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nLab > nLab General Discussions: nPOV on Complexity Theory?

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    • CommentRowNumber1.
    • CommentAuthorSaalHardali
    • CommentTimeJan 21st 2017
    • (edited Jan 21st 2017)
    Recently I noticed that most (if not all) nLab pages about complexity theory and related topics are essentially empty. This happened after I thought a bit about how to put Turing machines into a category. Later I realized (after reading a couple of answers to questions on Theoretical CS stack-exchange) that there doesn't yet seem to be a natural categorical home for complexity theory (or at least not a consensus about such a thing).

    I was wandering how accurate this observation of mine is. Has there been any insights in this area ("categorical complexity theory")? Or is this topic, like the nLab page displays, essentially empty as of yet.

    I apologize in advance if this is the wrong sub for this post. I'm new to this forum.

    Disclaimer: I must warn anyone who attempts to answer this that I know very little about complexity theory and so wouldn't be of much use in a discussion.
    • CommentRowNumber2.
    • CommentAuthormaxsnew
    • CommentTimeJan 21st 2017
    • (edited Jan 21st 2017)

    I came across this abstract Monoidal Computer II in the past, that may be of interest. Haven’t read it myself, though.

    • CommentRowNumber3.
    • CommentAuthorThomas Holder
    • CommentTimeJan 21st 2017
    • (edited Jan 21st 2017)

    On Turing machines there is old work by Khalil and Walters which is documented in a section of Bob Walters, Categories and Computer Science , Cambridge UP 1991.

    On complexity theory proper there has presumably not done much from a categorical perspective but one can give the following two a try:

    • Basu, Simpson, Mass problems and higher-order intuitionistic logic , arXiv:1408.2763 (2014).

    • Basu, Izik, Categorical Complexity , arXiv:1610.07737 (2016).

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJan 21st 2017

    You might be interested in Turing categories. Also, there are some variants of linear logic (which in general is closely connected to monoidal categories) that are designed to compute only certain complexity classes, although I don’t know that anyone has studied them categorically.

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