Automata 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.

The Myhill–Nerode Theorem

Fri, 06 Aug 2027 12:00:00 +0100

Symptom

You need to prove a language is not regular. Everyone points you at the pumping lemma, and you spend an afternoon losing to it.

The statement is a nest of quantifiers: for every regular language there exists a pumping length $p$ such that for every string $w$ with $|w| \ge p$ there exists a decomposition $w = xyz$ with $|xy| \le p$ and $|y| > 0$ such that for all $i \ge 0$, $xy^i z$ is in the language. To use it you negate all of that and play a game against an adversary who picks $p$ and the decomposition while you pick $w$ and $i$.

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.