title

abstract

`mist`: Artifact for the ECOOP21 Paper _Refinements of Futures Past_

`mist` is a tiny language for teaching and experimenting with refinement types, in the style of [LiquidHaskell](https://github.com/ucsd-progsys/liquidhaskell). We use it as a platform for experimenting with and as a demonstration of implicit refinement types as presented in the ECOOP21 paper _Refinements of Futures Past: Higher-Order Specification with Implicit Refinement Types_. We start with the parser and AST we use to teach our undergradute compilers class, and layer upon it a refinement type checker directly translated from the typing rules presented in that paper, which produces constraints that are solved with the `liquid-fixpoint` horn clause solver. We present source code and binaries for `mist` in a container image that includes installations of the competing tools we compare to.

Initial build, install, and running all tests

You can use the Docker image or install mist manually. The Docker image also includes the tools we compare against.

Using `docker`

Windows and Mac users: Make sure your docker container has at least 4GB of RAM

The following command will download an image containing mist, fstar, and mochi, run the full mist test suite, and then drop you into an interactive shell at the root of the mist code repository.

$ docker run -it atondwal/mist

If you want to skip the test suite, instead run

$ docker run -it atondwal/mist /bin/bash

You can then (re)run all of the tests in the tests/ directory (perhaps after editing some) at any time by running

$ stack test

Juggling containers

You can use docker ps to see the running container and open another shell to it using docker exec, e.g.:

$ docker ps
CONTAINER ID      IMAGE             STATUS            NAMES
696b2221e3ad      atondwal/mist     Up 45 seconds     vibrant_leavitt
$ docker exec -it vibrant_leavitt bash
ecoop21@696b2221e3ad:~/mist$

You can use docker start to restart exited containers

$ docker ps -a
CONTAINER ID      IMAGE             STATUS                     NAMES
696b2221e3ad      atondwal/mist     Exited (137) 5 seconds ago vibrant_leavitt
$ docker start vibrant_leavitt
vibrant_leavitt
$ docker exec -it vibrant_leavitt bash
ecoop21@696b2221e3ad:~/mist$

Manually

You'll need git, z3 version 4.8.10, and stack.

$ git clone -b ecoop21 --recursive https://github.com/ucsd-progsys/mist
$ cd mist
$ stack install
$ export PATH=$HOME/.local/bin/:$PATH

You can then run the full mist test suite (which is located in the tests/ directory).

$ stack test

Running specific tests

You can run a specific test by calling mist on the test file, e.g.

$ mist tests/pos/incrState.hs

If you're using the docker image, you can also run tests for fstar:

$ fstar fstar-tests/incrState.fst

Benchmarks from the paper

Here's a table of where you can find each of the tests described in the paper:

Name	Mist test (tests/pos/)	Mochi (mochi-tests/)	Fstar (fstar-tests/)
incr	incr00.hs	incr00.ml	incr.fst
sum	sum.hs	sum.ml	sum.fst
repeat	repeat.hs	repeat.ml	x
d2	mochi-app-lin-ord2.hs	d2.ml	mochi-d2.fst

incrState	incrStatePoly.hs	incrState.ml	incrState.fst
accessControl	acl.hs	acl.ml	accessControl.fst
tick	tick-append.hs	x	tick.fst
linearDSL	linearTypes.hs	x	linearDSL.fst

pagination	paginationTokens.hs	x	x
login	idr_login.hs	x	x
twophase	twoPhaseCommit.hs	x	x
ticktock	ticktock3.hs	x	x
tcp	tcp_client.hs	x	x

As in the paper, an x indicates that the specification cannot be directly expressed with that tool.

Unfortunately, the version of MoCHi we compare against was made available as a web demo that is no longer functional. Since then, there has been a source release of MoCHi, but it does not support the -relative-complete verification mode that we compared against in our paper. We include a build of the latest version of MoCHi anyways, in case you want to play with it yourself and get an idea of how it works:

$ mochi mochi-tests/incrState.ml

A quick tutorial in writing mist

A note about UX: We demonstrate the ability of our type system to localize error messages in this prototype, but when it comes to the parser, we favor an easy to modify and understand grammar over one that provides the best user experience. As such...

When experimenting with mist, we recommend starting with one of the known working test cases, and then expanding on it to achieve the desired result, rather than starting from scratch in an empty text file. In this short tutorial we will take the same approach, starting from a minimal test case and building up to the pagination example from the ECOOP21 paper that demonstrates both implicit refinement function types and pair types.

How to read this tutorial

We recommend reading the pdf version of this tutorial as it is the easiest to read, but we also recommend keeping open a copy of the markdown source (~/mist/README.md) in your text editor as you follow along, as the markdown source includes the location of each snippet as range of lines in a test file, so you can open, edit, and rerun those tests yourself.

