Why is that surprising? Haskell (as typically implemented; I'm slightly less confident about the details of the report) is lazy and pure by default, but that can be violated locally or more globally if you ask for it loud enough. So the language, as a whole, is not pure in the sense required for those theoretical questions to really apply.
What's needed here is ST. It permits local O(1) mutable vectors, while enforcing referential transparency when viewed from outside the runST.
ST is clearly not pure in the sense that theory requires to make "can we implement this algorithm with the same complexity" an interesting question. It seems wrong to consider it part of "the unsafe stuff", however - it doesn't have the same kind of unenforced caveats as unsafePerformIO and unsafeCoerce and similar.
Separately, "the unsafe stuff" is still part of the language as used, and while it's certainly better practice to avoid it when you can, it's silly to exclude it when asking whether it's possible to do something - they are sometimes appropriate.
Are you suggesting to define purity by the inability to define O(1)
mutable vector-like operations? That would be an interesting
proposal. But I wonder if it captures the conceptual content of the
informal concept of purity. Maybe a time complexity restriction should
be provable from purity?
"Are you suggesting to define purity by the inability to define O(1) mutable vector-like operations?"
"Define", no, as that certainly doesn't encompass the full scope of purity. I think mutability runs counter to most people's notion of purity, particularly when they say "Haskell isn't pure". Of course, we can fake mutability at a cost of O(log n), which is what makes the O(1) notable.
If you do think ST is "pure" then it's not an open question whether you can implement update-heavy array based algorithms in a pure language without algorithmic penalty - you clearly can.
Looking over the PLDI paper, I get the impression that the key difference between ST and the usual State monad is (1) the ability to name state, and (2) the use of parametric polymorphism to prevent scope-extrusion of state.
Of course ML has scope-extrusion of state, and one of my questions/examples upthread was about the ability of Haskell's monadically controlled effects to implement such scope extrusion without 'tricks'.
All in all I feel that the notion of purity is controversial, insufficiently theorised, and that the community would benefit from a clearer understanding of the trade-offs involved in the different notions of purity
If you object to notions of purity, why did you bring it up in the first place?
The first place it was used in this thread was your citing a paper entitled "Pure versus Impure Lisp".
In that paper, it was used to mean prohibiting "set-car!" and "set-cdr!". ST provides the equivalent of "set-car!" and "set-cdr!", so in the presence of ST the result you were citing in the paper doesn't even apply to a strict version of Haskell.
Please just stop making claims about Haskell until you have some sense of what you're talking about (or at least add a caveat warning others that you might not).
I don't object to purity, I want to know what it means. I didn't
realise that this is not a well-understood concept. (I mentioned
Sabry's definition of purity which I found surprising.)
The original discussion started because somebody complained that
Haskell programmers often appear unable to transliterate imperative
algorithms. I assume that this complaint was about transliteration
into pure Haskell (whatever that may mean exactly). Clearly
transliteration into full Haskell with actual mutability is
possible. I mentioned Pippenger's article to bring to the debate the
idea that the question whether such transliteration is always possible
is an open research problem.
"I assume that this complaint was about transliteration into pure Haskell (whatever that may mean exactly). Clearly transliteration into full Haskell with actual mutability is possible."
Ah, if you'd made that assumption explicit I would not have objected.
I don't see anywhere the querent suggesting that his translation demand invoked any notion of purity. If it did, then certainly he was asking a harder question than he presented, although even in that case a naive translation could have trivially produced an O(m n lg(n)) solution and O(m^n) should quite reasonably be considered failure.
I think the issue in the main is that relatively novice Haskell developers are likely to be the most eager to try and translate arbitrary code on demand in #haskell.
I don't think anyone knows a solid definition of purity. It certainly refers to at least two things (1. independence of order of evaluation, 2. no side effects) and informally speaking we often "know" what it is in a particular context, but writing down a formal definition is very hard. I'm certainly not satisfied with Sabry's definition. I don't like the idea that you have to have both a strict and a non-strict evaluation for your language.
If nobody knows what purity means, then what are all those tortured discussions on whether Haskell is pure or not about? ...
Anyway, I'm a bit unhappy about the requirement of full evaluation order
independence, because even the purest of pure languages (e.g. the
untyped lambda-calculus or PCF) don't have full evaluation order
independence. I guess you mean evaluation order independence modulo
termination?
My intuition is that pure means that all language expressions M can always be
evaluated locally by just looking at the meaning of all subterms of
M. In terms of Hoare logic (for total correctness) that would mean
that the pre-condition speaks only about the termination of the free
variables of M (sorry for being so terse, I think this can be made
formal).
Usually this is the way words work - we notice a regularity in the environment we wish to refer to; we build a sense of it, usually motivated by some examples and counterexamples; we give it a handle; and we then try to find a definition that matches our intuition. There certainly seem to be regularities that "pure" is intended to point at in this context, but it definitely seems that we've not settled on a definition, and we may not even agree on examples.
The notions of purity I was pointing at was any that entailed the restrictions in the paper you'd linked. There are restrictions of Haskell that are pure in those senses, but Haskell taken as a whole is not, even without invoking anything unsafe.
"I get the impression that the key difference between ST and the usual State monad is (1) the ability to name state, and (2) the use of parametric polymorphism to prevent scope-extrusion of state."
The key difference between ST and the State monad is that the State monad is simply sugar for a function call of the form (s -> (a, s)), whereas ST allows for actual mutability. This is what changes the asymptotic complexity.
ST also does things to prevent references from leaking, and that's valuable and more useful in the context of ST, but that's irrelevant to our questions up-thread.
"Of course ML has scope-extrusion of state, and one of my questions/examples upthread was about the ability of Haskell's monadically controlled effects to implement such scope extrusion without 'tricks'."
That wasn't this thread.
I still think scope extrusion - without asking for it - would be a bug not a feature; particularly so in the case of a multithreaded, lazily evaluated language with inlining and CSE - where you really have little control over what gets executed when if you're ducking the features that guide what gets executed when.
This doesn't limit expressibility, because you do have the ability to ask for it, using those "tricks" that you keep excluding for no good reason. "How expressive is Haskell if it lacked those" is an interesting question, but not am immediately practical one, and you should be clear that that is the question you're asking.
Lazy is easy: it's call-by-name evaluation, i.e. (lambda x.M)N -> M[N/x] with caching, ie. each argument is evaluated at most once.
Purity is more subtle. I don't think there's an agreed upon definition. [1] essentially ties purity to the equivalency of
call-by-name, call-by-need and call-by-value evaluation orders (ignoring divergence and errors). I'm not totally sure that is the right definition.
[1] A. Sabry, What is a Purely Functional Language?
The most proximate question here, with respect to purity, is "can you do O(1) mutation (outside of the narrow case of replacing a thunk with its result)?" The theory is far more interesting when we say "no", but the answer in the case of Haskell is "yes".
