[Agda] What style of proof irrelevance do you prefer: annotated or inferred?
Andreas Nuyts
andreas.nuyts at cs.kuleuven.be
Mon Oct 29 12:32:06 CET 2018
Dear everyone,
Another dimension (or is it again the compile/run-time dimension?) is
that of type-level vs. value-level annotations.
We can consider `IrrFun = .(x : A) -> B x` a subtype of `RelFun = (x :
A) -> B x`. In this view, RelFun may be inhabited by lambdas that are in
fact known to be irrelevant and whose argument can therefore be erased,
both at compile and at run-time.
My (draft of a) suggestion is the following: when the Agda user types a
lambda-expression `λ (x : A) -> b[x]`, it is inferred from `b[x]`
whether this is an irrelevant lambda. Of course, if the type given by
the user is `IrrFun`, then this only type-checks if the lambda is
inferred to be irrelevant. However, if the type given by the user is
`RelFun` but the lambda is irrelevant, then this fact is remembered in
the form of an internal annotation on the lambda, as it allows erasure
of the argument both at compile and at runtime. Without the annotation
on the lambda, erasure could only happen after beta-reducing the lambda,
which is probably less efficient.
This behaviour would be in accordance to what happens if you explicitly
coerce a function `f : IrrFun` to `RelFun` by lambda-expanding. It then
becomes `λ (x : A) -> f _`.
When it comes to inferring type-level annotations: this is a problem
quite similar to inferring universe levels. When a user does not specify
a universe level, do you want to infer the smallest one, or do you want
to complain? (I think Agda currently complains about an unresolved meta,
although C-c C-= will fill out the smallest option?) Similarly, when a
user does not specify a modality, do you want to infer the most
informative one (irrelevance, if possible), or do you want to complain?
(I don't have any answers here, just pointing out the similarity of the
problems.)
Best regards,
Andreas Nuyts
(not the Andreas mentioned in Jesper's emails)
On 29/10/2018 09:28, Jesper Cockx wrote:
> Dear Jon,
>
> Thanks for your answer. I see these two not so much as two different
> kinds of proof irrelevance, but as two separate dimensions:
> compile-time vs run-rime irrelevance and annotated vs inferred
> irrelevance. For example, Agda has annotated compile-time irrelevance
> (irrelevant functions and Prop), inferred run-time irrelevance (type
> erasure, forcing analysis, detection of unused arguments), Andreas
> recently added annotated run-time irrelevance (the @erased
> annotation), but it has very little inferred compile-time irrelevance
> (the only thing that comes to mind is the unit type, which is
> automatically proof-irrelevant because of eta).
>
> It seems to me you like a combination of annotated compile-time
> irrelevance and inferred run-time irrelevance, is that correct?
>
> What Wolfram seemed to suggest however is to infer more *compile-time*
> irrelevance. For example, we could infer that the empty type is
> proof-irrelevant and automatically discard any equation at the empty
> type during conversion (i.e. have eta for the empty type). Or we could
> detect that the identity type has a single constructor and (assuming
> --with-K) replace all proofs of identity by primTrustMe during
> elaboration.
>
> The thing is, such automatically inferred irrelevance wouldn't work
> very well in some cases, and it would certainly not help for the fancy
> applications of irrelevance you describe (enforcing parametricity or
> coherence). However it *would* give some benefits of irrelevance to
> people who use Agda but don't want to add extra irrelevance
> annotations, or indeed even to people who don't know about irrelevance
> at all. So it has a much greater potential impact than the annotated
> style of irrelevance.
>
> Of course, much depends on how often we could actually detect
> irrelevance automatically, which in turn would depend greatly on the
> concrete codebase. But perhaps doing the experiment could give us a
> better idea.
>
> -- Jesper
>
> On Mon, Oct 29, 2018 at 1:33 AM Jon Sterling <jon at jonmsterling.com
> <mailto:jon at jonmsterling.com>> wrote:
>
> Dear Jesper,
>
> I feel that these are two separate issues getting conflated -- on
> the one hand, a compiler may infer that something is irrelevant,
> and then erase it in order to achieve a more efficient execution.
