Benjamin Redelings benjamin.redelings at gmail.com
Thu Jan 11 16:27:47 UTC 2018

```Hi,

0.  Does anyone know of any simple extensions of the HM type system to
non-deterministic functions?  The reason that I'm asking is that for
probabilistic programming in the lambda calculus, there are two ways of
writing expressions:

(a) stochastic: let x = sample \$ normal 0 1 in x*x

or simply (sample \$ normal 0 1)^2

(b) "mochastic": do {x <- normal 0 1; return (x*x)}

The "mo" in the second one refers to the use of monads.  That is the
approach taken in the paper "Practical Probabilistic Programming with
really enjoyed.

However, I am interested in the stochastic form here.  There are a
number of reasons, such as the fact that the monadic representation
forces serialization on things that need not be serial.  In fact,
though, I'm not trying to prove which one is best, I am just interested
in exploring the non-monadic approach as well.

1. So, is it possible to do a simple extension to the type system to
express non-determinism?  I found this paper (Implicit Self-Adjusting
Computation for Purely Functional Programs) that uses "level" tags on
types to express either (i) security or (ii) changeability.  The first
idea (for example) is that each type is tagged with one of two "levels",
say Public and Secure, so that we actually have Int[Public] or
Int[Secure].  Any function that consumes a Secure value must (i) must
return a Secure type and (ii) has the arrow in its type labelled with
[Secure].  I won't explain the "changeable" idea because its kind of
complicated, but I am very interested in it.

2. This is kind of tangential to the point of my question, but to
explain the examples below, it might be important to distinguish
sampling from a distribution from the distribution itself.  So, normal 0
1 won't generate a random sample.  Instead, normal 0 1 () will generate
a random sample.  This allows us to pass (normal 0 1) to another
function which applies it multiple times to generate multiple samples
from the same distribution.

-- sample from a distribution dist
sample dist = dist ()

--- take n samples from a distribution dist
iid n dist = take n (map sample \$ repeat dist)

Here we see some of the value of using the stochastic approach, versus
the "mochastic" approach: we can use normal Haskell syntax to handle
lists of random values!

3. So, I'm wondering if its possible to extend the HM type system to
handle non-determinism in a similar fashion by either (i) having some
function types be non-deterministic and/or (ii) having term types be
non-deterministic.  Taking the second approach, I suggest tagging each
type with level [D] (for deterministic) or [N] (for non-deterministic).
Notation-wise, if a determinism level is unspecified, then this means (I
think) quantifying over determinism levels.  A function that samples
from the normal distribution we would get a type like:

normal:: double -> double -> () -> double[N]

Our goal would be that an expression that consumes a non-deterministic
expression must itself be non-deterministic, and any function that takes
a non-deterministic input must have a non-deterministic output.   We
could implement that using rules something like this, where {a,b} are
type variables and {l1,l2} are level variables.

x:a[l1] :: a[l1]
\x:a[l1] -> E:b[l2]  :: a[l1] -> b[max l1 l2]
E[a[l1]->b[l2]] E[a[l]] :: b[l2]

The idea is that max l1 l2 would yield N (non-deterministic) if either
l1=N or l2=N, because N > D.

4. Putting non-determinism into the type system would affect GHC in a
few ways:

(a) we shouldn't pull non-deterministic expressions out of lambdas:

We should NOT change
\x -> let y=sample \$ normal 0 1 in y+x
into
let y = sample \$ normal 0 1 in \x -> y+x

(b) we should merge variables with identical values if the types are
non-deterministic.

For example it is OK to change
let {x=normal 0 1; y = normal 0 1 in (sample x * sample y)}
into
let {x=normal 0 1} in sample x

However it is NOT OK to change
let {x=sample \$ normal 0 1; y = sample \$ normal 0 1} in x*y
into
let {x=sample \$ normal 0 1} in x*x

Perhaps this would be useful in other contexts?

5. If what I've written makes sense, then the types of the functions
'sample' and 'iid' would be:

sample:: (()->a[N]) -> a[N]

iid:: Int -> (() -> a[N]) -> [a[N]]

6.  This is quite a long e-mail, so to summarize, I am interested in
whether or not there are any simple systems for putting non-determinism
into HM.  Is the use of tagged types known NOT to work?  Is there are
work on this that I should be aware of?

Any help much appreciated! :-)

take care,

-BenRI

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