paper/
: ICFP 2016 paper.
src/
: Implementation of Datafun in Racket. src/repl.rkt
is most useful.
What follows is an extremely out-of-date description of Datafun's type theory.
For more up-to-date information,
here's a paper preprint; or you can
clone the repository and run make
in the paper/
directory to produce
datafun.pdf
.
poset types A,B ::= bool | nat | A × B | A → B | A →⁺ B | Set A | A + B
lattice types L,M ::= bool | nat | L × M | A → L | A →⁺ L | Set A
expressions e ::= x | λx.e | e e
| (e, e) | πᵢ e
| true | false | if e then e else e
| inᵢ e | case e of in₁ x → e; in₂ x → e
| ∅ | e ∨ e | {e} | ⋁(x ∈ e) e
| fix x. e
contexts Δ ::= · | Δ,x:A
monotone ctxts Γ ::= · | Γ,x:A
Types correspond to partially ordered sets (posets):
bool
is booleans; false
< true
.
nat
is the naturals, ordered 0 < 1 < 2 < ...
A × B
is pairs, ordered pointwise:
(a₁,b₁) ≤ (a₂,b₂)
iff a₁ ≤ a₂
and b₁ ≤ b₂
.
A + B
is sums, ordered disjointly. in₁ a₁ ≤ in₁ a₂
iff a₁ ≤ a₂
, and
likewise for in₂
; but in₁ a
and in₂ b
are not comparable to one another.
Set A
is finite sets of A
s, ordered by inclusion:
x ≤ y
iff ∀(a ∈ x) a ∈ y
.
A → B
are functions, ordered pointwise: f ≤ g
iff ∀(x : B) f x ≤ g x
.
A →⁺ B
are monotone functions; for any f : A →⁺ B
, given x,y : A
such
that x ≤ y
we know that f x ≤ f y
. The type system enforces this
monotonicity. Monotone functions are ordered pointwise, just like regular
functions.
Lattice types L
are a subset of all types, defined so that every lattice type
happens to be unital semilattices (usls) — that is, join-semilattices with a
least element. Any lattice type is a type, but not all types are lattice types.
Semantics of expressions, in brief:
x
, (e₁, e₂)
, πᵢ e
, inᵢ e
, if
, true
, false
, and case
all do
what you'd expect.
λx.e
and e e
both do what you'd expect. However, it is left ambiguous
whether they represent ordinary or monotone function creation/application.
One could of course require the programmer to write ordinary and monotone
functions differently (or even ordinary and monotone function applications
differently). But for our purposes it's simplest to just give two typing rules
(ordinary and monotone) for λx.e
(and likewise e e
).
It is definitely possible to infer monotonicity in a bidirectional way, and possibly even in a Damas-Milner-ish way, but that's outside the scope of this README.
∅
represents the least element of a lattice type.
e₁ ∨ e₂
represents the least upper bound ("lattice join") of e₁
and e₂
.
{e}
represents the singleton set containing e
.
⋁(x ∈ e₁) e₂
is set-comprehension. e₁
must have a finite set type; e₂
must have a lattice type. For each x
in e₁
, we compute e₂
; then we
lattice-join together all values of e₂
computed this way, and that is our
result. This generalizes the "bind" operation of the finite-set monad.
fix x. e
finds the least fixed-point of the monotone function λx. e
.
Δ;Γ ⊢ e : A
Our typing judgment is Δ;Γ ⊢ e : A
We call Δ
our unrestricted context and Γ
our monotone context. Both
contexts obey the usual intuitionistic structural rules (weakening, exchange).
Δ,x:A; Γ ⊢ e : B Δ;Γ ⊢ e₁ : A → B Δ;· ⊢ e₂ : A
------------------ λ -------------------------------- app
Δ;Γ ⊢ λx.e : A → B Δ;Γ ⊢ e₁ e₂ : B
Δ; Γ,x:A ⊢ e : B Δ;Γ ⊢ e₁ : A →⁺ B Δ;Γ ⊢ e₂ : A
------------------- λ⁺ --------------------------------- app⁺
Δ;Γ ⊢ λx.e : A →⁺ B Δ;Γ ⊢ e₁ e₂ : B
NB. The monotone context of e₂
in the rule app
for applying ordinary
functions must be empty! Since A → B
represents an arbitrary function, we
cannot rely on its output being monotone in its argument. Thus its argument must
be, not monotone in Γ, but constant.
The typing rules for tuples, sums, and booleans are mostly boring:
Δ;Γ ⊢ eᵢ : Aᵢ Δ;Γ ⊢ e : A₁ × A₂
----------------------- ------------------
Δ;Γ ⊢ (e₁,e₂) : A₁ × A₂ Δ;Γ ⊢ πᵢ e : Aᵢ
Δ;Γ ⊢ e : bool Δ;Γ ⊢ eᵢ : A
----------------- ------------------ -------------------------------
Δ;Γ ⊢ true : bool Δ;Γ ⊢ false : bool Δ;Γ ⊢ if e then e₁ else e₂ : A
Δ;Γ ⊢ e : Aᵢ
---------------------
Δ;Γ ⊢ inᵢ e : A₁ + A₂
However, there are two eliminators for sum types:
TODO
The typing rules get more interesting now:
Δ;Γ ⊢ eᵢ : L
----------- -----------------
Δ;Γ ⊢ ∅ : L Δ;Γ ⊢ e₁ ∨ e₂ : L
Δ;· ⊢ e : A Δ;Γ ⊢ e₁ : Set A Δ,x:A; Γ ⊢ e₂ : L
----------------- ------------------------------------
Δ;Γ ⊢ {e} : Set A Δ;Γ ⊢ ⋁(x ∈ e₁) e₂ : L
Δ; Γ,x:L ⊢ e : L L equality
----------------------------- fix
Δ;Γ ⊢ fix x.e : L
In the last rule, for fix
, the premise L equality
means that the type L
at
which the fixed-point is computed must have decidable equality.
Alternative, two-layer formulation:
set types A,B ::= U P | A ⊗ B | A ⊕ B | A ⊃ B
poset types P,Q ::= bool | nat | P × Q | P →⁺ Q | Set A
| Disc A | P + Q
lattice types L,M ::= bool | nat | L × M | P →⁺ M | Set A
expressions e ::= x | λx.e | e e | (e, e) | πᵢ e
| inᵢ e | case e of in₁ x → e; in₂ x → e
| U u
lattice exprs u ::= x | λx.u | u u | (u, u) | πᵢ u
| ∅ | u ∨ u | {e} | ⋁(x ∈ u) u
| fix x. u
| D e | U⁻¹ e | let D x = u in u
Δ;· ⊢ u : P Δ ⊢ e : U P
------------- U --------------- U⁻¹
Δ ⊢ U u : U P Δ;Γ ⊢ U⁻¹ e : P
Δ ⊢ e : A Δ;Γ ⊢ u₁ : D A Δ,x:A; Γ ⊢ u₂ : P
--------------- D ----------------------------------- let-D
Δ;Γ ⊢ D e : D A Δ;Γ ⊢ let D x = u₁ in u₂ : P
I use ⊗
and ⊕
for set types not because they are linear, but simply to
distinguish them from the ×
and +
operations on poset types.
This version needs to be fleshed out more fully. In particular, we need some
axioms to ensure that U (P + Q) = U P ⊕ U Q
.