[seqfan] Re: Can we create a workable taxonomy for classifying all the various kinds of sequences in the OEIS?

[Veikko Pohjola]

Just a short comment,

Taxonomies are typically domain specific ad hoc systems which prove unsatisfactory when the domain evolves or is viewed in a larger perspective. I would suggest an adequate ontology based approach where the sequences forming a hierarchical whole are treated as objects with a generic set of attributes. I have myself a lot of experience of applying the PSSP ontology with success on various diffuse problem areas of the real world like knowledge management (even ignorance management) and why not to mention waste management, the project which started by defining the waste itself. Some of my students went crazy in searching most strange targets to test the approach like formulating zero and God as a PSSP object. So, I do not doubt its power in attacking the integer sequences, but who would have the necessary interest and time.

Veikko Pohjola


> Wayne VanWeerthuizen <waynemv at gmail.com> kirjoitti 6.4.2018 kello 2.39:
> 
> 
> I've long wondered whether it might be possible to develop a good taxonomy for classifying sequences. I'm not sure how well this can be achieved. A single sequence may fall into multiple categories. Getting all sequences into at least one appropriate category may mean we need a lot more categories than just the one I suggest below. And some of the categories, such as geometric, might be subjective, with no precise formal way to rigorously define what should be included. But anyway, if we can find a feasible approach, I'd appreciate seeing a classification of all or most sequences in the OEIS into categories somewhat like these:
> 
> Geometric Sequences:
>   Sequences whose primary or simplest interpretation is geometric.
> 
>   Examples: triangular numbers (A000217), river crossings (A005316)
> 
> 
> Combinatoric Sequences: 
>   Sequences a(N) answering how many ways to create a construction of a certain size or order N such that it meets a given criteria
> 
>   Example: Number of ways to have N red, white, or blue beads in a string such no red and blue beads are adjacent. (A078057)
> 
> 
>  Discriminatory Sequences: 
>    Sequence lists all which pass a given test for compliance with some criteria.
>    That is, for each possible k, if boolean test f(k) is true, then k is part of the sequence.
>    Such sequences should be strictly increasing: if j and k are both in the sequence, j should be listed before k if j<k.
> 
>    Examples: Prime numbers (A000040), squarefree numbers (A005117), palindromes (A002113), The Positive Integers (A000027)
> 
> 
>  Least-qualifying-value sequences:
>    Sequences a(n)=k, such that for each n, and a given boolean function f(n,k), k is the lowest positive value such that f(n,k) is true.
>    Examples: A246491, A215537
> 
> 
> Physically-based Sequences:
>    Sequences based on experiments in the actual world.
> 
>    Example: Decimal expansion of electron mass in kg. (A081801)
> 
> 
> Recursive Sequences:
>    Sequences whose most natural definition is based on a recursively applying a given formula. 
>    a(n)=f( a(n-1), a(n-2), ... )
> 
>    Example: Fibonacci numbers (A000045)
> 
> 
>  Polynomial Sequences:
>    Sequences whose terms are defined by a single polynomial P, such that a(n)=P(n).
> 
>    Example: square numbers, triangular numbers
> 
> 
> Off the top of my head, I don't know any interesting Polynomial sequences that are not also Geometric sequences. Recursive Sequences might be a poorly chosen category, given many sequences in the database can be generated using recursive functions, not all of which look recursive at first glance. But without such a category, I am not sure how otherwise to classify sequences such as the Fibonacci and Lucas numbers. This example gets to my core question of whether a classification scheme like this is even viable. I don't expect these few categories to cover all existing sequences in the database, I think many more categories would be needed for that. And again, a single sequence could fall into multiple categories. I'm also not yet suggesting any hierarchy of subcategories of categories, but that could also come into play later if we developed a more refined and detailed taxonomy. Also, I think even if the big goal of finding sufficient categories to cover all the sequences in the database proves too difficult, it might still be valuable to have a few categories and to tag just those sequences which can be easily classified, accepting that not all sequences would be tagged.
> 
> Anyway, has anyone else here thought much about how to classify sequences in this sort of way? How might the categories I've suggested be more clearly and rigorously defined? Can they be given better names? What additional categories might one suggest? What percentage of the sequences in the OEIS do you estimate can be classified into one or more of the categories I've suggested above?
> 
> 
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