Executable
Relational Specifications of
Polymorphic Type Systems
using Prolog
Ki Yung Ahn
kyagrd@gmail.com
( this slide is available online at http://slides.com/kyagrd/TIPERdundee )
Andrea Vezzosi
vezzosi (AT) chalmers (DOT) se
http://www.cse.chalmers.se/~vezzosi/
Static Types are Cool
 (Connections to all the cool theories, fun for research)
 Efficiency
 provide basic information at compile time for compilers to optimize code generation
 Safety
 using typesafety to guarantee that program does not reach bad state, before running the program
 provides basic safeguard while refactoring :
when you change the type of a function, you can follow all the type errors to track where that change is being propagated without looking through whole source or running regression tests
Static Types are Scarce
 Apart from a few exeptions such as
F# (MS), Swift (Apple),
Scala, Elm, PureScript (community based)
that tries to benefit from type system research
 Most of the recent major buzz langauges we hear
about are untyped, especially on web/cloud platforms.
e.g., Javascript, Python, Ruby, ...
however, in the real world ...
Why on earth there are so many
dynamicallytyped untyped languages
and still more piling up every new year?
Langugae Construction toolstack avialble today
We need a TIPER
Language System Frontend Construction toolstack

Lexer generator

Parser generator

TIPER :
Type Inference Prototyping Engine
from Relational specifications
of type systems
Outline

