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  • Q: So the underlying type of spec in Examples.hs is [Trigger]. Is the idea that any Copilot program should include a trigger, since triggers are how you cause some change to the outside world?

    In other words, does your framework allow compiling something like:

    fib :: Stream Word64
    fib = [1, 1] ++ fib + drop 1 fib
    • A: Currently triggers are not only the only way to cause some change to the outside world, they are also the only way to observe the behaviour of a Copilot program.

      As all streams are anonymous we can no longer observe them. We could implement a mechanism for observing streams, e.g. for debugging purposes:

      spec :: Spec ()
      spec =
          trigger "f" booleans [ triggerArg fib, triggerArg (fib + fib) ]
          trigger "g" booleans [ triggerArg nats ]
          observe "x" (fib + fib + 4)

      In the generated C99-code the observe function binds the value of fib + fib + 4 to the C99 variable x.

  • Q: Are GADTs necessary in core/src/Copilot/Core/Stream.hs?

    • A: I assume that you mean copilot-language/src/Copilot/Core/Stream.hs? We don't need to use GADTs; for the sake of our users, however, we use GADTs as this makes the interface a lot cleaner.

      Using GADTs we can write

      fib :: Stream Word64
      fib = [1, 1] ++ fib + drop 1 fib
      counter :: (Num a, Typed a) => Stream Bool -> Stream a
      counter reset = y
          zy = [0] ++ y
          y  = if reset then 0 else zy + 1

      ...instead of

      fib :: Stream s => s Word64
      fib = [1, 1] ++ fib + drop 1 fib
      counter :: (Stream s, Num a, Typed a) => s Bool -> s a
      counter reset = y
          zy = [0] ++ y
          y  = if reset then 0 else zy + 1

      This is a minor detail, and if you prefer we can use type-classes in copilot-language instead of GADTs, in the same way as we do in copilot-core.

  • Q: Why are data types replicated between copilot-language/src/Copilot/Language/Stream.hs and copilot-core/src/Copilot/Core/Spec.hs?

    • A: The reason data types are replicated is that in copilot-language a Copilot program is simply a list of triggers, whereas in copilot-core a Copilot program consists of a list of triggers and a list of canonized streams. I haven't exlained to you yet what a canonized stream is, but I think it is to complicated to do here. Breifly, in copilot-language a stream is a possible recursive expression, whereas in copilot-core a stream consists of a buffer and a non-recursive expression.
  • Q: What are the "Observers" defined in copilot-core/src/Copilot/Core/Spec.hs?

  • Q: What model of sharing is used in Copilot?

    • A: Efficiency is important for an eDSL, but there's a tradeoff against complexity in the compiler design. There are two "levels" in the Copilot language: (1) Stream (Copilot/Language/Stream.hs in the copilot-language package), which define the basic datatype for bulding computational elements, (2) and Spec (Copilot/Language/Spec.hs in copilot-language) that contains a list of triggers and observers (for debugging).

      Second, there are (at least) two reasons to identify sharing in a language: (1) to identify "back-edges", which is necessary for doing things like compilation for deeply-embedded languages and (2) improving efficiency, so that common sub-expressions aren't re-evaluated. Let us call (1) "pointer sharing" and (2) "efficiency sharing". The same techniques can be used to address both kinds of concerns.

      In Copilot, we implement pointer sharing at the Stream level. We use the techniques described in Andy Gill's "Type-Safe Observable Sharing in Haskell", but we do not use his data-reify package itself

      For simplicity sake, we do not implement efficiency sharing at the Spec level of Copilot. For example, in

      spec :: Spec spec = do trigger "f" strA [ arg strB, arg strC ] trigger "g" strA [ arg strB ] observer "strA" strB

      the expressions strA and strB are not shared between any of the triggers or observers.

      As for the Stream level, we implement efficiency sharing explicitly using a local variable bindings, inside the Copilot language. For example,

      majority :: (Num a, Typed a) => [Stream a] -> Stream a majority [] = error "Error in majority: list must be nonempty." majority ls = majority' 0 ls 0 0

      majority' :: (Num a, Typed a)
        => Int -> [Stream a] -> Stream a -> Stream Word32 -> Stream a
      majority' _ []     candidate _   = candidate
      majority' k (x:xs) candidate cnt = 
        local (if cnt == 0 then x else candidate) $ \ candidate' ->
          local (if cnt == 0 || x == candidate then cnt+1 else cnt-1) $ \ cnt' ->
            majority' (k+1) xs candidate' cnt'

      local takes and expression and a one-place function and binds the value of the expression to the function's argument.

      Our hope is that the use of locals is not commonly needed. We use it in Copilot library functions. By building on these libraries, we hope the user can largely ignore efficiency issues. (As an aside, because we hope to rarely use locals, we don't build a monadic wrapper for streams, which is somewhat heavy-weight.)

      Currently, Copilot does not implement common-subexpression elimination (CSE). Implementing CSE for the core-language (package copilot-core) is a bit more involved as we don't use a graph-representation. One approach might be convert copilot-core specifications into a graph-representation, then do the CSE, and then convert the reduced graph back again into the core-representation (and use local variables for representing shared expressions).

      In practice, the only library function we have encountered so far that requires better efficiency sharing is the majority-vote function (shown above). One reason is that Copilot already provides some modularity at the stream level, by allowing users to compose streams, which can be evaluated modularly. For example,

      even :: (P.Integral a, Typed a) => Stream a -> Stream Bool
      even x = x mod 2 == 0

      odd :: (P.Integral a, Typed a) => Stream a -> Stream Bool
      odd = not . even

      Using local variables is only really necessary when generating Copilot expressions. If more substantial Copilot expressions are required in library functions, implementing CSE may be necessary.

      As a side note, sharing can be used to control both (1) the time required to evaluate an eDSL and (2) the size of the code generated (when compiling). While we anticipate Copilot having a variety of back-ends, our current main back-end is Atom Atom implements CSE itself, and it has a number of optimizations to reduce the complexity of expressions when values are statically known. In practice, we have not experienced overly-large C programs being generated; our issues have just been with compilation time (and more trivially, size of expressions when pretty-printing them).

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