We recommend running a continuous build in a terminal while you experiment with a mist file. For example, this sets up entr to run mist on a file every time it gets written (whether or not it's a mist file).

$ find tests | entr mist /_
(in another window - see above for docker instructions)
$ vim tests/.../mytest.hs

Bits of syntax that are potential sources of confusion or frustration are highlighted in grey boxes. If you're struggling to make your code parse, checking to see if you've stepped on one of these Legos is a good place to start.

Refinement Types

We start from an extremely simple example that demonstrates the concrete semantics of mist's refinement type system.

twelve :: { v : Int  | v == 12 }
twelve = 12

Here, we have a top-level binder for the constant int. Each top level binder includes a type signature (line 1), and a body (line 2). The body of int simply states that it's equal to the integer constant 12. This type signature is a minimal example of a refinement type: we refine the base type Int, binding its values to v, and taking the quotient of this type by the proposition v == 12. This results in a singleton type that checks against the body of twelve.

$ mist tests/pos/Int00.hs
SAFE

If we had used a different value in the type and body:

twelve :: { v : Int  | v == 14 }
twelve = 12

We'd see a type error:

$ mist tests/neg/Int01.hs
Working 150% [=================================================================]
Errors found!
tests/neg/Int01.hs:2:7-9: Expected (VV##0 == 14) :

         2|  int = 12
                   ^^^

Functions and polymorphism

We can extend this to writing functions in mist:

incr :: x:Int -> {v:Int | v == x + 1}
incr = \x -> x + 1

moo :: {v:Int | v == 8}
moo = incr 7

This program checks that incrementing 7 results in 8.

Here, the binder x:Int binds x in the type on the right-hand side of ->. Similarly, at the value level, \ denotes a lambda.

Functions can also be polymorphic:

id :: rforall a. {v:a | True} -> {v:a | True}
id = \x -> x

bar :: {v:Int | v == 8}
bar = incr (id 7)

$ mist tests/pos/Inc02.hs
SAFE

All function applications that are not directly under a binder or a function abstraction should be enclosed in parentheses.

Here, rforall denotes that the function id is refinement polymorphic in the type variable a. That is, a stands in for any refined type, so we know that the result of applying id to any value will always result in a value of the same refinement type; i.e. one for which all the same propositions are true. The only function of this type is id.

Later we will also see forall, which allows functions to be polymorphic over base types.

Implicit function types

We're ready for our first example of a feature introduced in this paper! We write an implicit function type the same way as a normal function, but using the squiggly arrow ~> instead of the straight arrow ->:

incr :: n:Int ~> (Int -> { v : Int | v == n }) -> { v : Int | v == n + 1 }
incr = \f -> (f 0) + 1

test1 :: { v : Int | v == 11 }
test1 = incr (\x -> 10)

test2 :: m:Int -> { v : Int | v == m+1 }
test2 = \mv -> incr (\x -> mv)

$ mist tests/pos/incr00.hs
SAFE

Note the parentheses around (f 0) --- there are no precedence rules for infix primitives.

Given a constant function, incr increments the result. This is straightforward at the value level, but encoding it at the type level requires the use of implicit parameters. Here, n in bound at the type level, but has no corresponding binder at the value level in the surface syntax. The body of the function must typecheck for all values of n, but each call to the function need only be valid for some particular choice of n. n is picked at the call site by the implicit instantiation algorithm for refinement types described in the paper, such that the function application typechecks.

Here, for the call to incr on line 5 inside test1, n takes on the value 10, and on line 8, it takes on the value mv.

Datatypes, axioms, and measures

Mist supports user-defined datatypes by axiomatising their constructors. In this section we're going to demonstrate specification and verification with the List datatype, which in Haskell one might write:

data List a = Nil | Cons a (List a)

Datatypes

In mist, List a is spelled List >a

There are two things of note here:

As in Haskell, Mist datatypes are written in TitleCamelCase.
Unlike Haskell, datatypes carry variance annotations with them that tell you if they're co- or contra-variant in a given argument. Having these around can be helpful when you're debugging or reading code with complex subtyping relationships.

Here, the variance annotation > indicates that a appears covariantly in List (that is, List contains things that are subtypes of a). If it appeared contravariantly, we would have written List <a (a List of supertypes of a). Or in other words, a List >a behaves like a List that might produce as when it is used up, whereas a List <a behaves as a list that might consume as to use up.

