The course on typed tagless-final embeddings of domain-specific languages has been presented at the Spring School on Generic and Indexed Programming (SSGIP) http://www.comlab.ox.ac.uk/projects/gip/school.html at Wadham College, Oxford, UK on 22nd to 26th March 2010. This page collects the notes for the course in the form of the extensively commented Haskell and OCaml code.
The topic of the course is the embedding of domain-specific languages (DSL) in a host language such as Haskell or OCaml. We will often call the language to embed 'the object language' and the host language 'the metalanguage'. All throughout the course we will repeatedly encounter the following points:
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Multiple interpretations:
- writing a DSL program once, and interpret it many times, in standard and non-standard ways;
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Extensibility:
- enriching the syntax of the object language, re-using but not breaking the existing interpreters;
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Types
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Typed implementation language
- getting the typechecker to verify properties of interpreters, such as not getting stuck;
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Typed object language
- getting the metalanguage typechecker to enforce properties of DSL programs, such as being well-typed;
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Connections with logic
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Final
- preferring lower-case
- preferring elimination over introduction
- connecting to denotational semantics
We will be talking about ordinary data types and (generic) operations on them. The expression problem will make its appearance. The first-order case makes it easier to introduce de-serialization and seemingly non-compositional operations.
Intro1.hs [2K] Algebraic data type/initial representation of expressions; Constructor functions: the intimation of the final representation (or, shallow embedding)
Intro2.hs [3K] Symantics: parameterization of terms by interpreters
Intro3.hs [2K] Initial and Final, Deep and Shallow, First-class
ExtI.hs [<1K] Algebraic data types are indeed not extensible
ExtF.hs [2K] Adding a new expression form to the final view: solving the expression problem
Serialize.hs [4K] Serialization and de-serialization
SerializeExt.hs [4K] De-serializing the extended language
We demonstrate several encodings of extensible first-order languages in OCaml. Objects turn out handy in emulating the composition of type class dictionaries.
final_obj.ml [2K] The traditional application of objects to represent extensible data types. Alas, the set of operations on these data types is not extensible.
final_mod.ml [3K] Tagless-final embedding using modules
final_dic.ml [3K] Tagless-final embedding with objects emulating type-class dictionaries. Both the language and the set of its interpretations are extensible.
Interpreters are well suited for compositional, fold-like operations on terms. The tagless-final representation of terms makes writing interpreters quite convenient. One may wonder about operations on terms that do not look like fold. Can we even pattern-match on terms represented in the tagless-final style? Can we compare such terms for equality? We answer the first question here, deferring the equality test till the part on implementing a type checker for a higher-order language. Our running examples are term transformations, converting an expression into a simpler, more optimal, or canonical form. The result is an uncrippled expression, which we can feed into any of the existing or future interpreters. Our sample term transformations look like simplified versions of the conversion of a boolean formula into a conjunctive normal form.
PushNegI.hs [3K] Pushing the negation down: the initial implementation
PushNegF.hs [3K] Pushing the negation down: the final implementation
PushNegFExt.hs [4K] Pushing the negation down for extended tagless-final terms
FlatI.hs [2K]
FlatF.hs [3K] Flattening of additions, the initial and the final implementations
PushNegFI.hs [3K] The final-initial isomorphism, and its use for implementing arbitrary pattern-matching operations on tagless-final terms.
http://www.comlab.ox.ac.uk/ralf.hinze/SSGIP10/Slides.pdf
Ralf Hinze, in Part 7 of his Spring School course, has derived this 'initial-final' isomorphism rigorously, generally and elegantly from the point of view of Category Theory. In the first-order case, both 'initial' and 'final' are the left and the right views to the same Initial Algebra. The 'final' view is, in the first-order case, ordinary Church encoding.
Higher-order languages are data types with binding, so to speak. In the first part, only the interpreters were typed; we could get away with our object language being unityped. Now, the object language itself becomes typed, bringing the interesting issues of interpreting a typed language in a type language ensuring type preservation. It is this part that explains the attributes 'typed' and 'tagless' in the title of the course.
Using simply-typed lambda-calculus with constants as a sample DSL, we demonstrate its various embeddings into Haskell. We aim at a type-preserving embedding and efficient and type-preserving evaluations. The tagless-final embedding not only achieves this goal, it also makes the type-preservation patently clear. Tagless-final evaluators are well-typed Haskell programs with no pattern-matching on variant types. It becomes impossible for the evaluators to get stuck. Since the type preservation of the evaluators is apparent not only to us but also to a Haskell compiler, the evaluators can be efficiently compiled. Tagless-final embeddings are also extensible, letting us add to the syntax of the DSL, preserving and reusing old interpreters.
