So that first step already relies on IO (where the two are equivalent). Come again?
The first step in your implication chain was (without the return) throw (ErrorCall "urk!") <= 1 ==> evaluate (throw (ErrorCall "urk!")) <= evaluate 1 but, using evaluation only (no context-sensitive IO), we have
throw (ErrorCall "urk") <= evaluate (throw (ErrorCall "urk"))
Sure enough.
meaning that first step replaced a smaller with a bigger item on the smaller side of the inequation. Unless the reasoning includes context- sensitive IO rules, in which case the IO rule for evaluate brings the throw to the top (evaluate (throw ..) -> (throw ..)), making the two terms equivalent (modulo IO), and hence the step valid (modulo IO). Unless you just rely on
But throwIO (ErrorCall "urk") /= _|_: Control.Exception> throwIO (ErrorCall "urk!") `seq` () ()
in which case that step relies on not invoking IO, so it can't be mixed with the later step involving IO for catch (I think).
This is very delicate territory. For instance, one might think that this 'f' seems to define a "negation function" of information content
f x = Control.Exception.catch (evaluate x >> let x = x in x) (\(ErrorCall _)->return 0) >>= print
and hence violates monotonicity
(_|_ <= ()) => (f _|_ <= f ())
since
*Main> f undefined 0 *Main> f () Interrupted.
But that is really mixing context-free expression evaluation and context-sensitive execution of io operations. Most of our favourite context-free equivalences only hold within the expression evaluation part, while IO operations are subject to additional, context-sensitive rules.
Could you elaborate on this? It sounds suspiciously like you're saying Haskell's axiomatic semantics is unsound :: IO.
Not really unsound, if the separation is observed. One could probably construct a non-separated semantics (everything denotational), but at the cost of mapping everything to computations rather than values. Then computations like that 'f' above would, eg, take an extra context argument (representing "the world", or at least aspects of the machine running the computation), and the missing information needed to take 'f _|_'[world] to '()'[world'] would come from that context parameter (somewhere in the computational context, there is a representation of the computation, which allows the context to read certain kinds of '_|_' as exceptions; the IO rule for 'catch' takes that external information and injects it back from the computational context into the functional program, as data structure representations of exceptions). That price is too high, though, as we'd now have to do all reasoning in context-sensitive terms which, while more accurate, would bury us in irrelevant details. Hence we usually try to use context-free reasoning whenever we can get away with it (the non-IO portions of Haskell program runs), resorting to context-sensitive reasoning only when necessary (the IO steps of Haskell program runs). This gives us convenience when the context is irrelevant as well as accuracy when the context does matter - we just have to be careful when combining the two kinds of reasoning.
For instance, without execution
*Main> f () `seq` () () *Main> f undefined `seq` () ()
but if we include execution (and the context-sensitive equivalence that implies, lets call it ~),
So
a ~ b = `The observable effects of $(x) and $(y) are equal'
?
Observational equivalence is one possibility, there are various forms of equivalences/bi-similarities, with different ratios of convenience and discriminatory powers (the folks studying concurrent languages and process calculi have been fighting with this kind of situation for a long time, and have built up a wealth of experience in terms of reasoning). The main distinction I wanted to make here was that '=' was a context-free equivalence (valid in all contexts, involving only context-free evaluation rules) while '~' was a context-sensitive equivalence (valid only in IO contexts, involving both context-free and context-sensitive rules).
we have
f () ~ _|_ <= return 0 ~ f _|_
so 'f' shows that wrapping both sides of an inequality in 'catch' need not preserve the ordering (modulo ~)
If f _|_ <= f (), then it seems that (<=) is not a (pre-) order w.r.t. (~). So taking the quotient of IO Int over (~) gives you a set on which (<=) is not an ordering (and may not be a relation).
As I said, mixing '=' and '~', without accounting for the special nature of the latter, is dangerous. If we want to mix the two, we have to shift all reasoning into the context-sensitive domain, so we'd have something like f () [world] ~ _|_ [world''] <= return 0 [world'] ~ f _|_ [world] (assuming that '<=' is lifted to compare values in contexts). And now the issue goes away, because 'f' doesn't look at the '_|_', but at the representation of '_|_' in the 'world' (the representation of '_|_' in GHC's runtime system, say).
- its whole purpose is to recover from failure, making something more defined (modulo ~) by translating _|_ to something else. Which affects your second implication.
If the odd properties of 'f' capture the essence of your concerns, I think the answer is to keep =, <=, and ~ clearly separate, ideally without losing any of the context-free equivalences while limiting the amount of context-sensitive reasoning required. If = and ~ are mixed up, however, monotonicity seems lost.
So
catch (throwIO e) h ~ h e
but it is not the case that
catch (throwIO e) h = h e
? That must be correct, actually:
Control.Exception> let x = Control.Exception.catch (throwIO (ErrorCall "urk!")) (\ (ErrorCall _) -> undefined) in x `seq` () ()
So catch is total (even if one or both arguments is erroneous), but the IO executor (a beast totally distinct from the Haskell interpreter, even if they happen to live in the same body) when executing it feels free to examine bits of the Haskell program's state it's not safe for a normal program to inspect. I'll have to think about what that means a bit more.
Yes, exactly!-)
[Totally OT tangent: How did operational semantics come to get its noun? The more I think about it, the more it seems like a precís of the implementation, rather than a truly semantic part of a language specification.]
There's bad taste associated with the term, stemming from older forms of operational semantics that were indeed unnecessarily close to the implementations (well, actually, such close resemblance can still be useful to guide implementations, or to prove things about implementations, so there are many forms of operational semantics, varying in levels of abstraction to accommodate the target areas of study). Modern forms of operational semantics (when studying languages, not implementations) are much more abstract than that, closer to inference rules of a programming logic. Oversimplified: they study equivalence classes of semantics, using syntactic terms as canonical representatives. This use of syntax tends to confuse denotational semantics adherents, who say that syntax should be irrelevant in order to achieve sufficiently abstract semantics. But if we have two denotational semantics, S1 and S2, mapping constructs of language L to D1 and D2, respectively, then the only thing they are going to have in common are the constructs of L and, hopefully, the relations between the things these constructs are mapped to. So, if we want to abstract over the specific denotational semantics Sx, and its specific domain Dx, we just talk about [| L |] or, by abuse of notation, about L. So, when abstract operational semantics talk about 'X ~ Y' for some X,Y in L, they are really talking about 'forall semantics S :: L -> D. S[| X |]::D ~ S[| Y |]::D', without the repetitive details. In other words, when abstract operational semantics focus on syntax, they only focus on those aspects of syntax that are relevant to all semantics, such as composition and relations. Hey, who put me on that hobby-horse again?-) Claus