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
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