data Stream (s a : Data) where MkStream : (s -> Step s a) -> s -> Stream s a data Step (s a : Data) where Yield : a -> s -> Step s a Skip : s -> Step s a Done : Step s a -- | Apply a function to every element of a stream. smap (f : a -> b) (ss : Stream s a) : Stream s b = case ss of MkStream stepA sA0 -> let stepB q = case stepA q of Yield x sA1 -> Yield (f x) sA1 Skip sA2 -> Skip sA2 Done -> Done in MkStream stepB sA0 -- | Take the given number of elements from the front of a stream. stake (n : Nat) (ss : Stream s a) : Stream (Tup2 s Nat) a = case ss of MkStream fA sA0 -> let stepB q = case q of T2 sA ix -> if ix >= n then Done else case fA sA of Yield x sA2 -> Yield x (T2 sA2 (ix + 1)) Skip sA3 -> Skip (T2 sA3 ix) Done -> Done in MkStream stepB (T2 sA0 0)As we can see, work on larger demos is starting to be hampered by the lack of pattern matching sugar in the source language, so I'm going to add that next. After adding pattern matching, I'm going to spend a couple of weeks bug fixing, and should get the DDC 0.5 release out in early February 2015.

# Disciple Development

More info at the The Disciplined Disciple Compiler (DDC) Development Wiki.

Easy tickets to get started with: on the trac.

## Tuesday, December 30, 2014

### Higher order functions and interface files.

Work on DDC progresses, though recently it's been straightforward compiler engineering rather than anything novel, so I haven't been blogging.
DDC now drops interface files when compiling modules, multi-module compilation works, and the front-end supports unicode string literals. I've also implemented a lambda lifter and support for higher order functions. This part is well baked enough to implement simple stream functions, as used in the Haskell Data.Vector library, though the compiler doesn't do the associated fusion yet.
For example:

## Sunday, March 2, 2014

### Civilized error messages

I spent the weekend fixing bugs in preparation for the upcoming 0.4 release. I've finished all the necessary compiler hacking, but still need to update the cabal files, tutorial, and wiki before releasing it proper.
I've made sure to fix all the bugs that would result in a poor user experience for people that just want to spend an afternoon kicking the tyres. Even though DDC still has lots of missing features (garbage collection, compilation for higher order functions etc), the stuff that is implemented is reasonably well baked. If one tries to do something unsupported it should at least give a civilized error message.

For example:

Trying to use an anonymous type function (higher order functions are not supported yet)

Trying to use higher kinded type arguments (which need an 'Any' region to implement safely, in general)

Trying to partially apply a primitive operator:

In contrast, the old ddc-alpha compiler (which I wrote for my PhD work), would signal its displeasure with partially applied primops by producing a program that segfaulted at runtime. We turn our backs on the bad old days.

For example:

Trying to use an anonymous type function (higher order functions are not supported yet)

module Test with letrec id [a : Data] (x : a) : a = x foo (_ : Unit) : Unit = do id (/\a. \(x : a). x) () Fragment violation when converting Tetra module to Salt module. Cannot convert expression. Cannot convert type abstraction in this context. The program must be lambda-lifted before conversion. with: /\(a : Data). \(x : a). x

Trying to use higher kinded type arguments (which need an 'Any' region to implement safely, in general)

module Test data List (a : Data) where Nil : a -> List a Cons : a -> List a -> List a with letrec foo [a : Data ~> Data] [b : Data] (x : b) : b = x bar (_ : Unit) : Nat# = foo [List] [Nat#] 5# Cannot convert expression. Unsupported type argument to function or constructor. In particular, we don't yet handle higher kinded type arguments. See [Note: Salt conversion for higher kinded type arguments] in the implementation of the Tetra to Salt conversion. with: [List]

Trying to partially apply a primitive operator:

module Test with letrec thing (_ : Unit) : Nat# -> Nat# = add# [Nat#] 5# Fragment violation when converting Tetra module to Salt module. Cannot convert expression. Partial application of primitive operators is not supported. with: add# [Nat#] 5#

In contrast, the old ddc-alpha compiler (which I wrote for my PhD work), would signal its displeasure with partially applied primops by producing a program that segfaulted at runtime. We turn our backs on the bad old days.

