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# Introduction: Evan Patterson

Hi! I'm a PhD student in Statistics at Stanford. I'm both a student and a researcher of applied category theory. I find the whole subject fascinating but I'm especially interested in applications to data science and knowledge representation.

I'm involved in several activities in this area. I've written a paper about "relational ologs", inspired by David Spivak's paper on ologs. I'm working on a Julia library for computational category theory called Catlab. And I've got some other fun projects in the pipeline.

I'm excited to participate in this course. I can tell from the introductions that a lot of different perspectives are represented, which is great.

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relational ologs are triples?

Comment Source:relational ologs are triples?
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Hi Evan, liked your paper. I'm also interested in this area. I'm thinking this days on how to combine several ideas: Ologs, the Grothendiek construction, and Wille's formal concept analysis (coincidentally an example of Galois Connection that we are now seeing) and one extension called relational concept analysis to arrive at a mathematically motivated ontology in knowledge representation.

Comment Source:Hi Evan, liked your paper. I'm also interested in this area. I'm thinking this days on how to combine several ideas: Ologs, the Grothendiek construction, and Wille's formal concept analysis (coincidentally an example of Galois Connection that we are now seeing) and one extension called relational concept analysis to arrive at a mathematically motivated ontology in knowledge representation.
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edited March 2018

Hi WebTubTel, by "triples" do you mean RDF triples? If so, then no, relational ologs are not triples. However, you can think of instance data for a relational olog as a collection of RDF triples (Sec 6.2 of my paper).

Comment Source:Hi WebTubTel, by "triples" do you mean RDF triples? If so, then no, relational ologs are not triples. However, you can think of instance data for a relational olog as a collection of RDF triples (Sec 6.2 of my paper).
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Hi Jesus, thanks, I'm glad you liked the paper! I don't know formal concept analysis but I've been meaning to learn it for a while now. Do know a good place to read about it?

Comment Source:Hi Jesus, thanks, I'm glad you liked the paper! I don't know formal concept analysis but I've been meaning to learn it for a while now. Do know a good place to read about it?
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edited March 2018

What worked for me was first to understand what they are good for, and only after read the mathematic fireworks. I always liked [1]. I think Ganter also offered copies of the main reference book by personal request. I find nice that with FCA and given a bag of triplets saying what entities have what attributes (only one fixed predicate, varying subject and object), FCA discovers the best T-Box (concept hierarchy) to organize them.

Comment Source:What worked for me was first to understand what they are good for, and only after read the mathematic fireworks. I always liked [1]. I think Ganter also offered copies of the main reference book by personal request. I find nice that with FCA and given a bag of triplets saying what entities have what attributes (only one fixed predicate, varying subject and object), FCA discovers the best T-Box (concept hierarchy) to organize them. [1] http://www.math.tu-dresden.de/~ganter/psfiles/FingerExercises.pdf
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Jesus, thanks, I'll take a look.

Comment Source:Jesus, thanks, I'll take a look.
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Right, everything is a triple. So olog is just adding Prolog to triples, hence one can group the triples any way one wants to using Prolog rules.

Comment Source:Right, everything is a triple. So olog is just adding Prolog to triples, hence one can group the triples any way one wants to using Prolog rules.
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edited March 2018

WebHubTel, I agree that relational ologs provide a formalism for imposing logical constraints on a collection of triples, but I'd hesitate to say that it's "just adding Prolog to triples". For one thing, Prolog is usually interpreted in a special, fixed model (the minimum Herbrand model). But we even if we drop that restriction and interpret a Prolog program more liberally, like a typical first-order theory, it's not clear how regular logic and Prolog compare. On the face of it they look different, but I don't know Prolog very well, so maybe I'm wrong. It might be interesting to investigate further.

Comment Source:WebHubTel, I agree that relational ologs provide a formalism for imposing logical constraints on a collection of triples, but I'd hesitate to say that it's "just adding Prolog to triples". For one thing, Prolog is usually interpreted in a special, fixed model (the minimum Herbrand model). But we even if we drop that restriction and interpret a Prolog program more liberally, like a typical first-order theory, it's not clear how regular logic and Prolog compare. On the face of it they look different, but I don't know Prolog very well, so maybe I'm wrong. It might be interesting to investigate further.
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SWI-Prolog is what I use

Comment Source:SWI-Prolog is what I use 
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Hi, Evan! It's great to see you here, and great to see ideas are starting to bubble in this conversation. I"m hoping some people here will start to work in groups, or at least get into serious discussions that lead to new research.

I don't know much about Formal Concept Analysis but from my brief glances at it, it looked startlingly similar to classical predicate logic, where a model associates to each predicate a set of entities obeying that predicate. I was disappointed, in the treatments I saw, by a certain lack of mathematical sophistication. I felt that a little dose of category theory could work wonders here. But this was just a rough first impression, and I probably didn't read the best stuff. (That is, the stuff aimed at people who already know lots of math.)

Prolog is usually interpreted in a special, fixed model (the minimum Herbrand model). But we even if we drop that restriction and interpret a Prolog program more liberally, like a typical first-order theory, it's not clear how regular logic and Prolog compare.

I've never studied Herbrand semantics? Is that the realm where the "minimum Herbrand model" lives?

