Olog Explained
The theory of ologs is an attempt to provide a rigorous mathematical framework for knowledge representation, construction of scientific models and data storage using category theory, linguistic and graphical tools. Ologs were introduced in 2012 by David Spivak and Robert Kent.[1]
Etymology
The term "olog" is short for "ontology log". "Ontology" derives from onto-, from the Greek ὤν, ὄντος "being; that which is", present participle of the verb εἰμί "be", and -λογία, -logia: science, study, theory.
Mathematical formalism
An olog
for a given domain is a
category whose
objects are boxes labeled with phrases (more specifically, singular indefinite noun phrases) relevant to the domain, and whose
morphisms are directed arrows between the boxes, labeled with verb phrases also relevant to the domain. These noun and verb phrases combine to form sentences that express relationships between objects in the domain.
In every olog, the objects exist within a target category. Unless otherwise specified, the target category is taken to be
, the
category of sets and functions.The boxes in the above diagram represent objects of
. For example, the box containing the phrase "an amino acid" represents the set of all amino acids, and the box containing the phrase "a side chain" represents the set of all side chains. The arrow labeled "has" that points from "an amino acid" to "a side chain" represents the function that maps each amino acid to its unique side chain.
of the power set monad. Given an
,
is then the power set of A. The
natural transformation
maps
to the
singleton
, and the natural transformation
maps a set of sets to its union. The
Kleisli category
is the category with the objects matching those in
, and morphisms that establish
binary relations. Given a morphism
, and given
and
, we define the morphism
by saying that
whenever
. The verb phrases used with this target category would need to make sense with objects that are subsets: for example, "is related to" or "is greater than".
Another possible target category is the Kleisli category of probability distributions, called the Giry monad. This provides a generalization of Markov decision processes.
Ologs and databases
An olog
can also be viewed as a
database schema. Every box (object of
) in the olog is a
table
and the arrows (morphisms) emanating from the box are columns in
. The assignment of a particular instance to an object of
is done through a
functor
. In the example above, the box "an amino acid" will be represented as a table whose number of rows is equal to the number of types of amino acids and whose number of columns is three, one column for each arrow emanating from that box.
Relations between ologs
"Communication" between different ologs which in practice can be communication between different models or world-views is done using functors. Spivak coins the notions of a 'meaningful' and 'strongly meaningful' functors. Let
and
be two ologs,
,
functors (see the section on ologs and databases) and
a functor.
is called a schema mapping. We say that a
is
meaningful if there exists a natural transformation
(the
pullback of J by F).
Taking as an example
and
as two different scientific models, the functor
is meaningful if "predictions", which are objects in
, made by the first model
can be translated to the second model
.
We say that
is
strongly meaningful if given an object
we have
. This equality is equivalent to requiring
to be a natural isomorphism.
Sometimes it will be hard to find a meaningful functor
from
to
. In such a case we may try to define a new olog
which represents the common ground of
and
and find meaningful functors
}:\mathcal\to \mathcal and
}:\mathcal\to \mathcal.
If communication between ologs is limited to a two-way communication as described above then we may think of a collection of ologs as nodes of a graph and of the edges as functors connecting the ologs. If a simultaneous communication between more than two ologs is allowed then the graph becomes a symmetric simplicial complex.
Rules of good practice
Spivak provides some rules of good practice for writing an olog whose morphisms have a functional nature (see the first example in the section Mathematical formalism). The text in a box should adhere to the following rules:
- begin with the word "a" or "an". (Example: "an amino acid").
- refer to a distinction made and recognizable by the olog's author.
- refer to a distinction for which there is well defined functor whose range is
, i.e. an instance can be documented. (Example: there is a set of all amino acids).
- declare all variables in a compound structure. (Example: instead of writing in a box "a man and a woman" write "a man
and a woman
" or "a pair
where
is a man and
is a woman").
The first three rules ensure that the objects (the boxes) defined by the olog's author are well-defined sets. The fourth rule improves the labeling of arrows in an olog.
Applications
This concept was used in a paper published in the December 2011 issue of BioNanoScience by David Spivak and others to establish a scientific analogy between spider silk and musical composition.[2]
See also
External links
Notes and References
- Spivak . David I. . Kent . Robert E. . Ologs: A Categorical Framework for Knowledge Representation . . 31 January 2012 . 7 . 1 . e24274 . 10.1371/journal.pone.0024274 . 22303434 . 3269434 . 2012PLoSO...724274S . 1102.1889 . free .
- 1111.5297. Giesa . Tristan . Spivak . David I. . Buehler . Markus J. . Reoccurring patterns in hierarchical protein materials and music: The power of analogies. 2011 . 10.1007/s12668-011-0022-5 . 1 . 4 . BioNanoScience . 153–161. 5178100 .
