My comments on Lance's Favorite Theorems

In Lance's last post (see here) he listed his favorite theorems from 1965 to 2024.
There are roughly 60 Theorems. I mostly agree with his choices and omissions. I will point out where I don't.

I could make a comment on every single entry; however, that would be madness! Madness I say!

Instead, here are some random thoughts.  (Is that Random as in Random Restriction or Random as in Random Access Machine?  I leave that an exercise for the reader.)

1) 1965-1974

MANY BASIC RESULT WITH EASY PROOFS.
EXAMPLE:
The Cook-Levin Theorem. P, NP, and SAT is NPC

ANSWERS A QUESTION:
Ladner: Answers a very important question: YES, if P NE NP there are
intermediary sets. The set is constructed for the sole point of not being in P or NPC. Graph Isom and Factoring are natural candidates for being intermediary.

SEEMS TO HAVE BEEN FORGOTTEN:
Blum: Abstract Complexity Theory. Seems to not be taught anymore. I give a corollary for our young readers who might not know it:

There is a decidable set A such that If A is in DTIME(T(n)) then A is in DTIME((log T(n)). Hence A cannot be assigned a complexity. (The set A is constructed for the sole point of having this property. There are no natural examples or even candidates for sets that have this behavior.)

I might disagree with putting this on the list. It has not stood the test of time; however, it still seems important. 

II) 1975-1984.

This may be my favorite decade on the list; however, its been said
that everyone thinks that the best music was when they were a teenager.

EXAMPLES:

INTERESTING THEOREMS WITH INTERESTING PROOFS:
Everything on the list is in this category but I pick out three:

Baker-Gill-Solovay Oracles: The basic paper for my thesis work.

Furst-Saxe-Sipser Parity is not in constant depth.  A meaningful lower bound on a natural problem! Motivated by an Oracle open question (Sep PH from PSPACE) however, circuit complexity quickly became a field onto itself.  What is more interesting the circuit lower bound or the oracle-corollary? I would vote for the circuit lower bound. The issue was discussed here.

Valiant-Permanent. Perm is #P-complete is a theorem I've learned and forgotten many times. Scott much later gave a proof that may be more intuitive for some people (I am not one of them) see here.  The only theorem I've learned-and-forgotten more is the Hales-Jewitt Theorem.

 
III) 1985-1994.

A Decade of Surprises!

Barrington: Branching programs more powerful than we thought!

Toda: #P is more powerful then we thought!

LFKN: IP is more powerful than we thought! Bonus: used non-rel methods! (This result was not on the list but would have been if anybody except L or F or K or N had written the list.)

Nisan: Randomization is less powerful than we thought!

Lance did not have Factoring in Quantum P on the list for 1985-1994. It came out in 1994 towards the end of the year so it ended up missing both the 1985-1994 list and the 1995-2004 list. Reminds me of the Betty White awards, see here.  I do not think we disagree on the importance and merit of the result, though we disagree about altering the past- I would have included it in the post he did recently and explain that it was a late add to an old list.

IV) 1995-2004.

In 1990 a theorist told me that he could teach EVERYTHING known in complexity theory in a year-long graduate course.  Even then, that was not quite right, and may have really meant he could teach everything he thought was important. By 1995 this was no longer true. The PCP result alone would take a few months.

Theory begins to get really hard. Most of the papers before 1992 I have read and understood. Most of the papers after 1992 I have not, though I know what's in them. Sometimes not even that!

PCP: Many problems are hard to approximate.  Good to know, bad that its true. Proofs are hard!

Raz's Parallel Repetition: Really useful in later non-approx results (my favorite: Set Cover Lower bounds, see this survey here) but also Really Hard to read.

Most of the other papers are also hard and important.

AKS- Primality in P- not that hard. Indeed, surprising that it was not proven until 2002.

Lance did not include the work on Natural Proofs by Razborov and Rudich. He says why in a blog post here. I disagree with him- I would have put it in a top theorems list.

VI) 2005-2014.

Some upper bounds and some lower bounds. By now it was hard to have a surprising result since our intuitions were not so firm as to be surprised. (There is one exception in the 2015-2024 list.)

Reingold-Undirected Connectivity in Log Space: great result! I wish the proof was easier. I think people thought this would be true.

Lots of interesting lower bounds: Nash Equilibrium, Unique Game Conj, new PCP proof. None of which was surprising, though perhaps that we could proof things about these concepts is surprising.

JJUW-QIP=PSPACE. Really! I was shocked to find out that was true. No, I wasn't. I didn't understand QIP well enough. Was this surprising or not? Was the fact that this could be proven surprising or not?

VII) 2015-2024.

No real theme here though they all have hard proofs. I discuss a few.

Babai-Graph Isomorphism is the only result in this decade that I can explain to Darling. And she has a Masters Degree in SE so she knows stuff. (I recently told her the result that for all 2-colorings of R^6 there is a mono unit square  (see here). She was unimpressed.)

BZ- Dichotomy: Excellent result and explains the lack of natural intermediary problems.

CZ-Extracting Ramsey Graphs: An example of TCS helping to prove a result in math, though it also shows that the border between the two is thin. Obviously a favorite of mine.

JNVWY- MIP* = RE. This surprised people, including me.