Re: "Databases are Categories"

Re: "Databases are Categories"

[Mattias Wikström]

I think it makes sense to regard database schemas as theories and
databases as models of such theories. Given a theory T that models
some database schema S, a term in the language of T may be thought of
as a "database query" (to obtain the result of the query, simplify the
term), while a statement that two terms in the language of T are equal
may be thought of as a "database constraint" that one may want to add
to S. (In practise, though, one may want to formulate ones queries and
contraints not in the language of T but in languages somehow obtained
from that language.)

What sort of theory should a database schema be? This surely depends
on what exactly one is trying to model: A schema in Company A's DBMS
(database management system) is rarely the same thing as a schema in
Company B's DBMS, and in any case one probably wants to work with some
idealised mathematical model.

David Spivak seems to offer two different answers. On the one hand, a
database schema may be the same thing as a category. On the other
hand, a database schema may be a labeled simplicial set. Both answers
may be found at http://www.uoregon.edu/~dspivak/cs/ .

Mattias Wikstrom

----------------------------------------
> Date: Mon, 9 Aug 2010 11:12:34 -0500
> Subject: categories: "Databases are Categories" (again)
> From: vigalchin@gmail.com
> To: categories@mta.ca
>
> Hello,
>
> I stumbled across this tech talk:
> http://www.galois.com/blog/2010/05/27/tech-talk-categories-are-databases/ I
> was wondering
> what others in this mail list think about Spivak's thesis. I apologize if
> already posted.
>
>
> Regards,
>
> Vasili
>

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Re: "Databases are Categories"

[Pym, Professor David J.]

John Cartmell (who introduced contextual categories and generalized algebraic theories as models of dependent types in his 1978 Oxford D.Phil thesis and a 1986 Ann. Pure App. Logic paper) worked on topics related to this thread some years ago. Another 1986 paper, 'Formalizing the Network and Hierarchical Data Models --- an Application of Categorical Logic', CATEGORY THEORY  AND COMPUTER PROGRAMMING
LNCS, 1986, Volume 240/1986, 466-492, DOI: 10.1007/3-540-17162-2_138, can be found via the following
link:

         http://www.springerlink.com/content/y31234tkk63wp56k/

'Abstract. We have noted that data modelling and conceptual modelling have content and performance as their concerns. For the different data models, the Network and the Hierarchic, we have given logics involving the operations which are physically supported according to the data model. The logics are sensitive to performance in a way that classical logic is not. We have now suggested how we might formalise this. Network and Hierarchical databases  have the functional inverse or family as their primitive of organisation. To formalise the Network model we have given a general definition of network category which seems to generalise correctly the hierarchical logic of contextual categories.'

There is also relevant work on categories and logic programming.

David Pym




















--
Professor David J. Pym, MA, PhD, ScD, FBCS, CITP, FIMA, CMath, CSci
6th Century Chair in Logic
University of Aberdeen
Scotland

+44 (0)1 224 27 4577
d.j.pym@abdn.ac.uk
http://www.abdn.ac.uk/~csc335






The University of Aberdeen is a charity registered in Scotland, No SC013683.


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Re: Re: "Databases are Categories"

[Zinovy Diskin]

2010/8/14 Mattias Wikström <mattias.wikstrom@gmail.com>:
> I think it makes sense to regard database schemas as theories and
> databases as models of such theories. Given a theory T that models

yes, this is the basic underlying assumption in a series of papers of
Michael  Johnson and the moderator of this list.

> some database schema S, a term in the language of T may be thought of
> as a "database query" (to obtain the result of the query, simplify the
> term),

a query is a formula (defining a  derived table) rather than a term

while a statement that two terms in the language of T are equal
> may be thought of as a "database constraint" that one may want to add
> to S.

there are many types of non-equational constraints, for example,
inclusion P=>Q with P,Q formulas

>
> What sort of theory should a database schema be? This surely depends
> on what exactly one is trying to model: A schema in Company A's DBMS
> (database management system) is rarely the same thing as a schema in
> Company B's DBMS, and in any case one probably wants to work with some
> idealised mathematical model.
>

a good question is what the place of SQL on the scale of logical
doctrines is. Since models of SQL-theories include predefined infinite
domains with operations (think of Integer and String), and  finite
domains for tables, model theory for SQL is not classical. I'm not
sure but it seems model-theorists call it "metafinite model theory"
(Gradel and Gurevich).

> David Spivak seems to offer two different answers. On the one hand, a
> database schema may be the same thing as a category. On the other
> hand, a database schema may be a labeled simplicial set. Both answers
> may be found at http://www.uoregon.edu/~dspivak/cs/ .
>

I took a look at the slides of  "databases are categories" talk. It
seems to be an overly simplified model.  Since an employee may work
for several or none departments, data should be modeled by a functor
into Rel rather than Set. 2-category structure may be important, and
we often have a lax functor F(a;b)<F(a);F(b). For example, a Person
_owns_ a Car, which is _parked_ at an Address, where the person
_lives_, but a Person may not have a Car but still _live_ at an
Address (arrow names are underlined, _lives=owns;parked_).
Considering all columns as foreign keys, that is, disregarding the
difference between value-valued and object-valued attributes, would
look very strange for a practitioner.