### Nested Data Types

Pleasingly, the new type inferencer does seem to work with some non-standard programs -- even though these don't compile all the way to object code yet. Here is a Core Tetra program using nested data types:module Test data Tuple2 (a b : Data) where T2 : a -> b -> Tuple2 a b data Nest (a : Data) where NilN : Nest a ConsN : a -> Nest (Tuple2 a a) -> Nest a with letrec thing (_ : Unit) = ConsN 7# (ConsN (T2 1# 2#) (ConsN (T2 (T2 6# 7#) (T2 7# 4#)) NilN))This example is based on one from the Bird and Meertens 1998 paper. Note that the second argument of the ConsN constructor takes a Nest where the element type is more complex than the original parameter. The type inference algorithm in the alpha compiler would have diverged on this program.

### Higher Rank Types

I've also tried out some simple examples with higher ranked types, here is one:module Test with letrec thing1 (blerk : ([a : Data]. a -> a) -> Nat#) : Nat# = blerk (/\a. \(x : a). x) thing2 (u : Unit) = thing1 (\(f : [a : Data]. a -> a). f 5#)thing1 has a rank-3 type because it is a function, that takes a function, that takes a function which is polymorphic. There is a quantifier that appears at depth 3 contravariantly. Writing the type of thing1 in full makes this easier to see:

thing1 :: ((([a : Data]. a -> a) -> Nat#) -> Nat#Again, I can't compile this example to object code yet because code generation for higher order functions isn't finished. However, type inference is a separate concern, and I don't know of any remaining problems in this part.

## Tuesday, February 11, 2014

### Bidirectional type inference for DDC

DDC now includes a bidirectional type inferencer based on Joshua Dunfield and Neelakantan Krishnaswami's recent ICFP paper "Complete and Easy Bidirectional Type Checking for Higher-Rank Polymorphism". I've extended the base algorithm to infer the kinds of type parameters, as well as to automatically insert type abstractions and applications into the result.

The DDC command-line interface allows one to apply the type inferencer to such a source file, producing a core file with explicit type annotations, like so:

### Type inference for Disciple Source Tetra

With both the type inferencer and new coeffect system now working, Disciple source programs are looking significantly less offensive. For example, some simple list functions:data List (a : Data) where Nil : List a Cons : a -> List a -> List a singleton (x : a) : List a = Cons x Nil append (xx : List a) (yy : List a) : List a = case xx of Nil -> yy Cons x xs -> Cons x (append xs yy) mapS [a b : Data] [e : Effect] (f : a -> S e b) (xx : List a) : S e (List b) = box case xx of Nil -> Nil Cons x xs -> Cons (run f x) (run mapS f xs)etc. etc. The above 'mapS' function is the version using the coeffect system I described in my last post. Using effects currently requires the user to add explicit 'box' and 'run' casts, though these will be automatically inserted in the next DDC release.

The DDC command-line interface allows one to apply the type inferencer to such a source file, producing a core file with explicit type annotations, like so:

$ bin/ddc -infer -load demo/Tetra/Lists/Lists.dst ... module Lists data List (a : Data) where { Nil : List a; Cons : a -> List a -> List a; } with letrec { singleton : [a : Data].a -> List a = /\(a : Data). \(x : a). Cons [a] x (Nil [a]); append : [a : Data].List a -> List a -> List a = /\(a : Data). \(xx yy : List a). case xx of { Nil -> yy; Cons (x : a) (xs : List a) -> Cons [a] x (append [a] xs yy) }; mapS : [a b : Data].[e : Effect].(a -> S e b) -> List a -> S e (List b) = /\(a b : Data)./\(e : Effect). \(f : a -> S e b).\(xx : List a). box case xx of { Nil -> Nil [b]; Cons (x : a) (xs : List a) -> Cons [b] (run f x) (run mapS [a] [b] [e] f xs) } } ...Such a file can then be converted to C or LLVM for compilation to object code. In the current implementation higher-order programs will type-check, but cannot be compiled all the way to object code. I need to finish the runtime support for that.

### Type inference for Disciple Core Salt

In practical terms, work on the runtime system is already being helped by the new type inferencer. The DDC runtime is written in a first-order imperative fragment named 'Disciple Core Salt', which compiled with DDC itself. Here is the code that allocates a boxed heap object on a 64-bit platform:allocBoxed [r : Region] (tag : Tag#) (arity : Nat#) : Ptr# r Obj = do -- Multiple arity by 8 bytes-per-pointer to get size of payload. bytesPayload = shl# arity (size2# [Addr#]) bytesObj = add# (size# [Word32#]) (add# (size# [Word32#]) bytesPayload) -- Check there is enough space on the heap. case check# bytesObj of True# -> allocBoxed_ok tag arity bytesObj False# -> fail# allocBoxed_ok [r : Region] (tag : Tag#) (arity : Nat#) (bytesObj : Nat#) : Ptr# r Obj = do addr = alloc# bytesObj tag32 = promote# [Word32#] [Tag#] tag format = 0b00100001w32# header = bor# (shl# tag32 8w32#) format write# addr 0# header -- Truncate arity to 32-bits. arity32 = truncate# [Word32#] [Nat#] arity write# addr 4# arity32 makePtr# addrSome type annotations are still required to specify data formats and the like, but all the day-to-day annotations can be inferred.

