Generic-case — decidable on almost all inputs

Parent: relaxing-undecidability

Worst-case undecidable ≠ hard everywhere. Often an algorithm decides correctly on a set of inputs of density 1, and only fails (or fails to halt) on a negligible remainder.

Generic-case decidability (Kapovich–Myasnikov–Schupp–Shpilrain, 2003)

A problem is generically decidable if some partial algorithm halts with the correct answer on a set of inputs of asymptotic density 1, and may diverge on the rest.

Examples: the word problem in many groups, and the halting problem for "most" Turing machines, are generically decidable — the undecidability is concentrated on a density-0 knife-edge, even though it's real.

Control use. "Most" controllers/systems have an obvious convergence verdict (a Lyapunov function is easy to find, or the system is trivially unstable); only carefully-tuned boundary systems are genuinely hard. So a practical convergence-checker can be right almost always.

Caveat: genericity is about typical inputs — an adversary can still hand you a density-0 hard instance. It doesn't help when the specific system you care about is the knife-edge.