Logic on CS Theorems

Resolution and CDCL

Fri, 10 Dec 2027 12:00:00 +0100

Symptom

SAT is NP-complete (T027). It is the NP-complete problem, the one everything else reduces to. The textbook conclusion is that you should avoid it.

Meanwhile: a hardware verification team routinely discharges instances with two million variables and ten million clauses in under a minute. Package managers solve dependency resolution with SAT. Every bounded model checker (T049) is a SAT solver with a front end. The planning community abandoned custom search for SAT encodings and got faster.

Hoare Logic and Weakest Preconditions

Fri, 19 Nov 2027 12:00:00 +0100

Symptom

You changed a loop. It looks right. The tests pass. You are not confident.

The reason you are not confident is that the tests check a finite set of inputs and the loop runs on all of them. You know this. So you stare at the loop and try to convince yourself by simulating it in your head, which works for three iterations and then stops working.

The Curry–Howard Correspondence

Fri, 17 Sep 2027 12:00:00 +0100

Symptom

You write a function signature and it looks like a sentence:

f :: (a -> b) -> (b -> c) -> (a -> c)

Read it aloud: if $a$ implies $b$, and $b$ implies $c$, then $a$ implies $c$. That is transitivity of implication, a rule of logic you have known since school. And the implementation, f g h = h . g, is the proof of it: given evidence for $a$, apply the first, then apply the second.

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 Cook–Levin Theorem

Fri, 19 Mar 2027 12:00:00 +0100

Symptom

You know reductions (T017): to prove your problem hard, reduce a known-hard problem to it. Fine. But that begs the obvious question, and it is the question a sharp colleague asks the first time you use the technique.

Godel's Incompleteness Theorems

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

Symptom

The proof assistant is failing you. You are trying to verify a piece of concurrent code in Coq or Lean, and the tool will not accept your termination argument. A colleague says, without much conviction, “well, Gödel — you can’t prove everything anyway.”

Kleene's Recursion Theorem

Fri, 15 Jan 2027 12:00:00 +0100

Symptom

Write a program that prints its own source code. No file I/O, no reading __file__, no cheating.

The first attempt fails instantly. To print the source you must contain the source, and then the containing text is also part of the source, so you must contain that too. The regress is obviously infinite, and most people conclude after ten minutes that the task is impossible.

The Halting Problem

Fri, 09 Oct 2026 12:00:00 +0200

Symptom

Your CI has a test that hangs. Not fails — hangs. Somebody suggests the obvious fix: before running a test, check whether it terminates, and skip it if not.

Cantor's Diagonal Argument

Fri, 02 Oct 2026 12:00:00 +0200

Symptom

Somebody proposes a tool that will find all the bugs. Not most bugs. All of them. The pitch has the shape: we enumerate the failure modes, we write a checker for each, we keep adding checkers, and the set of undetected bugs shrinks toward zero.