$ bin/ddc -infer -load packages/ddc-code/salt/runtime64/Object.dcs [slightly reformatted] ... allocBoxed : [r : Region].Tag# -> Nat# -> Ptr# r Obj = /\(r : Region). \(tag : Tag#).\(arity : Nat#). let bytesPayload : Nat# = shl# [Nat#] arity (size2# [Addr#]) in let bytesObj : Nat# = add# [Nat#] (size# [Word32#]) (add# [Nat#] (size# [Word32#]) bytesPayload) in case check# bytesObj of { True# -> allocBoxed_ok [r] tag arity bytesObj; False# -> fail# [Ptr# r Obj] }; allocBoxed_ok : [r : Region].Tag# -> Nat# -> Nat# -> Ptr# r Obj = /\(r : Region). \(tag : Tag#).\(arity bytesObj : Nat#). let addr : Addr# = alloc# bytesObj in let tag32 : Word32# = promote# [Word32#] [Tag#] tag in let format : Word32# = 33w32# in let header : Word32# = bor# [Word32#] (shl# [Word32#] tag32 8w32#) format in let _ : Void# = write# [Word32#] addr 0# header in let arity32 : Word32# = truncate# [Word32#] [Nat#] arity in let _ : Void# = write# [Word32#] addr 4# arity32 in makePtr# [r] [Obj] addr; ...

### Upcoming 0.4 Release

There are still some bugs to fix before I can enable type inference by default, but when that is done I will release DDC 0.4, which I'm hoping to get done by the end of the month.## Sunday, December 8, 2013

### Capabilities and Coeffects

Over the past couple of months I've had "a moment of clarity" regarding some problems with Disciple and DDC that have been bothering me for the last six years or so. Itemized, they include:

The solution is to say that (Write r1) is NOT an effect type -- it's a coeffect type! Whereas an effect type is a description of the

When we take the view of coeffects instead of effects (or as well as effects), then the problem of type safe freezing becomes straightforward. If there is no (Write r1) capability in the calling context, then the type system prevents us from executing computations which require that capability. Here is a complete example:

- In the source language, the fact that every function type carries an effect and closure annotation upsets Haskell programmers. They want a function of type (a -> b) to be just a pure function. In Disciple, the function type constructor actually has two other parameters. We write (a -(e|c)> b) where (a) and (b) are the argument and return types as usual, (e) is the effect type of the function and (c) is the closure type. The effect type describes the side effects that the function might have, and the closure type describes what data might be shared via the function closure. Although DDC provides enough type inference that you don't need to write the effect and closure type if you don't want to, this still isn't cool with some people (for understandable reasons).

- The fact that the function type has an added effect and closure parameter means that it has a different kind relative to the Haskell version. We've got (->) :: Data -> Data -> Effect -> Closure -> Data, where 'Data' is the (*) kind from Haskell. This means that Haskell programs which are sensitive to the kind of (->) can't be written directly in Disciple.

- Even though Disciple has mutability polymorphism, there isn't a safe way to destructively initialize a mutable structure and then freeze it into an immutable one. In Haskell, the client programmer would use a function like (unsafeFreezeArray :: MutableArray a -> IO (Array a)) at their own peril, but you'd think in a language with mutability polymorphism we'd be able to do better than this.

### Run and Box

As you might have hoped, all these problems are now solved in the head version of DDC (henceforth known as "new DDC"), and you can expect a new release sometime in January when it's properly baked. I've got the typing rules worked out and partially implemented, but the modified core language doesn't yet compile all the way to object code. Here is an example in the core language:/\(r : Region). \(ref : Ref# r Nat#). box do { x = run readRef# [r] [Nat#] ref ; run writeRef# [r] [Nat#] ref (add# [Nat#] x x) }which has type

