The code below implements this approach. We use Scheme
notation for the source language (we could just as well supported ML
or Haskell-like notations). The notation for type terms is infix, with
the right-associative arrow. As an example, the end of the file
type-inference.scm shows the derivation for call/cc
, shift
and reset
from their types in the
continuation monad. Given the type:
(define (cont a r) `((,a -> . ,r) -> . ,r)) (((a -> . ,(cont 'b 'r)) -> . ,(cont 'b 'b)) -> . ,(cont 'a 'b))within 2 milli-seconds, we obtain the term for shift:
(lambda (_.0) (lambda (_.1) ((_.0 (lambda (_.2) (lambda (_.3) (_.3 (_.1 _.2))))) (lambda (_.4) _.4))))
From Curry-Howard correspondence, determining a term for a type is tantamount to proving a theorem -- in intuitionistic logic as far as our language is concerned. We formulate the proposition in types, for example:
(define (neg x) `(,x -> . F)) (,(neg '(a * b)) -> . ,(neg (neg `(,(neg 'a) + ,(neg 'b)))))This is one direction of the deMorgan law. In intuitionistic logic, deMorgan law is more involved:
NOT (A & B) == NOTNOT (NOT A | NOT B)The system gives us the corresponding term, the proof:
(lambda (_.0) (lambda (_.1) (_.1 (inl (lambda (_.2) (_.1 (inr (lambda (_.3) (_.0 (cons _.2 _.3))))))))))
The de-typechecker can also prove theorems in classical logic, via the double-negation (aka CPS) translation. We formulate a proposition:
(neg (neg `(,(neg 'a) + ,(neg (neg 'a)))))
and, within 403 ms, obtain its proof:
(lambda (_.0) (_.0 (inr (lambda (_.1) (_.0 (inl _.1))))))The proposition is the statement of the Law of Excluded Middle, in the double-negation translation.
Programming languages can help in the study of logic.
let
, sums and products -- and its
type. The code contains many comments that explain the notation, the
incremental composition of the typechecking relation and taking the
advantage of the first-class status of Kanren relations for
typechecking of polymorphic let
.logic.scm [9K]
Kanren code that illustrates the application of the inverse typechecker to proving theorems in intuitionistic and classical logics.
The Kanren project < http://kanren.sourceforge.net >
Deriving a program from its incomplete specification such as type using constraints, background knowledge, heuristics or training set is the subject of inductive programming
< http://www.inductive-programming.org >
int
. A type system is sound if it
correctly approximates the dynamic behavior and predicts its outcome:
if the static semantics predicts that a term has the type int
,
the dynamic evaluation of the term, if it terminates, will
yield an integer.
Conventional type-checking is big-step evaluation in the
abstract: to find a type of an expression, we fully `evaluate' its
immediate subterms to their types. We propose a different kind of type
checking that is small-step evaluation in the abstract: it unzips (pun
intended: cf. Huet's zipper) an expression into a context and a
redex. The type-checking algorithms in the paper are implemented in
Twelf; the complete code is available. In particular, the
well-commented file lfix-calc.elf
implements, by way of
introduction and comparison, small-step type-checking for simply typed
lambda calculus.
The benefit of small-step typechecking is that, by `following' the control flow, it is more suitable for effectful computations. A more interesting result comes from the fact that static evaluation, unlike the dynamic one, goes `under lambda'. Thus the small-step static evaluation context is a binding context.
Colon becomes the new turnstile! Since the evaluation context is neither commutative or associative but is subject to focusing rules, we obtain the framework for expressing various substructural and modal logics.
Joint work with Chung-chieh Shan.
small-step-typechecking.tar.gz [10K]
Commented Twelf code with small small-step typechecking algorithms and many sample derivations.
Abstract interpretation in the denotational context:
Patrick Cousot, Types as abstract interpretations. POPL 1997, 316-331.
Abstract interpretation as an aid during program development of logical programs:
Manuel Hermenegildo, Germa'n Puebla, and Francisco Bueno: Using global analysis, partial specifications, and an extensible assertion language for program validation and debugging.
In: The logic programming paradigm: A 25-year perspective, ed. Krzysztof R. Apt, Victor W. Marek, Mirek Truszczynski, and David S. Warren, 161-192, 1999. Berlin: Springer-Verlag.
Interpreting types as abstract values: A tutorial on Hindley-Milner type inference
fork(2)
,
splitting the current computational process into several, one per
choice alternative. Each process has its own memory, whose contents is
inherited from the parent but can be mutated without affecting the
other processes -- thus ensuring that independent non-deterministic
choices are indeed evaluated independently. As in Unix, making copies
of process memory is best done lazily, as `copy-on-write'. To manage
process memory and copy-on-write, the Unix kernel maintains a special data
structure. First-class memory is like such data structure, letting an
ordinary programmer do the kernel-like memory management.First-class memory has at least the following interface (described below in Haskell):
type Memory -- the abstract type of first-class memory empty :: Memory -- allocate new, initially empty memory size :: Memory -> Int -- get the current memory size (the number of allocated cells) type Ref a -- the abstract reference to a memory cell holding a value of type a new :: a -> Memory -> (Ref a, Memory) get :: Ref a -> Memory -> a set :: Ref a -> a -> Memory -> MemoryThe interface is similar to the familiar
Data.STRef
interface,
with newSTRef
, readSTRef
, writeSTRef
operations. However, our new
, get
and set
explicitly use
the Memory
in which the cell is (to be) located.
