The paper introduces the continuation+state monad for code generation with let-insertion, and briefly describes abstract interpretation for generating efficient code without its intensional analysis. These are the foundational techniques for typeful multi-staged (MetaOCaml) programming.
Joint work with Walid Taha, Kedar N. Swadi, and Emir Pasalic.
Dynamic Programming Benchmark: The corresponding MetaOCaml source code
< http://www.metaocaml.org/examples/dp/ >
Shifting the Stage: Staging with Delimited Control
An alternative, notationally more convenient approach relying on delimited control rather than monads
Tagless-staged is such a front-end -- a (tagless-final) library of
code generation combinators built on top of Template Haskell.
The library statically assures not only that all generated code is
well-formed and well-typed but also that all generated variables are
bound lexically as expected. Such assurances are crucial for code
generators to be written by domain experts rather than compiler
writers, because the most profitable optimizations are domain-specific
ones. Unlike MetaOCaml, the static guarantees are maintained in the
presence of arbitrary effects, including IO. Since Tagless-staged gets
by without the
Q monad of Template Haskell, it questions that monad's
Joint work with Chung-chieh Shan and Yukiyoshi Kameyama.
< http://www.cs.tsukuba.ac.jp/techreport/data/CS-TR-11-17.pdf >
Computational Effects across Generated Binders: Maintaining future-stage lexical scope.
Technical Report CS-TR-11-17, Department of Computer Science, Graduate School of Systems and Information Engineering, University of Tsukuba.
A difficult-to-understand explanation of the library
Examples and Sample code
We report on the status of the ongoing MetaOCaml project, focusing on the gap between theory and practice and the difficulties that arise in a full-featured staged language rather than an idealized calculus. We highlight foundational problems in type soundness and cross-stage persistence that demand investigation. We also suggest a lightweight implementation of a two-stage dialect of OCaml, as syntactic sugar.
Joint work with Chung-chieh Shan.
Chung-chieh Shan. Slides of the talk at the ML workshop. Baltimore, MD, September 26, 2010.
< http://www.cs.rutgers.edu/~ccshan/metafx/metaocaml-slides.pdf >
BER MetaOCaml: a streamlined version of MetaOCaml
Here is a sample MetaOCaml code and its Scheme translation
# let eta = fun f -> .<fun x -> .~(f .<x>.)>. in .<fun x -> .~(eta (fun y -> .<x + .~y>.))>. - : ('a, int -> int -> int) code = .<fun x_1 -> fun x_2 -> (x_1 + x_2)>. (let ((eta (lambda (f) (bracket (lambda (x) (escape (f (bracket x)))))))) (bracket (lambda (x) (escape (eta (lambda (y) (bracket (+ x (escape y)))))))))Translating MetaOCaml code into Scheme seems trivial: code values are like S-expressions, MetaOCaml's bracket is like quasiquote, escape is like unquote, and `run' is eval. If we indeed replace bracket with the quasiquote and escape with the unquote in the above Scheme code and then evaluate it, we get the S-expression
(lambda (x) (lambda (x) (+ x x))), which is a wrong result, quite different from the code expression
.<fun x_1 -> fun x_2 -> (x_1 + x_2)>.produced by MetaOCaml. The latter is the code of a curried function that adds two integers. The S-expression produced by the naive Scheme translation represents a curried function that disregards the first argument and doubles the second. This demonstrates that the often mentioned `similarity' between the bracket and the quasiquote is flawed. MetaOCaml's bracket respects alpha-equivalence; in contrast, Scheme's quasiquotation, being a general form for constructing arbitrary S-expressions (not necessarily representing any code), is oblivious to the binding structure.
Our implementation still uses S-expressions for code values (so we can print them and
eval-uate them) and treats escape as unquote. To maintain the hygiene, we need to make sure that every run-time evaluation of a bracket form such as
(bracket (lambda (x) x)) gives
'(lambda (x_new) x_new) with the unique bound variable
x_new . Two examples in the source code demonstrate why static renaming of manifestly bound identifiers is not sufficient. We implement the very clever suggestion by Chung-chieh Shan and represent a staged expression such as
.<(fun x -> x + 1) 3>. by the sexp-generating expression
`(,(let ((x (gensym))) `(lambda (,x) (+ ,x 1))) 3) which evaluates to the S-expression
((lambda (x_1) (+ x_1 1)) 3) . Thus
bracket is a complex macro that transforms its body to the sexp-generating expression, keeping track of the levels of brackets and escapes. The macro
bracket is implemented as a CEK machine with the defunctionalized continuation. In our implementation, the Scheme translation of the eta-example yields the S-expression
(lambda (g6) (lambda (g8) (+ g6 g8))) . Just as in MetaOCaml, the result represents the curried addition.