:: [r : Region].Ref# r Nat# -> S (Read r + Write r) UnitThe [r : Region] in the type signature means "forall", while [..] in the expression itself indicates a type argument. Things ending in # are primitives. The readRef# and writeRef# functions have the following types:

readRef# :: [r : Region]. [a : Data]. Ref# r a -> S (Read r) a writeRef# :: [r : Region]. [a : Data]. Ref# r a -> a -> S (Write r) UnitA type like (S e a) describes a suspended computation with effect (e) and result type (a). The S is a monadic type constructor, though in new DDC it is baked into the type system and not defined directly in the source language. In the above program the 'run' keyword takes a computation type and "runs it" (yeah!), which will unleash the associated effects. The 'box' keyword takes an expression that has some effects and then packages it up into a suspended computation. Critically, the type checker only lets you abstract over pure expressions. This means that if your expression has a side effect then you must 'box' it when using it as a function body. This restriction ensures that all side effect annotations are attached to 'S' computation types, and the functions themselves are pure. The idea of using 'run' and 'box' comes from Andrzej Filinski's "monadic reification and reflection" operators, as well as the paper "A judgemental reconstruction of modal logic" by Frank Pfenning, both of which I wish I had known about a long time ago. Note that the mechanism that

*combines*the two atomic effect types (Read r) and (Write r) into the compound effect (Read r + Write r) is part of the ambient type system. You don't need to hack up effect composition in the source language by using type classes or some such. In summary, new DDC gives you composable effect indexed state monads, with none of the pain of existing Haskell encodings.### Type safe freezing

The following example demonstrates type safe freezing./\(r1 : Region). \(x : Unit). box extend r1 using r2 with {Alloc r2; Write r2} in do { x = run allocRef# [r2] [Nat#] 0#; run writeRef# [r2] [Nat#] x 5#; x; }This code creates a new region 'r2' which extends the existing region 'r1', and says that the new one supports allocation and writing. It then allocates a new reference, and destructively updates it before returning it. The overall type of the above expression is:

:: [r1 : Region].Unit -> S (Alloc r1) (Ref# r1 Nat#)Note that the returned reference is in region 'r1' and NOT region 'r2'. When leaving the body of 'extend', all objects in the inner region (r2) are merged into the outer region (r1), so the returned reference here is in (r1). Also, as far as the caller is concerned, the visible effect of the body is that a new object is allocated into region r1. The fact that the function internally allocates and writes to r2 is not visible. Importantly, we allow objects in the inner region (r2) to be destructively updated, independent of whether it is possible to destructively update objects in the outer region (r1). Once the body of the 'extend' has finished, all writes to the inner region are

*done*, so it's fine to merge freshly initialized objects into the outer region, and treat them as constant from then on.### Coeffects

Of the previous example one might ask: what would happen if we also returned a function that can destructively update objects in region 'r2', even after the 'extend' has finished? Here we go:/\(r1 : Region). \(x : Unit). box extend r1 using r2 with {Alloc r2; Write r2} in do { x = run allocRef# [r2] [Nat#] 0#; f = \(_ : Unit). writeRef# [r2] [Nat#] x 5#; T2# [Ref# r2 Nat#] [Unit -> S (Write r2) Unit] x f; }this has type:

:: [r1 : Region] . Unit -> S (Alloc r1) (Tuple2# (Ref# r1 Nat#) (Unit -> S (Write r1) Unit))Here, 'T2#' is the data constructor for 'Tuple2#'. The result tuple contains a reference in region r1, as well as a function that produces a computation that, when run, will destructively update that reference. The worry is that if the calling context wants to treat the returned reference as constant, then we can't allow the computation that updates it to ever be executed.

The solution is to say that (Write r1) is NOT an effect type -- it's a coeffect type! Whereas an effect type is a description of the

*action*a computation may have on its calling context when executed, a coeffect type is a description of what*capabilities*the context must support to be able to execute that computation. It's a subtle, but important difference. A boxed computation with an*effect*can be executed at any time, and the computation does whatever it says on the box. In contrast, a boxed computation with a*coeffect*can only be executed when the bearer has the capabilities listed on the box. To put this another way, the*effect type*(Write r) says that the computation might write to objects in region 'r' when executed, but the*coeffect type*(Write r) says that the context must support writing to region 'r' for the computation to be executed. There is also a duality: "the context of a computation must support writing to region 'r', because when we execute that computation it might write to region 'r'". Most importantly, though, is to CHECK that the context supports some (co)effect before we run a computation with that (co)effect.When we take the view of coeffects instead of effects (or as well as effects), then the problem of type safe freezing becomes straightforward. If there is no (Write r1) capability in the calling context, then the type system prevents us from executing computations which require that capability. Here is a complete example:

let thingFromBefore = /\(r1 : Region). \(x : Unit). box extend r1 using r2 with {Alloc r2; Write r2} in do { x = run allocRef# [r2] [Nat#] 0#; f = \(_ : Unit). writeRef# [r2] [Nat#] x 5#; T2# [Ref# r2 Nat#] [Unit -> S (Write r2) Unit] x f; } in private r with {Alloc r; Read r} in case run thingFromBefore [r] () of { T2# r f -> run f () }In the above example, 'private' introduces a new region that is not an extension of an existing one. In the case-expression we unpack the tuple and get a computation that wants to write to an object in region 'r'. Because we have not given 'r' a (Write r) capability, then the type system stops us from running the computation that needs it:

