L3 on CS Theorems

Model Checking, LTL, and Buchi Automata

Fri, 26 Nov 2027 12:00:00 +0100

Symptom

Your distributed lock service deadlocks once a month. You cannot reproduce it.

The logs show three nodes in states that, as far as you can reconstruct, should not co-occur. You add tracing. It does not happen again for six weeks, then happens twice in a day during a deploy. You write a stress test that runs a million random interleavings overnight and finds nothing.

Abstract Interpretation and Galois Connections

Fri, 22 Oct 2027 12:00:00 +0100

Symptom

Rice’s theorem (T011) told you every non-trivial semantic property is undecidable. Your static analyzer must therefore be unsound, incomplete, or non-terminating. In practice it is incomplete: it reports things that cannot happen.

IP = PSPACE

Fri, 08 Oct 2027 12:00:00 +0100

Symptom

You have a claim you cannot check. A cloud provider ran a computation on a petabyte you do not have and reports an answer. A model was trained on data you cannot access. A database returned an aggregate over rows you are not permitted to read.

The CALM Theorem and CRDTs

Fri, 03 Sep 2027 12:00:00 +0100

Symptom

Every distributed result so far has been a prohibition. Two Generals: no agreement over a lossy channel. FLP: no deterministic asynchronous consensus. Byzantine: not below $3f+1$. CAP: not all three.

BPP, Pseudorandomness, and Derandomization

Fri, 23 Jul 2027 12:00:00 +0100

Symptom

You need to test whether two arithmetic circuits compute the same polynomial. The deterministic approach is to expand both into normal form, and the expansion is exponentially large.

The randomized approach takes four lines. Pick a random point, evaluate both circuits there, compare. If they differ as polynomials, the Schwartz-Zippel lemma says a random point catches it with probability at least $1 - d/|S|$. Repeat a few times and the error is negligible.

Parity Is Not in AC^0

Fri, 09 Jul 2027 12:00:00 +0100

Symptom

You have just read T030 and you are demoralized. Relativization kills the techniques that treat machines as black boxes. Natural proofs kill the combinatorial ones, assuming pseudorandom functions exist. Between them they appear to rule out everything anyone knows how to do, and the honest summary of fifty years is that nobody can prove any interesting problem is hard.

FLP Impossibility

Fri, 11 Jun 2027 12:00:00 +0100

Symptom

Your cluster of five nodes elects a leader. It works. It has worked for a year.

Then one afternoon a garbage collection pause on the leader runs for twelve seconds. The followers time out, start an election, and elect a new leader. The old leader wakes up, has no idea it was deposed, and keeps serving writes. For a few hundred milliseconds you have two leaders, and if your fencing is not airtight, two writes that should have been ordered land in an order nobody chose.

The PCP Theorem and Inapproximability

Fri, 07 May 2027 12:00:00 +0100

Symptom

You gave up on exact (T028) and went looking for an approximation.

Vertex cover: you find a 2-approximation in four lines. Knapsack: an FPTAS, any accuracy you want. Encouraged, you go looking for the same for max-clique, and you find nothing. Not “nothing good” — the best known ratio is around $n/(\log n)^2$, which on a 10,000-vertex graph means the algorithm might return a clique nearly 60 times smaller than the true maximum. That is not an approximation, it is a rumour.

The Barriers: Relativization and Natural Proofs

Fri, 02 Apr 2027 12:00:00 +0100

Symptom

You have read T026 and T027 and you are wondering the obvious thing.

Fifty years. Thousands of researchers. A million-dollar prize. And P vs NP has not moved. That is strange — most famous problems yield partial results, near misses, special cases that suggest the shape of the answer. Here there is almost nothing. We cannot even prove that NP requires more than linear-size circuits, which is a laughably weak statement compared to what everyone believes.

The Noisy-Channel Coding Theorem

Fri, 26 Feb 2027 12:00:00 +0100

Symptom

Your link drops 1% of bits. You need reliable delivery, so you triple every bit and take a majority vote. Now an error needs two of three bits to flip, which happens with probability about $3 \times 10^{-4}$ — better, but you paid 3x in bandwidth and the error rate is still not zero.