This notation is intended to evoke a function arrow ->: Just as you can use a function that returns any subtype of the type you need, and that accepts any supertype of the arguments you have, if you're a type variable on the pointy end of the variance annotation (or function arrow) you're a covariant type variable, and if you're on the other end you're contravariant.

If you try to pass a List >a as a List <a, that is a (base/unrefined) type error --- variance annotations are essentially a part of the type name.

You do not need to declare datatypes before using them in type signatures.

Some such datatypes (Int, Bool, Set, and Map) have special meaning when used in types, as they come with primitives (such as +, which we saw above) that have meaning to the solver's theories of arithmetic, sets, maps, etc.

Axioms

Mist relies on axioms to introduce data constructors. An axiom in Mist is written with "as" (assumed types) instead of "::" (checked types):

exFalsoQuodlibet as rforall a. False -> a
exFalsoQuodlibet = ...

Whatever we put for ... is taken to be the witness of the axiom, and executed when the axiom is used in code that is run.

You need to provide a body for every binding, axioms or otherwise.

To use the List datatype, we need constructors, and projections from these constructor (or induction principles, but let's keep it simple for the tutorial). To introduce axioms for each of these, we write something like

nil as forall a. List >a
nil = ...
cons as forall a. a -> List >a -> List >a
cons = ...
first as forall a. List >a -> a
first = ...
rest as forall a. List >a -> List >a
rest = ...

where ... can be e.g. Boehm-Beraraducci (1985) encoding of constructors and projection operators, but since we're focused on testing the typechecker here, we generally set them equal to 0 as the witnesses to axioms don't matter so far as the typechecker is concerned.

We can use axiomatized constructors to define a datatype Lin which is the type of terms of a linear DSL. Here, we use Set primitives.

var as x:Int -> (Lin >{v:Set >Int | v = setPlus emptySet x})

var constructs a term that is a variable mention. It checks that the variable is in the environment. $\frac{x \in \Gamma}{\Gamma \vdash \texttt{var } x}$

fun as env:(Set >Int) ~> n:{v:Int | (v ∈ env) ≠ True} -> (Lin >{v:Set >Int | v = setPlus env n}) -> (Lin >{v:Set >Int | v = env})

fun constructs a lambda term, while checking that the variable it binds is fresh. $\frac{x\not\in\Gamma \quad \Gamma \cup {x} \vdash e}{\Gamma \vdash \texttt{fun } x, e}$

It may not always be obvious when you need parenthesis around type constructor applications. Some rules of thumb: parenthesize them on both sides of ->, but in the left-hand side of a refinement type (the binder), "(" may not follow ":"

Note that the not-equals operator is the unicode symbol, not a multi-character sigil --- infix operators in Mist are single characters.

app as env1:(Set >Int) ~> env2:{v:Set >Int | env1 ∩ v = emptySet} ~> (Lin >{v:Set >Int | v = env1}) -> (Lin >{v:Set >Int | v = env2}) -> (Lin >{v:Set >Int | v = env1 ∪ env2})

app applies a function to a value, checking that no variable is used more than once in either argument. $\frac{\Gamma_1 \cap \Gamma_2 = \emptyset\quad \Gamma_2 \vdash e \quad \Gamma_1 \vdash f}{\Gamma_1 \cup \Gamma_2\vdash \texttt{app } f, e}$

typecheck as (Lin >{v:Set >Int | v = emptySet}) -> (Lin >(Set >Int))

typecheck simply checks that a term is closed $\frac{\emptyset\vdash e}{\vdash e}$

program2 :: Lin >(Set >Int)
program2 = typecheck (fun 1 (fun 2 (app (var 1) (var 2))))

$ mist tests/pos/linearTypes.hs
SAFE

State

We can define a State Monad datatype! See section 2.3 of the paper for more explanation. put takes a world (called wp), put updates the state to one where the state of the world is now wp.

put as wp:Int -> ST <Int >{p:Int|p==wp} >Unit

get takes a boolean and ignores it, then leaves the state of the world unchanged, but returns its value in the ST monad.

get as wg:Int ~> Bool -> ST <{gi:Int|gi==wg} >{go:Int|go==wg} >{gr:Int|gr==wg}

And then we have the standard monadic interface:

-- Monadic Interface
ret as rforall a. wr:Int ~> x:a -> ST <{ri:Int|ri==wr} >{ro:Int|ro==wr} >a

bind as rforall a, b. w1:Int ~> w2:Int ~> w3:Int ~> (ST <{v:Int|v==w1} >{v:Int|v==w2} >a)
  -> (unused:a -> ST <{v:Int|v==w2} >{v:Int|v==w3} >b)
  -> ST <{v:Int|v==w1} >{v:Int|v==w3} >b

Using this, we can verify a more stateful version of the incr example from before.