Effect of computation not supported by context. Effect: Write r with: run f ()Effect typing is old news, coeffect typing is where it's at.

## Friday, May 3, 2013

### First working program using the Repa plugin

### Haskell Source

repa_process :: R.Stream k Int -> Int repa_process s = R.fold (+) 0 s + R.fold (*) 1 s

### Actual well formed GHC Core code

.. or at least the parts that get printed with -dsuppress-all.repa_process repa_process = \ @ k_aq6 arg_s2Rf -> let { (# x1_s2R6, x1_acc_s2R5 #) ~ _ <- newByteArray# 8 realWorld# } in let { __DEFAULT ~ x2_s2R8 <- writeIntArray# x1_acc_s2R5 0 0 x1_s2R6 } in let { (# x3_s2Rd, x3_acc_s2Rc #) ~ _ <- newByteArray# 8 x2_s2R8 } in let { __DEFAULT ~ x4_s2RS <- writeIntArray# x3_acc_s2Rc 0 1 x3_s2Rd } in let { Stream ds1_s2Rs ds2_s2Rj ~ _ <- arg_s2Rf } in let { Vector rb_s2Rz _ rb2_s2Ry ~ _ <- ds2_s2Rj `cast` ... } in letrec { go_s2RP go_s2RP = \ ix_s2Rr world_s2Ru -> case >=# ix_s2Rr ds1_s2Rs of _ { False -> let { (# x7_s2RF, x0_s2RC #) ~ _ <- readIntArray# x1_acc_s2R5 0 world_s2Ru } in let { __DEFAULT ~ sat_s2SV <- +# rb_s2Rz ix_s2Rr } in let { __DEFAULT ~ wild3_s2RD <- indexIntArray# rb2_s2Ry sat_s2SV } in let { __DEFAULT ~ sat_s2SU <- +# x0_s2RC wild3_s2RD } in let { __DEFAULT ~ x8_s2RH <- writeIntArray# x1_acc_s2R5 0 sat_s2SU x7_s2RF } in let { (# x9_s2RN, x1_s2RL #) ~ _ <- readIntArray# x3_acc_s2Rc 0 x8_s2RH } in let { __DEFAULT ~ sat_s2ST <- *# x1_s2RL wild3_s2RD } in let { __DEFAULT ~ x10_s2RR <- writeIntArray# x3_acc_s2Rc 0 sat_s2ST x9_s2RN } in let { __DEFAULT ~ sat_s2SS <- +# ix_s2Rr 1 } in go_s2RP sat_s2SS x10_s2RR; True -> world_s2Ru }; } in let { __DEFAULT ~ x11_s2RU <- go_s2RP 0 x4_s2RS } in let { (# x12_s2RY, x1_s2S2 #) ~ _ <- readIntArray# x1_acc_s2R5 0 x11_s2RU } in let { (# _, x2_s2S3 #) ~ _ <- readIntArray# x3_acc_s2Rc 0 x12_s2RY } in let { __DEFAULT ~ sat_s2SW <- +# x1_s2S2 x2_s2S3 } in I# sat_s2SWIm using ByteArrays to fake imperative variables. The next job is to convert the arrays x1_acc_s2R5 and x3_acc_s2Rc into proper accumulation variables, and attach them to the go_s2RP loop.

### x86 assembly code

This is just for the inner loop.LBB11_2: movq (%rcx), %rdi addq %rdi, 16(%rdx) imulq 16(%rsi), %rdi movq %rdi, 16(%rsi) addq $8, %rcx incq %r14 cmpq %r14, %rax jg LBB11_2The extra memory operations are there because the LLVM optimiser doesn't know that the two ByteArrays I'm using to fake imperative variables don't alias. This should turn into optimal code once it uses pure accumulators.

## Wednesday, May 1, 2013

### Array fusion using the lowering transform

I'm getting back into blogging. This is current work in progress.