Comment Source:Hi, Evan! It's great to see you here, and great to see ideas are starting to bubble in this conversation. I"m hoping some people here will start to work in groups, or at least get into serious discussions that lead to new research. I don't know much about Formal Concept Analysis but from my brief glances at it, it looked startlingly similar to classical predicate logic, where a model associates to each predicate a set of entities obeying that predicate. I was disappointed, in the treatments I saw, by a certain lack of mathematical sophistication. I felt that a little dose of category theory could work wonders here. But this was just a rough first impression, and I probably didn't read the best stuff. (That is, the stuff aimed at people who already know lots of math.) > Prolog is usually interpreted in a special, fixed model (the minimum Herbrand model). But we even if we drop that restriction and interpret a Prolog program more liberally, like a typical first-order theory, it's not clear how regular logic and Prolog compare. I've never studied [Herbrand semantics](http://logic.stanford.edu/herbrand/herbrand.html)? Is that the realm where the "minimum Herbrand model" lives? 
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John. There was a Dagstuhl meeting back in ?2004 at which there were people working on FCA and also on Chu spaces, domains etc.(I think the reference was Spatial Representation: Discrete vs. Continuous Computational Models, Dagstuhl Seminar Proceedings (04351)). The conclusion of some of the participants (and I was one) was FCA and Domain theory were closely linked and the Concept lattices were the same as Chu spaces. I added some work on finite spaces and Chu ideas. This relates to the observation of spatial phenomena and hence to their representation and manipulation via AI type systems.

Comment Source:John. There was a Dagstuhl meeting back in ?2004 at which there were people working on FCA and also on Chu spaces, domains etc.(I think the reference was Spatial Representation: Discrete vs. Continuous Computational Models, Dagstuhl Seminar Proceedings (04351)). The conclusion of some of the participants (and I was one) was FCA and Domain theory were closely linked and the Concept lattices were the same as Chu spaces. I added some work on finite spaces and Chu ideas. This relates to the observation of spatial phenomena and hence to their representation and manipulation via AI type systems.
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I have found the reference: J. Gratus, T. Porter / Theoretical Computer Science 365 (2006) 206 – 215, in case that is useful to someone.

Comment Source:I have found the reference: J. Gratus, T. Porter / Theoretical Computer Science 365 (2006) 206 – 215, in case that is useful to someone.
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edited April 2018

Hi John, the least Herbrand model seems to be closely related to the Herbrand semantics you mentioned, but I understand it through the usual Tarski semantics.

The interpretation (called the Herbrand structure on Wikipedia) is built out of syntactic material: it takes the universe to be the set of all ground terms and interprets constants and function terms as themselves. The construction is reminiscent of classifying categories in categorical logic. In the minimum Herbrand model, the true ground atoms are exactly those entailed by the axioms. This leads to a classic feature of Prolog called "negation as failure": the negation of a ground atom is true in the minimum Herbrand model if the atom is not provable.

Because Prolog is a programming language first and a logical system second, it makes sense to fix the model. Otherwise, a Prolog program would have infinitely many interpretations! But you could drop this requirement and ask how Prolog works as a logical system, compared to say regular logic. I don't know Prolog well enough to know the answer, but this discussion has made me curious. When I have time I'll look into it.

Comment Source:Hi John, the least Herbrand model seems to be closely related to the Herbrand semantics you mentioned, but I understand it through the usual Tarski semantics. The interpretation (called the [Herbrand structure](https://en.wikipedia.org/wiki/Herbrand_structure) on Wikipedia) is built out of syntactic material: it takes the universe to be the set of all ground terms and interprets constants and function terms as themselves. The construction is reminiscent of classifying categories in categorical logic. In the minimum Herbrand model, the true ground atoms are exactly those entailed by the axioms. This leads to a classic feature of Prolog called ["negation as failure"](https://en.wikipedia.org/wiki/Negation_as_failure): the negation of a ground atom is true in the minimum Herbrand model if the atom is not provable. Because Prolog is a programming language first and a logical system second, it makes sense to fix the model. Otherwise, a Prolog program would have infinitely many interpretations! But you could drop this requirement and ask how Prolog works as a logical system, compared to say regular logic. I don't know Prolog well enough to know the answer, but this discussion has made me curious. When I have time I'll look into it.
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edited April 2018

Prolog works well enough for many applications.

Serving up web pages

Comment Source:Prolog works well enough for many applications. Serving up web pages ![graph](https://imageshack.com/a/img923/2040/JlklI1.png) 
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edited April 2018

The goto place for a mathematically formal account would be the book I referred to before, ISBN 9783540627715. Though rigorous is perhaps shallow. A theorem there, that is shown in the slides I referred to above says: The concept lattice of a relation is complete, and any complete lattice arises as the concept lattice of its associated order.

More interestingly in the abstract of arxiv:1407.0512:

...We show that the category of complete lattices with complete homomorphisms is (up to a natural isomorphism) a full reflective subcategory of the category of contexts with so-called conceptual morphisms; the reflector associates with each context its concept lattice.

Comment Source:The goto place for a mathematically formal account would be the book I referred to before, ISBN 9783540627715. Though rigorous is perhaps shallow. A theorem there, that is shown in the slides I referred to above says: The concept lattice of a relation is complete, and any complete lattice arises as the concept lattice of its associated order. More interestingly in the abstract of arxiv:1407.0512: > ...We show that the category of complete lattices with complete homomorphisms is (up to a natural isomorphism) a full reflective subcategory of the category of contexts with so-called conceptual morphisms; the reflector associates with each context its concept lattice.