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• CommentRowNumber1.
• CommentAuthorLarryB
• CommentTimeOct 20th 2017
I would like to start a page on descriptive complexity. Complexity theory is an area of research in computer science that aims to determine the amount of resources (time and space usually) that an algorithm needs to decide a problem. Descriptive complexity is a branch of complexity theory that uses finite model theory (first-order, second-order, and some in-between operators) to describe the problems. It turns out that these languages directly corresponds to known complexity classes.

My aim is to translate the logic used into an equivalent topos theory of finite logic, or even do away with logic altogether and directly map complexity classes with a corresponding category. The goal of this is to reframe questions in complexity theory as questions in category theory.

For example: we can describe the existential second order quantifier as the left adjoint of the diagonal functor (context extension?) from $P(P(X)) \to P(P(X))\times P(P(X))$. I'm not sure whether $X$ is the ccc with all the propositional formulas, or if it's the finite model. I'm still learning.

Would it be appropriate to add to the nLab? I'm autodidactic with category theory, so I'm not sure if I can accurately describe things, but I'd still like a space to share my ideas.
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeOct 20th 2017

Would it be appropriate to add to the nLab?

It sounds good. You should start somewhere and then we can give more substantial feedback.

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeOct 21st 2017

So Larry began descriptive complexity. We generally put in links to pages for concepts mentioned, e.g., I’ve put one in for your opening complexity theory. Of course that latter page needs enormous expansion, but at the very least could have your opening sentence.

Also you can use bold font to emphasise the new concept, as I’ve just done.

Looking about the nForum, there was a brief discussion of category theory and complexity theory here and geometric complexity theory here.

• Noson S. Yanofsky, Computability and complexity of categorical structures, arXiv:1507.05305

might be worth a look.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeOct 22nd 2017

I have hyperlinked yet more keywords.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeOct 22nd 2017

Just a note to Larry that it is customary at the nLab to leave a note at the nForum if pages are created or edited, except in minor cases like fixing typos. Thanks for starting this!