IntroHOT.hs [3K] The illustration of problems of embedding a typedDSL into a typed metalanguage
Either the Universal type (and hence spurious partiality, type tags and inefficiency), or fancy type systems seem inevitable. The problem stems from algebraic data types' being too broad: they express not only well-typed DSL terms but also ill-typed ones.
Term.agda [2K] http://www.iis.sinica.edu.tw/~scm/2008/typed-lambda-calculus-interprete/
Shin-Cheng Mu: Typed Lambda-Calculus Interpreter in Agda. September 24, 2008
Shin-Cheng Mu solves the problem of the type-preserving tagless interpretation of simply-typed lambda-calculus, relying on dependent types and type functions in full glory.
IntroHOIF.hs [6K] Tagless-initial and Tagless-final evaluators
TTFdB.hs [7K] Typed, tagless, final, with de Bruijn indices: Expressing the type system of simply-typed lambda-calculus in Haskell98. No dependent types are necessary after all. The types of methods in the Symantics type class read like the axioms and inference rules of the implication fragment of minimal logic.
TTF.hs [7K] Typed, tagless, final, in the higher order abstract syntax (HOAS). We illustrate extending the DSL with more constants, types, and expression forms.
TTIF.hs [8K] Initial-final isomorphism, in the higher-order case
Since we represent DSL expressions as well-typed Haskell terms, we can place DSL terms in Haskell code or enter at the GHCi prompt. However, we also want to interpret DSL expressions that are read from files or received from communication pipes. We no longer can then rely on GHC to convert DSL expressions from a text format into the typed embedding. We have to do the type-checking of DSL expressions ourselves. Our goal is to type-check an expression once, during de-serialization, and evaluate the result many times. Since a type checker needs to represent types and reason about type equality, we develop type representations and type safe cast. We regard the language of types, too, as a typed DSL, which we embed in Haskell in the tagless-final style.
TypeCheck.hs [12K]\ De-serialization: (Dynamic) Type Checking
In contrast to an earlier version of the type
checker, we use de Bruijn indices and obtain a
much clearer code. The code is quite similar to
Baars and Swierstra's ''Typing Dynamic Typing''
(ICFP02). However, the result of our type-checking
is an embedded DSL expression that can be
interpreted many times and in many ways, rather
than being merely evaluated. The set of possible
interpretations is open. Also, our code is written
to expose more properties of the type-checker for
verification by the Haskell type-checker; for
example, that closed source terms are
de-serialized into closed target terms.
Typ.hs [8K] Type representation, equality and the type-safe generalized cast
We present an above-the-board version of
Data.Typeable, in the tagless-final style. Our
implementation uses no GHC internal operations, no
questionable extensions, or even a hint of unsafe
operations.
http://www.comlab.ox.ac.uk/projects/gip/school/tc.hs
Stephanie Weirich some time ago wrote a very similar type-checker, but in the initial style, using GADTs. The comparison with the tagless-final style here is illuminating.
We demonstrate ordinary and administrative-redex--less call-by-value Continuation Passing Style (CPS) transformation that assuredly produces well-typed terms and is patently total. Our goal here is not to evaluate, view or serialize a tagless-final term, but to transform it to another one. The result is a tagless-final term, which we can interpret in multiple ways: evaluate, view, or transform again. We first came across transformations of tagless-final terms when we discussed pushing the negation down in the simple, unityped language of addition and negation. The general case is more complex. It is natural to require the result of transforming a well-typed term be well-typed. In the tagless-final approach that requirement is satisfied automatically: after all, only well-typed terms are expressible. We require instead that a tagless-final transformation be total. In particular, the fact that the transformation handles all possible cases of the source terms must be patently, syntactically clear. The complete coverage must be so clear that the metalanguage compiler should be able to see that, without the aid of extra tools.
Since the only thing we can do with tagless-final terms is to interpret them, the CPS transformer is written in the form of an interpreter. It interprets source terms yielding transformed terms, which can be interpreted in many ways. In particular, the terms can be interpreted by the CPS transformer again, yielding 2-CPS terms. CPS transformers are composable, as expected.
A particular complication of the CPS transform is that the type of the result is different from the type of the source term: the CPS transform translates not only terms but also types. Moreover, the CPS type transform and the arrow type constructor do not commute. For that reason, we have to introduce an extended Symantics class, ESymantics, which makes the meaning of function types dependent on a particular interpreter. As it turns out, we do not have to re-write the existing Symantics terms: we can re-interpret any old terms in the extended Symantics. Conversely, any extended Symantics term can be interpreted using any old, legacy, Symantics interpreter. The CPS transform is an extended Symantics interpreter proper.
