Compilers on CS Theorems

Register Allocation Is Graph Coloring

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

Symptom

You write a function with forty local variables. The x86-64 machine it runs on has sixteen general-purpose registers, two of which are effectively spoken for.

Something must decide which values live in registers and which live in memory. Get it right and the function runs at register speed. Get it wrong and every operation costs a load and a store, and on a modern machine an L1 hit is around 4 cycles against roughly 1 for a register, with an L2 miss costing 12 to 20 and a trip to DRAM 200 or more.

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.

Kleene's Theorem: Regexes Are Finite Automata

Fri, 30 Jul 2027 12:00:00 +0100

Symptom

Your service goes down. CPU pinned at 100% on one core, no memory growth, no error logs, requests timing out.

The cause is a validation regex — something reasonable-looking like ^(a+)+$, or in real life a URL or email validator with nested quantifiers — matched against a 30-character string that happens not to match. The engine is exploring an exponential number of ways to split the input among the nested groups, and it will get there eventually, some time after the heat death of everything.

The Chomsky Hierarchy

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

Symptom

The most upvoted answer in Stack Overflow history is a refusal to answer. Someone asked how to match nested HTML tags with a regular expression, and the reply is a page of escalating horror about the centre not holding. It is funny, and it is also a theorem, which is not obvious from reading it.