CSP poses another implementation problem. In MetaScheme we can write the MetaOCaml expression
.<fun x -> x + 1>. as
(bracket (lambda (x) (+ x 1))) , which yields an S-expression
(lambda (g1) (+ g1 1)) . When we pass this code to eval, the identifier
+ will be looked up in the environment of eval, which is generally different from the environment that was in effect when the original bracket form was evaluated. That might not look like much of a difference since
+ is probably bound to the same procedure for adding numbers in either environment. This is no longer the case if we take the following MetaOCaml code and its putative MetaScheme translation:
let test = let id u = u in .<fun x -> id x>. (define test (let ((id (lambda (u) u))) (bracket (lambda (x) (id x)))))The latter definition binds
testto the S-expression
(lambda (x) (id x))that contains a free identifier
id, unlikely to be bound in the environment of eval. Our code values therefore should be `closures', being able to include values that are (possibly locally) bound in the environment when the code value was created. Incidentally, the problem of code closures closed over the environment of the generator also appears in syntax-case macros. R6RS editors made a choice to prohibit the occurrence of locally-bound identifiers in the output of a syntax-case transformer.
MetaOCaml among other staged systems does permit the inclusion of values from the generator stage in the generated code. Such values are called CSP; they are evident in the output of the MetaOCaml interpreter for the above test:
val test : ('a, 'b -> 'b) code = .<fun x_1 -> (((* cross-stage persistent value (as id: id) *)) x_1)>.In MetaScheme, we have to write CSP explicitly:
(% e), which is often called `lift'.
(define test (let ((id (lambda (u) u))) (bracket (lambda (x) ((% id) x)))))One may think that such a lifting is already available in Scheme:
(define (lift x) (list 'quote x)). Although this works in the simple cases of
xbeing a number, a string, etc., it is neither universal nor portable: attempting this way to lift a closure or an output-port could be an error. According to R5RS, the argument of a quotation is an external representation of a Scheme datum. Closures, for example, are not guaranteed to have an external representation. For portability, we implement CSP via a reference in a global array, taking advantage of the fact that both the index (a number) and the name of the array (an identifier) have external representations and hence are trivially liftable by quotation. This is precisely the mechanism used by the current version of MetaOCaml.
The source code contains many more examples of the translation of MetaOCaml code into MetaScheme -- including the examples with many stages and CSP.
Joint work with Chung-chieh Shan.
D a => a -> awhere
Floating-like class, and produce a function of the same type that is the symbolic derivative of the former. The original function can be given to us in a separately compiled module with no available source code. The derived function is an ordinary numeric Haskell function and can be applied to numeric arguments -- or differentiated further. The Floating-like class
Dcurrently supports arithmetic and a bit of trigonometry. We also support partial derivatives.
test1f x = x * x + fromInteger 1 test1 = test1f (2.0::Float) -- 5.0 test1f' x = diff_fn test1f x test1' = test1f' (3.0::Float) -- 6.0 test1f'' x = diff_fn test1f' x test1'' = test1f'' (10.0::Float) -- 2.0The original function
test1fcan be evaluated numerically,
test1, or differentiated symbolically,
test1f'. The result is again an ordinary numeric function (i.e.,
2x), which can be applied to a numeric argument, see
test1f'can be differentiated further.
The original function is emphatically not represented as an algebraic data type -- it is a numeric function like
tan . Still, we are able to differentiate it symbolically (rather than numerically or automatically). The key insight is that Haskell98 supports a sort of reflection -- or, to be precise, type-directed partial evaluation and hence term reconstruction.
Our approach also shows off the specification of the differentiation rules via type classes (which makes the rules extensible) and the emulation of GADT via type classes. In 2006, we improved the approach by developing an algebraic simplifier and by avoiding any interpretative overhead.