Repa 4 will include a GHC plugin that performs array fusion using a version of Richard Waters's series expressions system, extended to support the segmented operators we need for Data Parallel Haskell.

The plugin converts GHC core code to Disciple Core, performs the fusion transform, and then converts back to GHC core. We're using Disciple Core for the main transform because it has a simple (and working) external core format, and the core language AST along with most of the core-to-core transforms are parametric in the type used for names. This second feature is important because we use a version of Disciple Core where the main array combinators (map, fold, filter etc) are primitive operators rather than regular function bindings.

The fusion transform is almost (almost) working for some simple examples. Here is the compilation sequence:

**Haskell Source**

process :: Stream k Int -> Int

process s

= fold (+) 0 s + fold (*) 1 s

In this example we're performing two separate reductions over the same stream. None of the existing short-cut fusion approaches can handle this properly. Stream fusion in Data.Vector, Repa-style delayed array fusion, and build/foldr fusion will all read the stream elements from memory twice (if they are already manifest) or compute them twice (if they are delayed). We want to compute both reductions in a single pass.

**Raw GHC core converted to DDC core**

repa_process_r2 : [k_aq0 : Data].Stream_r0 k_aq0 Int_3J -> Int_3J

= \(k_c : *_34d).\(s_aqe : Stream_r0 k_c Int_3J).

$fNumInt_$c+_rjF

(fold_r34 [k_c] [Int_3J] [Int_3J] $fNumInt_$c+_rjF

(I#_6d 0i#) s_aqe)

(fold_r34 [k_c] [Int_3J] [Int_3J] $fNumInt_$c*_rjE

(I#_6d 1i#) s_aqe)

**Detect array combinators from GHC core, and convert to DDC primops**

repa_process_r2 : [k_aq0 : Rate].Stream# k_aq0 Int# -> Int#

= /\(k_c : Rate).

\(s_aqe : Stream# k_c Int#).

add# [Int#]

(fold# [k_c] [Int#] [Int#] (add# [Int#]) 0i# s_aqe)

(fold# [k_c] [Int#] [Int#] (mul# [Int#]) 1i# s_aqe)

**Normalize and shift array combinators to top-level**

All array combinators are used in their own binding.

repa_process_r2 : [k_aq0 : Rate].Stream# k_aq0 Int# -> Int#

= /\(k_c : Rate).

\(s_aqe : Stream# k_c Int#).

let x0 = add# [Int#] in

let x1 = fold# [k_c] [Int#] [Int#] x0 0i# s_aqe in

let x2 = mul# [Int#] in

let x3 = fold# [k_c] [Int#] [Int#] x2 1i# s_aqe in

add# [Int#] x1 x3

**Inline and eta-expand worker functions**

This puts the program in the correct form for the next phase.

repa_process_r2 : [k_aq0 : Rate].Stream# k_aq0 Int# -> Int#

= /\(k_c : Rate).

\(s_aqe : Stream# k_c Int#).

let x1

= fold# [k_c] [Int#] [Int#]

(\(x0 x1 : Int#). add# [Int#] x0 x1) 0i# s_aqe in

let x3

= fold# [k_c] [Int#] [Int#]

(\(x2 x3 : Int#). mul# [Int#] x2 x3) 1i# s_aqe in

add# [Int#] x1 x3

**Do the lowering transform**

This is the main pass that performs array fusion. Note that we've introduced a single loop# that computes both of the fold# results.

repa_process_r2 : [k_c : Rate].Stream# k_c Int# -> Int#

= /\(k_c : Rate).

\(s_aqe : Stream# k_c Int#).

let x1_acc : Ref# Int# = new# [Int#] 0i# in

let x3_acc : Ref# Int# = new# [Int#] 1i# in

let _ : Unit

= loop# (lengthOfRate# [k_c])

(\(x0 : Nat#).

let x1 : Int# = next# [Int#] [k_c] s_aqe x0 in

let x0 : Int# = read# [Int#] x1_acc in

let _ : Void#

= write# [Int#] x1_acc (add# [Int#] x0 x1) in

let x2 : Int# = read# [Int#] x3_acc in

let _ : Void#

= write# [Int#] x3_acc (mul# [Int#] x2 x1) in

()) in

let x1 : Int# = read# [Int#] x1_acc in

let x3 : Int# = read# [Int#] x3_acc in

add# [Int#] x1 x3

**Assign imperative variable storage to arrays**

We need to convert the code back to GHC core, but we don't want to use IORefs because they can't hold unboxed values (of types like Int#). Instead, we use some new arrays to hold these values instead.

repa_process_r2 : [k_c : Rate].Stream# k_c Int# -> Int#

= /\(k_c : Rate).