> On the other hand, there is proof irrelevance which is used for
> *semantic* reasons: irrelevance is part of the specification of
> some function, and whether or not it executes in an irrelevant or
> erased way is really beside the point (of course, in such a case,
> the compiler *should* erase it because it is low hanging fruit).
>
> The latter (semantic) kind of proof irrelevance could be used, for
> instance, in order to achieve the following things:
>
> - to specify that some function is parametric in its indices: for
> instance, operations on vectors
>
> - relatedly, to force *coherence*: for instance, when defining
> some interpretation of some language, you can use proof
> irrelevence + some inversion lemmas to formalize the old pattern
> (from Streicher) of defining your function by recursion on the
> labels of the derivation rather than on the derivation itself. In
> old school math, this required defining a partial function and
> then establishing that it terminates, but in very fancy type
> theory we can use proof irrelevance to do this simultaneously and
> get a total function all at once.
>
> - and of course, to obtain more definitional equivalences where
> possible
>
> Let me unleash a kind of important point: as I mentioned, in all
> these cases, erasure should obviously be done, but there are cases
> where a good compiler would perform erasure even when it is not
> possible to type the term using the irrelevant types. Something
> might be erasable for global reasons. Anyway, it's not so
> important to me that Agda catch all of these cases, because I'm
> not using Agda for executing code (but others might be).
>
> What I'm saying is that inferring when something is
> computationally irrelevant must be treated as orthogonal to
> whether something can be typed with a proof irrelevance modality.
> Computational irrelevance is a property of code in the compilation
> target language (which could be anything), not a property of code
> in the Agda language.
>
> We should take compilation seriously, and not conflate it with
> typechecking and elaboration. Dependently typed languages have
> *two* computational semantics: the one which generates
> definitional equivalence of terms, and the one which executes.
>
> Best,
> Jon
>
>
>
>
> On Sun, Oct 28, 2018, at 2:01 PM, Jesper Cockx wrote:
> > Hi Agda people,
> >
> > In the comments on issue #3334
> >
> <https://github.com/agda/agda/issues/3334#issuecomment-433730080>,
> Wolfram
> > wrote:
> >
> > I consider it counterproductive to be able to declare
> irreleveance, and
> > > even more counterproductive to have to do it for efficiency
> reasons,
> > > because that encourages premature optimisations. The compiler
> (including
> > > the type checker) should detect it whenever possible, and
> optimise it away
> > > as soon as possible.
> >
> >
> > To which I replied:
> >
> > Often the fact that a certain term is irrelevant is obvious to
> *you* as the
> > > Agda programmer but there's no way for Agda to figure this out
> from the
> > > code. If it's possible that you will use the term at any point
> in the
> > > future, Agda cannot (and should not) erase it. So any
> automatic detection
> > > of irrelevance would probably be very disappointing. (The
> alternative where
> > > Agda would eagerly erase things without asking and then
> complaining later
> > > when you try to use them is arguably worse). Another advantage of
> > > annotating irrelevant things, is that Agda can warn you when
> you use it
> > > accidentally in a non-erased position, instead of removing the
> irrelevance
> > > silently and causing hard-to-explain performance regressions.
> > >
> >
> > But I realized this is based mostly on speculation and not hard
> facts.
> > Historically, Agda has focused on the annotated style of
> proof-irrelevance
> > (Andreas' irrelevant functions and irrelevant fields, and more
> recently my
> > implementation of Prop). Maybe inferred proof irrelevance would
> work better
> > than expected?
> >
> > So now my question to *you*, the Agda community is this: do you
> prefer
> > Agda's current style of annotated proof-irrelevance, or would
> you rather
> > have Agda infer things for you (and perhaps fail to do so in
> some cases)?
> >
> > I won't promise I will remove Prop and focus all my attention on
> inferring
> > irrelevance even if everyone votes for the latter option, but it
> would be
> > interesting to know going forward what you think.
> >
> > -- Jesper
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