Most optimal symbolic differentiation of compiled numeric functions
Our approach is reifying code into its `dictionary view', intensional analysis of typed code expressions, and staging so to evaluate under lambda. We improve the earlier, 2004 approach in algebraically simplifying the result of symbolic differentiation, and in removing interpretative overhead with the help of Template Haskell (TH). The computed derivative can be compiled down to the machine code and so it runs at full speed, as if it were written by hand to start with.
In the process, we develop a simple type system for a subset of TH code expressions (TH is, sadly, completely untyped) -- so that accidental errors can be detected early. We introduce a few combinators for the intensional analysis of such typed code expressions. We also show how to reify an identifier like
(+) to a
TH.Name -- by applying TH to itself. Effectively we obtain more than one stage of computation.
Our technique can be considered the inverse of the TH splicing operation: given a (compiled) numeric expression of a host program, we obtain its source view as a TH representation. The latter can be spliced back into the host program and compiled -- after, perhaps, simplification, partial evaluation, or symbolic differentiation. As an example, given the definition of the ordinary numeric,
Floating a => a->a function
test1f x = let y = x * x in y + 1 (which can be located in a separately compiled file), we reify it into a TH code expression, print it, differentiate it symbolically, and algebraically simplify the result:
*Diff> test1cp \dx_0 -> GHC.Num.+ (GHC.Num.* dx_0 dx_0) 1 *Diff> test1dp \dx_0 -> GHC.Num.+ (GHC.Num.+ (GHC.Num.* 1 dx_0) (GHC.Num.* dx_0 1)) 0 *Diff> test1dsp \dx_0 -> GHC.Num.+ dx_0 dx_0The output is produced by TH's pretty-printer. We can splice the result in a Haskell program as
$(reflectQC test1ds) 2.0and use it as ordinary, hand-written function
\x -> x+x.
The main implementation file, which defines the reification of code into TH representation, differentiation rules and algebraic simplification rules, all via the intensional analysis of the typed code. The file also includes many examples, including those of partial differentiation.
Running the splicing tests from Diff.hs. Due to the TH requirement, this code must be in a separate module.
This file introduces the type
Code a of typed TH code expressions. The (phantom) type parameter is the expression's type. The file defines combinators for building and analyzing these typed expressions.
Obtain the Name that corresponds to a top-level (Prelude-level) Haskell identifier by applying TH to itself.
The example is the following simple expression.
\i -> (if i then 2 else 3) + 4To partially evaluate it, we first annotate all sub-expressions as static
S(known at compile time) or dynamic
D(known only at run-time). That is, we perform the `binding-time analysis' (BTA) and add binding-time annotations. For clarity, we show all the steps of the analysis. First, we see no applications to our function and hence do not statically know
\i_D -> (if i_D then 2 else 3) + 4The if-test with the dynamic condition is dynamic:
\i_D -> (if i_D then 2 else 3)_D + 4and the addition expression with a dynamic operand is dynamic:
\i_D -> ((if i_D then 2 else 3)_D + 4)_DAll sub-expressions are dynamic and hence there are no opportunities for partial evaluation.
Let's re-write the original expression in CPS:
\i k -> let k' = \v -> k (v+4) in if i then k' 2 else k' 3and do the binding-time analysis again. As before, function arguments,
k, must be dynamic:
\i_D k_D -> let k' = \v -> k_D (v+4) in if i_D then k' 2 else k' 3The variable
k'is static since its value, the
\v-> ...abstraction, is statically known. Constants are obviously static:
\i_D k_D -> let k'_S = \v -> k_D (v+4) in if i_D then k'_S 2_S else k'_S 3_SThere are now two applications of a statically known argument to a statically known function. We can do them statically, at the specialization, or `compile'-time, obtaining:
\i k -> if i then k 6 else k 7which is certainly an improvement: there are no longer any additions left for the run-time.
Bondorf (1992) showed the alternative to writing the source program in CPS. We should instead write the specializer (partial evaluator) in CPS, and skillfully massage the continuations to achieve binding-time improvements. A partial evaluator becomes harder to write, but the source programs may remain in the original, direct style, and still benefit from improved specialization. Lawall and Danvy (1994) showed that delimited control helps keep even the partial evaluator in direct style. Thus a specializer capable of improvements shown in this message becomes only a little more difficult to write.
Julia L. Lawall and Olivier Danvy: Continuation-Based Partial Evaluation
Conf. Lisp and Function Programming, 1994
oleg-at-pobox.com or oleg-at-okmij.org
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