\(s_aqe : Stream# k_c Int#).

let x1_acc : Array# Int# = newArray# [Int#] 8# in

let _ : Void# = writeArray# [Int#] x1_acc 0# 0i# in

let x3_acc : Array# Int# = newArray# [Int#] 8# in

let _ : Void# = writeArray# [Int#] x3_acc 0# 1i# in

let _ : Unit

= loop# (lengthOfRate# [k_c])

(\(x0 : Nat#).

let x1 : Int# = next# [Int#] [k_c] s_aqe x0 in

let x0 : Int# = readArray# [Int#] x1_acc 0# in

let _ : Void#

= writeArray# [Int#] x1_acc 0#

(add# [Int#] x0 x1) in

let x2 : Int# = readArray# [Int#] x3_acc 0# in

let _ : Void#

= writeArray# [Int#] x3_acc 0#

(mul# [Int#] x2 x1) in

()) in

let x1 : Int# = readArray# [Int#] x1_acc 0# in

let x3 : Int# = readArray# [Int#] x3_acc 0# in

add# [Int#] x1 x3

**Thread state token through effectful primops**

The lowered code is naturally imperative, and GHC uses state threading to represent this.

repa_process_r2 : [k_c : Rate].Stream# k_c Int# -> World# -> Tuple2# World# Int#

= /\(k_c : Rate).

\(s_aqe : Stream# k_c Int#).\(x0 : World#).

caselet T2# (x1 : World#) (x1_acc : Array# Int#)

= newArray# [Int#] 8# x0 in

caselet T2# (x2 : World#) (_ : Void#)

= writeArray# [Int#] x1_acc 0# 0i# x1 in

caselet T2# (x3 : World#) (x3_acc : Array# Int#)

= newArray# [Int#] 8# x2 in

caselet T2# (x4 : World#) (_ : Void#)

= writeArray# [Int#] x3_acc 0# 1i# x3 in

caselet T2# (x11 : World#) (_ : Unit)

= loop# (lengthOfRate# [k_c])

(\(x0 : Nat#).\(x5 : World#).

caselet T2# (x6 : World#) (x1 : Int#)

= next# [Int#] [k_c] s_aqe x0 x5 in

caselet T2# (x7 : World#) (x0 : Int#)

= readArray# [Int#] x1_acc 0# x6 in

caselet T2# (x8 : World#) (_ : Void#)

= writeArray# [Int#] x1_acc 0# (add# [Int#] x0 x1) x7 in

caselet T2# (x9 : World#) (x2 : Int#)

= readArray# [Int#] x3_acc 0# x8 in

caselet T2# (x10 : World#) (_ : Void#)

= writeArray# [Int#] x3_acc 0# (mul# [Int#] x2 x1) x9 in

T2# x10 ()) x4 in

caselet T2# (x12 : World#) (x1 : Int#)

= readArray# [Int#] x1_acc 0# x11 in

caselet T2# (x13 : World#) (x3 : Int#)

= readArray# [Int#] x3_acc 0# x12 in

T2# x13 (add# [Int#] x1 x3)

Here, "caselet" is just sugar for a case expression with a single alternative.

**Covert back to GHC core**

repa_process_sTX

:: forall k_c.

Data.Array.Repa.Series.Stream k_c GHC.Types.Int

-> GHC.Prim.State# GHC.Prim.RealWorld

-> (# GHC.Prim.State# GHC.Prim.RealWorld, GHC.Prim.Int# #)

[LclId]

lowered_sTX =

\ (@ k_c)

(rate_sTY :: GHC.Prim.Int#)

(x_sTZ :: Data.Array.Repa.Series.Stream k_c GHC.Types.Int)

(x_sU0 :: GHC.Prim.State# GHC.Prim.RealWorld) ->

let { (# x_sU1, x_sU2 #) ~ scrut_sUv

<- newByteArray#_sU3 @ GHC.Prim.RealWorld 8 x_sU0

} in

let { __DEFAULT ~ x_sU8

<- writeIntArray#_sU4 @ GHC.Prim.RealWorld x_sU2 0 0 x_sU1

} in

let { (# x_sU5, x_sU6 #) ~ scrut_sUw

<- newByteArray#_sU7 @ GHC.Prim.RealWorld 8 x_sU8

} in

let { __DEFAULT ~ x_sUp

<- writeIntArray#_sU9 @ GHC.Prim.RealWorld x_sU6 0 1 x_sU5

} in

let { (# x_sUa, x_sUb #) ~ x_sUa

<- Main.primLoop

(Main.primLengthOfRate rate_sTY)