Motivation  need a TIPER
 why Relational Specification
 Prolog Specifications of HM
 HM + type constructor poly + kind poly
 Plans and Ideas on the TIPER project
Relational Specifcation
 It is the way how we usually
define type systems on paper  Specification for type checking and
type inference without duplication  Can be queried multiple ways on
logic programming systems
Relational Specifcation
 Type Checking
 Type Inference
 Type Inhabitation
? type([], lam(x,var(x)), A>A).
true .
? type([], lam(x,var(x)), T).
T = (_G123>_G123) .
? type([], E, A>A).
E = lam(_G234,var(_G234)) .
A functional specification would need several different functions for each functionality, which will inevitably cause duplication in the specification.
HM
type(C,var(X), T1) : first(X:T,C), instantiate(T,T1).
type(C,lam(X,E), A > B) : type([X:mono(A)C],E,B).
type(C,X $ Y, B ) : type(C,X,A > B), type(C,Y,A).
type(C,let(X=E0,E1), T ) : type(C,E0,A),
type([X:poly(C,A)C],E1,T).
first(K:V,[K1:V1Xs]) : K = K1, V=V1.
first(K:V,[K1:V1Xs]) : K\==K1, first(K:V, Xs).
instantiate(poly(C,T),T1) : copy_term(t(C,T),t(C,T1)).
instantiate(mono(T),T).
Type Constructor Polymorphism
(a.k.a. higherkinded polymorphism)
Haskell supports parametric polymorphism over type constructors as well as types, historically motivated to support the monad class, which classifies over type constructors, but useful for defining generic form of datatypes even when type classes aren't involved.
 Tree :: (* > *) > * > *
data Tree c a
= Leaf a  Leaf :: Tree c a
 Node (c (Tree c a))  Node :: (c(Tree c a)) > Tree c a
type BinTree a = Tree Pair a  two children on each node
type Pair t = (t,t)
type RoseTree a = Tree [] a  varying number of children
HM + TC Poly + Kind Poly
kind(KC,var(Z),K1) : first(Z:K,KC), instantiate(K,K1).
kind(KC,F $ G, K2) : kind(KC,F,K1 > K2), kind(KC,G,K1).
kind(KC,A > B,o) : kind(KC,A,o), kind(KC,B,o).
type(KC,C,var(X), T1) : first(X:T,C), inst_type(KC,T,T1).
type(KC,C,lam(X,E), A>B) : type(KC,[X:mono(A)C],E,B),
kind(KC,A>B,o).
type(KC,C,X $ Y, B) : type(KC,C,X,A>B), type(KC,C,Y,A).
type(KC,C,let(X=E0,E1),T) : type(KC,C,E0,A),
type(KC,[X:poly(C,A)C],E1,T).
% first and instantiate are same as before in HM
% inst_type does what instantiate does but adds more
% kind assertions for freshly instantiated variables
We wanted to write a specification like this.
Unfortunately, this does not work
because Prolog is not perfect ...
STLC + Kind
kind(KC,var(Z),K) : first(Z:K,KC).
kind(KC,F $ G,K2) : kind(KC,F,K1 > K2), kind(KC,G,K1).
kind(KC,A > B,o) : kind(KC,A,o), kind(KC,B,o).
type(KC,C,var(X), T) : first(X:T,C).
type(KC,C,lam(X,E), A>B) : type(KC,[X:AC],E,B),
kind(KC,A>B,o).
type(KC,C,X $ Y, B) : type(KC,C,X,A>B), type(KC,C,Y,A).
? type([], [], lam(x,var(x)), A>A).
ERROR: Out of local stack
Exception: (1,456,711) kind([], _G8740235, (_G8740238>
_G8740232>_G8740226>_G8740220> ... > ...)) ? abort
% Execution Aborted
HM + TC Poly + Kind Poly
A twist to make type & kind inference work in Prolog
using Definite Clause Grammar (DCG) rues as
a neat syntax for a writer monad that collects
kind assertions as a sideoutput to be called upon later.
kind(KC,var(Z),K1) : first(Z:K,KC), instantiate(K,K1).
kind(KC,F $ G, K2) : kind(KC,F,K1 > K2), kind(KC,G,K1).
kind(KC,A > B,o) : kind(KC,A,o), kind(KC,B,o).
type(KC,C,var(X), T1) > first(X:T,C), inst_type(KC,T,T1).
type(KC,C,lam(X,E), A>B) > type(KC,[X:mono(A)C],E,B),
[ kind(KC,A>B,o) ]. % delayed goal
type(KC,C,X $ Y, B) > type(KC,C,X,A>B), type(KC,C,Y,A).
type(KC,C,let(X=E0,E1),T) > type(KC,C,E0,A),
type(KC,[X:poly(C,A)C],E1,T).
HM + TC Poly + Kind Poly
? phrase(type([],[],lam(x,var(x)), Gs).
T = (_G2662>_G2662),
Gs = [kind([], (_G2662>_G2662), o)] .
Freeze the Prolog variables in types into
variables for kind inference and also
extend the kinding context with them
? kind([t1:mono(K1)], (var(t1)>var(t1)), o).
K1 = o
For further details (including pattern matching, Mendlerstyle recursion)
see
1st stage of
type inference
2nd stage of
kind inference
preprocessing for 2nd stage
Lessons
 Executable Relational Specifications are writable in Prolog
(pure) logic programming + extra logical operations
 Hacking in Prolog is fun experiment but not ideal as a tool
We need to design and implement TIPER
 Essential extralogical features
 variable test/comparison
 fresh variable/atom generation
 Undesirable extralogical features
 use of metainterpreter
Used for certain purposes such as handling delayed goals, but it is too powerful, not intending to support in TIPER
Contributions
 This work is a pioneering case study on the subject matter of how relational specification of type systems can be practically beneficial.
 we were able to obtain reference implementation of
advanced type systems directly from the specifications
 There are surprisingly few executable specifications of polymorphic type systems in LP/ATP.
 few for HM (type polymorphism)
 wasn't able to find any beyond type polymorphsm
 Majority of related examples in ATP/ITP systems were about type inference algorithms, but not type system specification. There can be many variations of type inference algorithms for the same type system.
What's next? TIPER
Type Inference Prototyping Engine
from Relational specifications
of type systems
 Continue experimenting with Prolog on various features
 to research what functionality is needed in TIPER
 Code generation for a target language of choice (Haskell)
 build TIPERu using microKanren as a backend
microKanren is an eDSL for LP, being ported to increasing number of host languages including Haskell
 build TIPERu using microKanren as a backend
 Study Sresolution in the context of TIPER development
 also develop an eDSL for Sres based LP
 Change TIPERs backend to Sresolution based eDSL
Executable Relational Specifications of Polymorphic Type Systems using Prolog
By 안기영 (Ahn, Ki Yung)
Executable Relational Specifications of Polymorphic Type Systems using Prolog
Talk @ Dundee, UK 20150828
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