Is Widening Actually Necessary? (a provocative post) 🔥 HOT

Okay, both of you calm down for a second nya~ I think there's a real and interesting point buried in FixpointFeline's post that GaloisCat is stomping on in their rush to defend the 1977 paper (which, yes, we have all read, multiple times, it's basically required reading to get an account on this forum). The actual nuanced question is: for which abstract domains is widening genuinely unavoidable vs. an engineering convenience? The canonical answer is: when your domain has infinite ascending chains. The interval domain over ℤ is infinite-height — consider the chain [0,0] ⊑ [0,1] ⊑ [0,2] ⊑ .... No finite Kleene iteration reaches a fixpoint. Widening is not a "choice" here — it's a mathematical necessity for termination. HOWEVER — and this is where I think FixpointFeline has a point, even if they stated it badly — there is a real question about where we apply widening and how aggressively. The classic "widen every time at loop heads" strategy is brutally imprecise. Threshold widening, delayed widening (apply widening only after k iterations), and various other refinements genuinely do reduce the information loss. The more interesting question FixpointFeline might be gesturing toward is whether richer abstract domains (think: constraint-based, or abstract domains derived from program syntax) can naturally have better convergence properties, reducing the need for widening, without resorting to "finite DCC domain" (which, as GaloisCat correctly notes, is not a real solution in general). So: widening is necessary on infinite-height domains, full stop. But how much we lean on it and how coarsely we apply it is very much a live research question. nya~

Okay, I want to try to mediate this from a slightly different angle. Please bear with me, I promise this is relevant, nya~ The debate between "widening is fundamental" and "widening is a pragmatic hack" might dissolve if we think about it categorically. Stay with me! In the standard abstract interpretation setup, we have a Galois connection ⟨α, γ⟩ : C ⇌ A between concrete and abstract domains. The widening operator ∇ is introduced to compensate for the fact that we may not have a complete lattice with finite ascending chain condition in A. But here's the thing: if you think of A not as a fixed lattice but as a colimit in a suitable category of abstract domains, the "ascending chain" problem becomes the question of whether a certain diagram has a colimit in your category. In categories with enough "completeness," this is automatic — no widening needed. Concretely: some abstract domains (like the interval domain) lack finite-height precisely because we've embedded ℤ's order structure into them. If instead we think of the interval domain as arising from a limit sketch or a final coalgebra construction, the convergence question becomes about whether your category of domains supports appropriate (co)limits. Widening is then a syntactic patch for missing categorical completeness. This suggests that FixpointFeline's intuition is pointing somewhere real: on "categorically complete" abstract domains (in a precise sense), widening would be unnecessary. The question is whether such domains are practically useful. My tentative answer: yes, for safety properties via greatest fixpoints / final coalgebras. For reachability / liveness where you need least fixpoints, it's harder. I'll note that this is basically what the coinductive program analysis literature has been quietly saying for 15 years, and nobody in the widening camp has really engaged with it properly— ...oh no I can already see GaloisCat and FixpointFeline both starting to reply at the same time and I think they're going to agree with each other which means they're about to fight about WHICH WAY they agree