Although Professor Fortnow's blog covers many topics in complexity theory, it is a pity that the blog rarely touches on the elegant connection between logic and complexity theory, which --- God knows why --- many finite model theorists think will be a key to proving that P does not equal NP. For example, here are some deep results in the area:
- Fagin's theorem: the set of properties of (relational) finite structures expressible in existential second-order logic (ESO) coincides with NP.
- Dual Fagin's theorem: the set of properties of finite structures expressible in universal second-order logic (ASO) coincides with co-NP.
- The set of properties of finite structures expressible in second-order logic (SO) coincides with PH.
So, in order to prove that P does not equal NP, it suffices to prove that the set of properties of finite structures expressible in ESO is different from that expressible in ASO. In fact, to prove this it is enough to show that the property 3-colorability is not expressible in ASO on the class of finite graphs. That is, we have reduced an important problem in complexity theory to a problem in finite model theory. The area of theoretical computer science that explores the connection between finite model theory and complexity theory is called descriptive complexity.
In view of these, I thought it would be a good idea for me to start a blog that aims to explore the deep connection between logic and complexity. Nonetheless, I realize that it is a formidable task to be a good blogger even for a small subarea of logic and complexity, especially for a newbie like myself. Thus, I will try to focus on areas that I will explore during my graduate studies: (finite and infinite) model theory and descriptive complexity, proof complexity, and perhaps bounded arithmetic. [Nonetheless, I will probably have to every now and then violate this rule and speak about other things that I find interesting and fun.] So, from now on, when I speak about 'Logic and Complexity', I will be referring to these subareas. There are already excellent blogs that cover logic, like this one maintained by Professor Restall at The University of Melbourne's Department of Philosophy, and logic and computation, like this one. However, none of these cover the stuff that I plan to talk about here such as finite model theory and descriptive complexity.
Anyway, for a starter, I plan to write a concise introduction to finite model theory that other mathematical readers, who know a bit of logic, can read. I will try to post this bit-by-bit every week. Any other ideas? Feel free to comment.