Z.

> ----------------------------------------
>> Date: Mon, 9 Aug 2010 11:12:34 -0500
>> Subject: categories: "Databases are Categories" (again)
>> From: vigalchin@gmail.com
>> To: categories@mta.ca
>>
>> Hello,
>>
>> I stumbled across this tech talk:
>> http://www.galois.com/blog/2010/05/27/tech-talk-categories-are-databases/ I
>> was wondering
>> what others in this mail list think about Spivak's thesis. I apologize if
>> already posted.
>>
>>
>> Regards,
>>
>> Vasili
>>


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Re: "Databases are Categories"

[Dr. Cyrus F Nourani]

Hello, thought I state the "obvious"?  since I had written an ACM review for
newer techniques to
model database and program transformation techniques a couple? years ago.
There were papers by our UCLA Semantics and TCS group two decades ago with
publications on database algebras. Further specificson databases and
algebras
were at TU Berlin.

I enclose an excerpt:

[There are publications at UCLA,JACM, and TU Berlin since 1979 that address
abstract algorithms, programs, morphisms, and database models and schemas
that are benefit the areas.There are  mapping techniques that can be applied
to the data processing realms: There is a concrete syntax route, e.g.,
untyped canonical mappings sparse trees using language independent
format-universal types,  generative mappings- metaprogram- GDK grammar
deployment GUI, XML data biding, and the problem of providing an object
model that is to represent an XML schema are stated. Graph Transformation
for Model Refactoring deploys models and transformations as primary
aritifcats. The techniques presented are to use graph model representations
and apply graph transformations at the model refactoring arena.
Refactoring (Opdyke) is changes to the internal program structure to improve
without changing the external introduce model refraction as a new
transformation. The Berlin school AGG is applied on graph grammars and to
specify model refactoring.]

CyrusFN

-- 
Akdmkrd.tripod.com
Projektakdmkrd.tripod.com


On Sun, Aug 15, 2010 at 11:25 AM, Pym, Professor David J. <
d.j.pym@abdn.ac.uk> wrote:

> John Cartmell (who introduced contextual categories and generalized
> algebraic theories as models of dependent types in his 1978 Oxford D.Phil
> thesis and a 1986 Ann. Pure App. Logic paper) worked on topics related to
> this thread some years ago. Another 1986 paper, 'Formalizing the Network and
> Hierarchical Data Models --- an Application of Categorical Logic', CATEGORY
> THEORY  AND COMPUTER PROGRAMMING
> LNCS, 1986, Volume 240/1986, 466-492, DOI: 10.1007/3-540-17162-2_138, can
> be found via the following
> link:
>
>         http://www.springerlink.com/content/y31234tkk63wp56k/
>
> 'Abstract. We have noted that data modelling and conceptual modelling have
> content and performance as their concerns. For the different data models,
> the Network and the Hierarchic, we have given logics involving the
> operations which are physically supported according to the data model. The
> logics are sensitive to performance in a way that classical logic is not. We
> have now suggested how we might formalise this. Network and Hierarchical
> databases  have the functional inverse or family as their primitive of
> organisation. To formalise the Network model we have given a general
> definition of network category which seems to generalise correctly the
> hierarchical logic of contextual categories.'
>
> There is also relevant work on categories and logic programming.
>
> David Pym
>

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Re: "Databases are Categories"

[David Spivak]

Hi all,

The basic idea of my talk was this: one can think of any category C as a
"database schema": the objects of C are called "tables" and an arrow f:A-->B
is called a "column of table A with values in table B".  Now a functor
C-->Sets is a "state" of that database: it fills every table with a set of
rows.  Leaf objects of C (objects with no outgoing arrows) correspond to
"pure data."  One can thus visualize a category as a system of tables;
commutative diagrams correspond to "rules" such as "the secretary of a
department must be in that department."

Using sketches instead of categories allows a little more flexibility, but
the basic idea is as above.  The model is nice for a variety of reasons,
most of all its simplicity.  In polite contradiction to one of Diskin's
claims, two database administrators (at two large multi-national
corporations) that I know think it is a viable model.  They do not balk at
the idea of data columns being considered as foreign keys.  It puts
everything on the same playing field.

The database=category idea furthermore allows us to "migrate data" from one
schema to another.  Given a morphism of schemas (functor F:C-->D) one pushes
and pulls data between the two categories using F^*, F_*, and F_!.  One can
thus achieve unions, joins, etc.  Use of F^*, F_*, and F_! may have a
variety of applications including a solution to the "view update problem."
  The Grothendieck construction applied to a database state C-->Set yields
its "RDF graph," which is an important component of the semantic web, and of
which conversions to and from databases is of high interest.  The point is
that its all very natural in this model.

