers/TPLP/n3logic-tplp.pdf<http://dig.csail.mit.edu/2006/Papers/TPLP/n3logic-tplp.pdf>

2) Pat Hayes has developed this notion of context in a number of articles of which the following:

⚬ This give an interesting history of the concepts from problems in both external language
and internal language views of language analysis.
Hayes, P. (1997, November).
Contexts in context.
In Context in knowledge representation and natural language, AAAI Fall Symposium.
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.98.4812&rep=rep1&type=pdf<http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.98.4812&rep=rep1&type=pdf>

⚬ The following article was very influential. It also is very practical and gives a lot
of good examples including cryptography for the use of named graphs.

Bizer, C., Carroll, J. J., Hayes, P., & Stickler, P. (2005).
Named Graphs, Provenance and Trust.
In Proceedings of the 14th international conference on World Wide Web.
http://wwwconference.org/2005a/cdrom/docs/p613.pdf<http://wwwconference.org/2005a/cdrom/docs/p613.pdf>

It is not clear why these are better than DataSets though. I think in a
discussion Tim Berners Lee pointed out that if something has a name, then
it can also fail to have a name. So there is something that needs explaining
as to why the naming was important. I remember a long time ago at TPAC
Jeremy Carroll said he thought there was a theoretic problem with non named
graphs.

⚬ Hayes, P. (2007). Context Mereology. In AAAI Spring Symposium: Logical Formalizations of Commonsense Reasoning (pp. 59-64).
https://pdfs.semanticscholar.org/4899/493c11d2e803bb86ef6b849fb7b3185be1e3.pdf<https://pdfs.semanticscholar.org/4899/493c11d2e803bb86ef6b849fb7b3185be1e3.pdf>
This is I thought a really neat paper.

3) One should also note that Guha who was somehow related from the beginning to RDF
wrote his thesis on Contexts
Guha, R. V. (1991).
Contexts: a formalization and some applications (Vol. 101).
Stanford, CA: Stanford University.
http://www-formal.stanford.edu/guha/<http://www-formal.stanford.edu/guha/>
I read it quickly a long time ago, and interpreted it as a way to formalize the quotation notion that appears in Tim Berners Lee's N3, with quotes within quotes within quotes. That is why I put the RDF/XML example in the mail, since it shows how this has been there since the beginning of rdf - just very unclearly. (But I would need to re-read this)

4) Amazingly Paul Smart of our Social Machines group wrote a paper where
they link both the views of concepts and tie in the Category Theory notion of Institutions
Bao, J., Tao, J., McGuinness, D. L., & Smart, P. (2010).
Context representation for the semantic web.
https://eprints.soton.ac.uk/270829/<https://eprints.soton.ac.uk/270829/>


5) Searching further I found this thesis from 2013 relating context to DL & RDF

Klarman, S. (2013). Reasoning with contexts in description logics.
Now what is really interesting here is that it is a 2 dimensional Description Logics.
I remember you saying that CT could help merge such logics. This seems to be exactly
what he is doing.

6) Pat Hayes worked on a language IKL that has a well developed notion
of contexts
https://www.slideshare.net/PatHayes/ikl-survey<https://www.slideshare.net/PatHayes/ikl-survey>
https://www.ihmc.us/users/phayes/IKL/GUIDE/GUIDE.html<https://www.ihmc.us/users/phayes/IKL/GUIDE/GUIDE.html>



(C) Category Theory Institutions
-----------------------------

Because I was worried about Pat Hayes' position I read up on Institutions to see
if that would give me tools in advance to see what would be needed to extend the
semantics to modal logics and show it could then be compatible. As it happens
there was a very nice paper on Institutions and the various RDF languages,
which for me was great to learn about this construct (which I don't find that
difficult now)
Lucanu, D., Li, Y. F., & Dong, J. S. (2006).
Semantic web languages–towards an institutional perspective.
In Algebra, Meaning, and Computation (pp. 99-123). Springer, Berlin, Heidelberg.
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.119.5368&rep=rep1&type=pdf<http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.119.5368&rep=rep1&type=pdf>

See also Paul Smart's article above

Bao, J., Tao, J., McGuinness, D. L., & Smart, P. (2010).
Context representation for the semantic web.
https://eprints.soton.ac.uk/270829/<https://eprints.soton.ac.uk/270829/>

I found a 2008 book on institutions that goes all the way to showing how they
could apply to Kripke Frames, so that looks like it could build on the previous
article

Diaconescu, R. (2008).
Institution-independent model theory.
Springer Science & Business Media.

But that is difficult.

I asked a perhaps not well enough formed question on Stack Exchange today to see
if I am understanding institutions correctly. There are many illustrations though
that people here will find easy to understand.

https://math.stackexchange.com/questions/2921477/what-does-the-category-of-rdf-models-look-like-in-institution-theory<https://math.stackexchange.com/questions/2921477/what-does-the-category-of-rdf-models-look-like-in-institution-theory>

(D) Indexicality of Actuality
-------------------------

The debate on modal logic at some point turned around indexicality. Temporal
logic is indexical, so are other modal logics. What is true now may not be true
tomorrow. So I argued that this supported the view that RDF was modal because
truth depends on what the actual world is. Truth is Indexical was argued by
David Lewis, as summarized by

Van Inwagen, P. (1980).
Indexicality and actuality.
The Philosophical Review, 89(3), 403-426.
http://andrewmbailey.com/pvi/Indexicality_and_Actuality.pdf<http://andrewmbailey.com/pvi/Indexicality_and_Actuality.pdf>

The first page has the relevant quotes the rest of the paper is a critique, which
did not look that relevant. I think Lewis' point could be re-written in coalgebraic
terms by pointing out that the state is a variable, and that what one can observe
does not completely determine the state one is in.


(E) RDF Modal Extension proposal
----------------------------

At the end I proposed a way to extend RDF to deal with the belief relation
in a way that seemed quite self evident, and compatible with the existing specs.
I am not sure if it would just require a definition of a relation or of an extension,
or if the proposed extensions don't already exist. It exists with Tim Berners-Lee's
N3, and that is how I defined it.

https://lists.w3.org/Archives/Public/semantic-web/2018Sep/0027.html<https://lists.w3.org/Archives/Public/semantic-web/2018Sep/0027.html>

The semantics I proposed there are probably not quite right, but it seems on
the right track, so I will investigate. Perhaps contexts are really important
or perhaps these are compatible. It looks like Paul Smart in his article sees contexts
as something that can be built on { ... } graphs.


(F) co-algebraic view of the web
----------------------------

David Lewis in Language and Languages and Convention I think takes account of the evolution
of language by considering a human language not as a grammar + syntax mapping to sets of possible
worlds - those are ideal languages - but rather as sets of sets of such ideal languages. As we invent
new worlds our language becomes more precise. This could be modeled co-algebraically for the web,
as it clearly is a process.