Alexander Solla wrote:
And denotational semantics is not just nice. It is useful. It's the best way to understand why the program we just wrote doesn't terminate.
Denotational semantics is unrealistic. It is a Platonic model of constructive computation. Alan Turing introduced the notion of an "oracle" to deal with what we are calling bottom. An oracle is a "thing" that (magically) "knows" what a bottom value denotes, without having to wait for an infinite number of steps. Does Haskell offer oracles? If not, we should abandon the use of distinct bottoms. The /defining/ feature of a bottom is that it doesn't have an interpretation.
Huh? I don't see the problem. Introducing bottom as a value is a very practical way to assign a well-defined mathematical object to each expression that you can write down in Haskell. See http://en.wikibooks.org/wiki/Haskell/Denotational_semantics It's irrelevant whether _|_ is "unrealistic", it's just a mathematical model anyway, and a very useful one at that. For instance, we can use it to reason about strictness, which gives us information about lazy evaluation and operational semantics. Best regards, Heinrich Apfelmus -- http://apfelmus.nfshost.com