The ordinary (Fischer or Plotkin) CPS transform introduces many administrative redices, which make the result too hard to read. Danvy and Filinski proposed a one-pass CPS transform, which relies on the metalanguage to get rid of the administrative redices. The one-pass CPS transform can be regarded as an example of the normalization-by-evaluation.
CPS.hs [12K] Ordinary and one-pass CPS transforms and their compositions
Olivier Danvy and Andrzej Filinski. Representing
Control: A Study of the CPS Transformation.
Mathematical Structures in Computer Science, 1992.
Olivier Danvy's original POPL96 paper on type-directed partial evaluation used an untyped target language, represented as an algebraic data type. Type preservation was not apparent and had to be proved. In our presentation, the result of reification is a typed expression, in the tagless-final form. Type preservation of reification is now syntactically apparent and is verified by the Haskell type-checker. In the tagless-final presentation, reification and reflection seem particularly symmetric, elegant and insightful.
TDPE.hs [6K] Tagless-final presentation of type-directed partial evaluation
ToTDPE.hs [<1K] The imported module with sample functions to reify. Compiling this module makes for a nicer example.
http://www.brics.dk/~danvy/tdpe-ln.pdf
Olivier Danvy: Lecture notes on type-directed partial evaluation. The lecture notes are based on his POPL96 paper.
One may think that we can embed in Haskell only
those DSL whose type system is a subset of that of
Haskell. We will now attempt to break that
impression by showing how to embed typed linear
lambda calculus. Any bound variable must be
referenced exactly once in abstraction's body. As
before, our embedding makes sure that only
well-typed and well-formed terms are
representable. In other words, Haskell as the
metalanguage will statically reject as ill-typed
representations of lambda-terms in which the bound
variable appears several times -- or, as in the K
combinator, never. Our approach relies on de
Bruijn representation for variables. We have
already discussed the tagless-final encoding for
the ordinary simply-typed lambda calculus with de
Bruijn indices. An object term of the type a was
represented as a value of the type repr h a
where the binary type constructor repr is a
member of the class Symantics. The type variable
h represents the variable (or, hypothetical)
environment, describing the types of term's free
variables. In linear lambda calculus, terms
associated with bound variables are regarded as
resources; referencing a variable consumes the
resource. We use the variable environment for
tracking the state of resources: available, or
consumed. Since evaluating an expression may
'consume' a resource associated with one of
expression's variables, the variable environment
becomes the variable state. We can then follow
the approach described in Variable (type)state'monad'.
We represent linear lambda terms by Haskell values
of the type repr hi ho a, where hi stands for
the variable state before evaluating the term and
ho stands for the state after evaluating the
term. To be more precise, hi and ho are the
types of the variable states. We can determine the
types and hence the state of the variables
statically: As usual, the type checker does
abstract interpretation. In our tagless-final
encoding, lam has the following type
lam :: repr (F a,hi) (U,ho) b -> repr hi ho (a->b)
The expression (lam e) has the type
repr hi ho (a->b) provided the body of
abstraction, e, has the type
repr (F a,hi) (U,ho) b. That is, in the
environment extended with a term of the type a,
the body must produce the value of type b. The
body must consume the term at the top of the
environment, changing the type of the first
environment cell from F a to U (the type of
the used variable). A trivial modification turns
the embedding of the linear lambda-calculus into
the embedding of the affine lambda-calculus,
permitting no references to the bound variable
within the abstraction body. K combinator becomes
expressible.
LinearLC.hs [11K]
Commented code defining the typed linear lambda
calculus and its two interpreters, to evaluate and
to show linear lambda terms. The code demonstrates
extending the linear calculus by adding general
abstractions imposing no constraints on the use of
bound variables.
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CBAny.hs [7K]
Parameterizing evaluators by the evaluation order: call-by-name, call-by-value, call-by-need
CB98.hs [7K]
The same, but in pure Haskell 98 -
PrintScanF.hs [8K]
Typed formatting: printf and scanf. The code is one of the implementations of Type-safe Formatted IO. -
Sandro Magi: Mobile Code in C# via Finally Tagless Interpreters. June 23, 2009.
http://higherlogics.blogspot.com/2009/06/mobile-code-in-c-via-finally-tagless.htmlSandro Magi shows that different interpretations of the same DSL term may not only involve different run-time systems but also occur on different hosts.
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Expressing sharing
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Non-standard and abstract interpretations
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Unusual calculi: shift/reset with effect typing; lambda-mu-mubar
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Abstract Categorial Grammars http://www.loria.fr/equipes/calligramme/acg
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