The interface comes with the performance requirement:
the execution time of get
and set
(or, more precisely, the number
of accesses to the real computer memory) should not exceed a small constant.
In other words, the implementation of Memory
should at least be as
efficient as the Data.IntMap
trie.
The challenge is implementing the interface efficiently -- and, at the same time, statically guaranteeing some degree of correctness. The correct implementation should satisfy the familiar laws:
let (r,m') = new x m in get r m' === x for all x. let m' = set r' x m in get r m' === get r m if r /= r', r' exists let m' = set r x m in get r m' === x if r exists let m' = set r x m in set r y m' === set r y mThat is: we get what we store; storing in a memory cell overrides its previous value without affecting the other cells. Ensuring these laws in types seems to require too complicated type system to be practical. Still, we would like the types at the very least guarantee an approximation of these laws, where
===
is understood as
equating all values of the same type. In other words, we want to ensure
that get
-ting and set
-ting of existing references
are total and type-preserving.
We want to use types so that the correctness guarantees are clear even
to the compiler, which would then optimize out redundant checks.
Implementing practical first-class memory with such typing guarantees is still
an open challenge.
Let's consider a straightforward non-solution. Recall, Memory
should store values of arbitrary types. The first impulse
is to represent memory cells as Dynamic
, `dynamically-typed' values.
Hence the implementation
type Memory = IntMap Int Dynamic type Ref a = Inta finite map from locations (
Int
) to values of some type.
Alas, the operation fromDynamic :: Dynamic -> Maybe a
to read from
this Dynamic
reference cell is partial. Dynamic
stores a value and
the representation of its type; fromDynamic
checks that the
(representation of the) requested
result type a
matches the type (representation) of the stored value.
Trying to extract from the Dynamic
a value
of a different type than was stored returns Nothing
. In the correctly
implemented Memory
, get
for an existing reference is total
and type-preserving. Therefore, the type-matching check of fromDynamic
should always succeed and hence redundant. The Dynamic
representation
is unsatisfactory because it imposes a run-time check that should
never fail. Again, the challenge is to ensure correctness statically and
avoid run-time assertions.
Besides the performance, Dynamic
is unsatisfactory theoretically.
A Dynamic
stores a value and a representation of its type. On the other hand,
the memory address -- the location of a memory cell --
uniquely identifying a cell also uniquely
describes the type of the stored value.
That is, within given Memory
, the location is itself a type
representation: location equality implies type equality. Therefore,
the type representation used in Dynamic
is redundant.
The pragmatic way to avoid redundant checks is to use unsafeCoerce
. Although helping performance, it forces us to rely on
pen-and-paper correctness proofs. We better make no mistakes in the
proof, or else risk a segmentation fault.
Let's recall the outline of the pen-and-paper correctness argument,
to see why it is so hard to express in types. First, the type of reference
cells Ref a
includes the type of the stored value. The type signatures
of new
, get
and set
show that
once a reference r
is created to hold the value of some type a
, get r
and set r
can only read and write values of that type a
.
To complete the proof we have to check that the reference r
created by new x m
is `fresh' -- it names a cell that
is different from any cell in the parent memory m
--
and get r m'
and set r y m'
really access that very
cell r
. The correctness of get
and set
is hence tied up
to the third function, new
: that new
really creates fresh references.
Expressing and using this freshness property in types is the real challenge.
Memory.hs [4K]
First-class memory module used in the probabilistic programming language Hakaru10.
In this implementation the reference Ref acts as a cache for the stored value.
Probabilistic programming using first-class stores and first-class continuations
gensyms
) a fresh type variable and watches for its
scope, following the proof rule for universal introduction. Due to
this implementation, rank-2 polymorphism has been used (some would say
abused) to express static capabilities and assure a wide variety of
safety properties statically, including:In each of these applications, the fresh types generated by the type checker seems to reflect fresh values generated by the program at run time. For example, when generating code as a first-order data structure, we need gensym not only at the type level to ensure lexical scope but also at the value level to make unique concrete variable names. Can dependent types expose and eliminate this reflection for the superfluous encoding that it is? In other words, can our code for generating fresh values be reused during type checking to assure their safety properties more directly? In particular, whereas generated types are only implicitly fresh and are equipped with no operations, generated values are explicitly fresh and are equipped with operations such as comparison. Can we use generated variable names to automate weakening?
Joint work with Chung-chieh Shan.
Lightweight Dependent-type Programming
Lightweight monadic regions
A few applications of fresh type variables, not counting the ST monad
oleg-at-okmij.org
Your comments, problem reports, questions are very welcome!
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