Finally, in reference to Diskin's objection below (regarding people owning
cars, and both having addresses) we can consider the category [1]x[1]:
(excuse the typography -- V's are the tips of downward arrows)

O----->P
  |         |
  |         |
V        V
C----->A

A functor from this category to sets is a set of ownerships, each of which
has an associated person and an associated car, each of which has an
associated address (and these addresses must agree).  The same idea can be
used to model situations where an employee could work for any number of
departments (as in Diskin's other objection).

If anyone wishes to engage me more personally either in support or denial of
these ideas, I suggest we talk by phone; email me and I'll send you my
number.  Of course I also don't mind continuing the conversation by email,
either privately or publicly.  Also, let me extend my appreciation to Vasili
for bringing this whole thing up!

Best regards,
David

PS. I've informed the people at Galois of your comments on the camera-work;
they said they'd take it into consideration in the future.

PPS. A simplicial set (or any presheaf) can be regarded as a database state
(on a schema with countably many tables).  Can anyone imagine a database
management system like Oracle being of use in answering mathematical
questions about such things?  In other words, could modern, incredibly fast
database technology be useful to a mathematician studying the dynamics of a
presheaf category such as simplicial sets?


2010/8/16 Zinovy Diskin <zdiskin@gsd.uwaterloo.ca>

> 2010/8/14 Mattias Wikström <mattias.wikstrom@gmail.com>:
>> I think it makes sense to regard database schemas as theories and
>> databases as models of such theories. Given a theory T that models
>
> yes, this is the basic underlying assumption in a series of papers of
> Michael  Johnson and the moderator of this list.
>
>> some database schema S, a term in the language of T may be thought of
>> as a "database query" (to obtain the result of the query, simplify the
>> term),
>
> a query is a formula (defining a  derived table) rather than a term
>
> while a statement that two terms in the language of T are equal
>> may be thought of as a "database constraint" that one may want to add
>> to S.
>
> there are many types of non-equational constraints, for example,
> inclusion P=>Q with P,Q formulas
>
>>
>> What sort of theory should a database schema be? This surely depends
>> on what exactly one is trying to model: A schema in Company A's DBMS
>> (database management system) is rarely the same thing as a schema in
>> Company B's DBMS, and in any case one probably wants to work with some
>> idealised mathematical model.
>>
>
> a good question is what the place of SQL on the scale of logical
> doctrines is. Since models of SQL-theories include predefined infinite
> domains with operations (think of Integer and String), and  finite
> domains for tables, model theory for SQL is not classical. I'm not
> sure but it seems model-theorists call it "metafinite model theory"
> (Gradel and Gurevich).
>
>> David Spivak seems to offer two different answers. On the one hand, a
>> database schema may be the same thing as a category. On the other
>> hand, a database schema may be a labeled simplicial set. Both answers
>> may be found at http://www.uoregon.edu/~dspivak/cs/ .
>>
>
> I took a look at the slides of  "databases are categories" talk. It
> seems to be an overly simplified model.  Since an employee may work
> for several or none departments, data should be modeled by a functor
> into Rel rather than Set. 2-category structure may be important, and
> we often have a lax functor F(a;b)<F(a);F(b). For example, a Person
> _owns_ a Car, which is _parked_ at an Address, where the person
> _lives_, but a Person may not have a Car but still _live_ at an
> Address (arrow names are underlined, _lives=owns;parked_).
> Considering all columns as foreign keys, that is, disregarding the
> difference between value-valued and object-valued attributes, would
> look very strange for a practitioner.
>
> Z.
>

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Re: "Databases are Categories"

[soloviev]

Hi,

then , in this setting, how coherence should be interpreted?
According to Mac Lane, a coherence theorem asserts commutativity
of a class of diagrams; some other authors mean rather decision
procedure/criteria  for commutativity of diagrams.

By the way, did you consider some kind of "basic" arrows
and generated "canonical maps"? Do there appear/be used
some known types of categories with structure
(cartesian, monoidal, cartesian closed, monoidal closed)?

Another interesting question could be isomorphism of objects.

Best wishes

Sergei Soloviev

  Hi all,
>
> The basic idea of my talk was this: one can think of any category C as a
> "database schema": the objects of C are called "tables" and an arrow
> f:A-->B
> is called a "column of table A with values in table B".  Now a functor
> C-->Sets is a "state" of that database: it fills every table with a set  of
> rows.  Leaf objects of C (objects with no outgoing arrows) correspond to
> "pure data."  One can thus visualize a category as a system of tables;
> commutative diagrams correspond to "rules" such as "the secretary of a
> department must be in that department."
>
> Using sketches instead of categories allows a little more flexibility, but
> the basic idea is as above.  The model is nice for a variety of reasons,
> most of all its simplicity.  In polite contradiction to one of Diskin's
> claims, two database administrators (at two large multi-national
> corporations) that I know think it is a viable model.  They do not balk  at
> the idea of data columns being considered as foreign keys.  It puts
> everything on the same playing field.
>

...

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