incr :: i:Int ~> ST <{v:Int|i==v} >{w:Int|w==i+1} >Unit
incr = bind (get True) (\x -> put (x+1))

$ mist tests/pos/incrState.hs
SAFE

Going forward, however, we're going to use a more polymorphic definition of state:

bind :: rforall a, b, p, q, r.
  ST <p >q >a ->
  (x:a -> ST <q >r >b) ->
  ST <p >r >b
bind = undefined

pure :: rforall a, p. x:a -> ST <p >p >a
pure = undefined

thenn :: rforall a, b, p, q, r.
  ST <p >q >a ->
  ST <q >r >b ->
  ST <p >r >b
thenn = \f g -> bind f (\underscore -> g)

fmap :: rforall a, b, p, q.
  (underscore:a -> b) ->
  ST <p >q >a ->
  ST <p >q >b
fmap = \f x -> bind x (\xx -> pure (f xx))

If a function type signature is failing to parse, try assigning a name to the argument (e.g. x:Int -> instead of Int ->).

Implicit pair types

Next, we demonstrate how to use another core feature unique to Mist: implicit pair types as described in the paper. In the paper the syntax is $[n:Int].\text{---}$, but in the implementation, we use exists n. -. Consider iterating through an infinite stream of token handlers. The API for getting the next token is:

nextPage as
  token:{v: Int | v ≠ done} ->
  (exists tok:Int.
    (ST <{v:Int | v = token}
        >{v:Int | (v = tok) /\ (v ≠ token)}
        >{v:Int | v = tok}))

Remember to parenthesize both sides of conjunctions (/\)!

That is, given a token, nextPage give you a state action where it picks a new token that's not equal to the old token, and updates the state of the world to reflect the new token.

client :: token:Int ->
  (ST <{v:Int | v = token} >{v:Int | v = done} >Int)
client = \token ->
  if token == done
  then pure 1
  else bind (nextPage token) (\tok -> client tok)

$ mist tests/pos/paginationTokens.hs
SAFE

And that concludes our short tutorial on mist. Go forth and verify!

\appendix

Appendix : Measures

But these List constructors are all a bit boring --- what good are user datatypes if we can't say anything about them at the type level?! Until now, we've only been able to form refinements out of variables and primitive functions such as + and ∩ on special types such as Int and Set. We use (purely) refinement-level functions called measures (Vazou et al, ICFP 2014) to extend the language of refinements and enrich the types of our constructors.

measure mNil :: List [>Int] -> Bool

Measures use a different syntax for types --- type applications have a list of parameters in [square, brackets, separated, by, commas], unlike applications of type constructors that produce refinement types, which use the usual space-separated syntax.

This declares a measure mNil that takes a List of Ints and returns a Bool. Measure have unrefined types.

If you get an error message about free vars that implicates the start of the file, you probably tried to use a measure you didn't declare. This will cause the solver print a list of unbound measures and crash with some mumbo-jumbo about --prune-unsorted. Measures always go at the top of the file.

We can use these measures in constructor axioms to effectively define structurally recursive functions over a datatype.

nil as {v: List >Int | (mNil v) /\ (mLength v = 0) /\ (not (mCons v))}

cons as x:Int -> xs:(List >Int) -> {v: List >Int | (mCons v) /\ (mLength v = mLength xs + 1) /\ (not (mNil v))}

first as {v: List >Int | mCons v} -> Int

rest as rs:{v: List >Int | mCons v} -> {v: List >Int | mLength v + 1 == mLength rs }

and we can then use them in verification!

append :: xs:(List >Int) -> ys:(List >Int) -> {v: List >Int | mLength v = (mLength xs) + (mLength ys)}
append = \xs -> \ys ->
  if empty xs
    then ys
    else cons (first xs) (append (rest xs) ys)

$ mist tests/pos/recursion.hs
SAFE

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README.md

README.md

Initial build, install, and running all tests

Using `docker`

Juggling containers

Manually

Running specific tests

Benchmarks from the paper

A quick tutorial in writing mist

How to read this tutorial

Refinement Types

Functions and polymorphism

Implicit function types

Datatypes, axioms, and measures

Datatypes

Axioms

State

Implicit pair types

Appendix : Measures

Files

README.md

Latest commit

History

README.md

File metadata and controls

Initial build, install, and running all tests

Using docker

Juggling containers

Manually

Running specific tests

Benchmarks from the paper

A quick tutorial in writing mist

How to read this tutorial

Refinement Types

Functions and polymorphism

Implicit function types

Datatypes, axioms, and measures

Datatypes

Axioms

State

Implicit pair types

Appendix : Measures

Using `docker`