[Agda] Performance issues with --sharing

[Peter Selinger]

Sergei Meshveliani wrote:
> 
> On Thu, 2015-09-24 at 13:57 -0300, Peter Selinger wrote:
> > Dear all,
> > 
> > last month, in the thread "type check performance", Wolfram Kahl
> > suggested that I try the experimental --sharing feature in the
> > development branch.
> > 
> > The issue at hand was Agda's extremely inefficient evaluation
> > strategy, which is normal order and not lazy: it recomputes the same
> > value a potentially unlimited number of times.
> > 
> > [..] 
> >
> > I've created a self-contained example below. In order to compare
> > different Agda versions without depending on different versions of the
> > standard library, I have made the example to be completely independent
> > of the library.
> > 
> > Here are the results:
> > [..]
> > 
> > ----------------------------------------------------------------------
> > -- Begin Agda Code
> > ----------------------------------------------------------------------
> > 
> > -- Natural numbers
> > data Nat : Set where
> >   zero : Nat
> >   succ : Nat -> Nat
> > 
> > _+_ : Nat -> Nat -> Nat
> > zero + m = m
> > succ n + m = succ (n + m)
> > 
> > _-_ : Nat -> Nat -> Nat
> > zero - m = zero
> > n - zero = n
> > succ n - succ m = n - m
> > 
> > {-# BUILTIN NATURAL Nat #-}
> > {-# BUILTIN NATPLUS _+_ #-}
> > {-# BUILTIN NATMINUS _-_ #-}
> > 
> > -- Lists
> > data List (X : Set) : Set where
> >   nil : List X
> >   _::_ : X -> List X -> List X
> > 
> > replicate : ∀ {X} -> Nat -> X -> List X
> > replicate zero x = nil
> > replicate (succ n) x = x :: replicate n x
> > 
> > foldl : ∀ {A B} -> (A -> B -> A) -> A -> List B -> A
> > foldl f a nil = a
> > foldl f a (x :: xs) = foldl f (f a x) xs
> > 
> > -- Test
> > test : Nat -> Nat
> > test n = foldl (\a x -> succ ((a + x) - a)) 0 (replicate n 1)
> 
> 
> But how precisely the test is run?
> 
> For  n : Nat   
>      n = 12345678,
> 
> how much  (show n)  will take?
> Will the test convert  n  into the unary system and back?
> 
> (I do not know what is the effect of the above BUILTIN).
> 
> Instead of measuring the true  foldl,  could happen so that it measures
> the conversion costs of  Decimal <--> Unary  for  n  ? 
> 
> In the performance tests, I do not use BUILTIN, and use the binary
> arithmetic of Bin for the counter like  n  above.
> And I do not know, may be BUILTIN is simpler.


You will notice that the function that is being folded here,

(\a x -> succ ((a + x) - a))

is the second projection, i.e., a+x-a = x. So no large numbers ever
appear in this example; in fact, test n always returns 2.

To eliminate any doubt, just fold a boolean function:

data Bool : Set where
  true : Bool
    false : Bool

and : Bool -> Bool -> Bool
and true true = true
and _ _ = false

test2 : Nat -> Bool
test2 n = foldl (\a x -> and a (and a x)) true (replicate n true)

You will get exactly the same timing behavior (up to a constant
factor). I used BUILTIN primarily so I could write "test 18" instead
of "test (succ (succ (succ ...)))".

-- Peter