(\ (x_sUc :: GHC.Prim.Int#)

(x_sUd :: GHC.Prim.State# GHC.Prim.RealWorld) ->

let { (# x_sUe, x_sU1 #) ~ x_sUe

<- Main.primNext_Int @ k_c x_sTZ x_sUc x_sUd

} in

let { (# x_sUf, x_sUc #) ~ x_sUf

<- readIntArray#_sUg x_sU2 0 x_sUe

} in

let { __DEFAULT ~ x_sUl

<- writeIntArray#_sUh

@ GHC.Prim.RealWorld x_sU2 0 (+#_sUi x_sUc x_sU1) x_sUf

} in

let { (# x_sUj, x_sU8 #) ~ x_sUj

<- readIntArray#_sUk x_sU6 0 x_sUl

} in

let { __DEFAULT ~ x_sUo

<- writeIntArray#_sUm

@ GHC.Prim.RealWorld x_sU6 0 (*#_sUn x_sU8 x_sU1) x_sUj

} in

(# x_sUo, GHC.Tuple.() #))

x_sUp

} in

let { (# x_sUq, x_sU1 #) ~ x_sUq

<- readIntArray#_sUr x_sU2 0 x_sUa

} in

let { (# x_sUs, x_sU5 #) ~ x_sUs

<- readIntArray#_sUt x_sU6 0 x_sUq

} in

(# x_sUs, +#_sUu x_sU1 x_sU5 #)

This doesn't work yet because I've forgotten to pass the type arguments to the unboxed tuple constructor (#,#), and maybe other problems as well. I'll post again when I have an actual program running.

## Tuesday, January 1, 2013

### Code Generators, Rewrite Rules, Aliasing and the Coq

DDC 0.3.1 was pushed onto Hackage just before Christmas.

The main features in this new release are:

For the impatient:

The main features in this new release are:

- Back end code generation via C and LLVM that accepts the new core language. There is enough implemented to compile first order programs, but we'll need to implement a lambda lifter for the new core language before we can compile higher order programs directly.
- Lots more program transformations. You can apply program transformations to core programs on the command line, and check that the code is being optimised the way it should be. There is a tutorial describing how to do this.
- A rewrite rule framework that understands the region and effect system. Rewrite rules work on effectful expressions, and can be given predicates so they only apply when particular effects are non-interfering. This lets us perform build-fold fusion when the builder and folder have (non-interferring) side effects. (more info in Amos Robinson's honours thesis)
- Propagation of no-aliasing meta-data to the LLVM code generator. We have extended the core language with witnesses of distinctness, that prove particular regions of the store are distinct (do not alias). When LLVM has this meta-data it can perform more optimisations on the generated code that it would otherwise be able to. (more info in Tran Ma's honours thesis)

The main missing feature is:

- No type inference, so the core programs contain lots of type annotations. We'll need to get the type inferencer back online before the language will be useful to end-users.

Our immediate plans are to get a paper together about the new core language and how it supports the rewrite rules and LLVM meta-data. There are interesting stories to tell about all of these things. After that, we'll work on the type inferencer.

For the paper, we'll need a proof that the core language does what it's supposed to. To get this, I've started adding the region and effect system to the Coq formalisation of SystemF2+MutableStore I did about a year ago. I hadn't touched Coq since then, so eased my way back into it by cleaning up the proofs I had already done, then packaged them up into the Iron Lambda collection.

Iron Lambda is a collection of Coq formalisations for functional core languages of increasing complexity, with the SystemF2+Regions+Effects language I'm working on now being the latest one. I made a wiki page for Iron Lambda because I think it will be useful in its own right, to help intermediate Coq users bridge the gap between the end of the Software Foundations course and what appears in current research papers. The biggest language in Software Foundations is Simply Typed Lambda Calculus (STLC)+Records+SubTyping, but it doesn't include polymorphism or indirect mutual recursion. To deal with SystemF style polymorphism we need a better approach to names than unique identifiers (ie, something like deBruijn indices). To deal with indirectly mutually recursive language definitions we need to know how to define custom own induction schemes. Iron Lambda shows how to do these things.

For the impatient:

**DDC 0.3.1**

$ cabal update $ cabal install ddc-tools

**Iron Lambda**$ darcs get http://code.ouroborus.net/iron/iron-head/

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