In this post I will introduce a proof-of-concept implementation for distributing typed code over the internet where the unit of compilation is individual expressions.

#### The core language

To motivate this post, consider this Haskell code:

```
data Bool = True | False
and :: Bool -> Bool -> Bool
and b1 b2 = if b1 then b2 else False
or :: Bool -> Bool -> Bool
or b1 b2 = if b1 then True else b2
data Even = Zero | SuccE Odd
data Odd = SuccO Even
four :: Even
four = SuccE (SuccO (SuccE (SuccO Zero)))
doubleEven :: Even -> Even
doubleEven (SuccE o) = SuccE (SuccO (doubleOdd o)
doubleEven Zero = Zero
doubleOdd :: Odd -> Even
doubleOdd (SuccO e) = SuccE (SuccO (doubleEven e)
```

I will encode each one of the above types, terms, and constructors as separate, closed, non-recursive expressions in the calculus of constructions. You can think of the calculus of constructions as a typed assembly language for functional programs which we will use to distribute program fragments over the internet. You can learn more about the calculus of constructions and other pure type systems by reading this clear paper by Simon Peyton Jones: "Henk: a typed intermediate language".

For example, here is how you encode the `True`

constructor in the calculus of constructions:

`λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True`

Note that the entire expression is the `True`

constructor, not just the right-hand side:

```
This is the True constructor
vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True
^^^^
Not this
```

I just chose the variable names so that you can tell at a glance what constructor you are looking at from the right-hand side of the expression.

Similarly, here is how you encode the type `Bool`

in the calculus of constructions:

```
This is the `Bool` type
vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
^^^^
Not this
```

Again, the entire expression is the `Bool`

type, but I chose the variable names so that you can tell which type you are looking at from the right-hand side.

You can learn more about the full set of rules for translating data types to System F (a subset of the calculus of constructions) by reading the paper: "Automatic synthesis of typed Λ-programs on term algebras". Also, I will soon release a compiler named `annah`

that automates this translation algorithm, and I used this compiler to translate the above Haskell code to the equivalent expressions in the calculus of constructions.

#### Distribution

We can distribute these expressions by hosting each expression as text source code on the internet. For example, I encoded all of the above types, terms and constructors in the calculus of constructions and hosted them using a static web server. You can browse these expressions by visiting sigil.place/post/0/.

Click on one of the expressions in the directory listing to see how they are encoded in the calculus of constructions. For example, if you click the link to `four`

you will find an ordinary text file whose contents look like this (formatted for clarity):

```
λ(Even : *) -- This entire
→ λ(Odd : *) -- expression is
→ λ(Zero : Even) -- the number `four`
→ λ(SuccE : ∀(pred : Odd) → Even) --
→ λ(SuccO : ∀(pred : Even) → Odd) --
→ SuccE (SuccO (SuccE (SuccO Zero))) -- Not just this last line
```

Each one of these expressions gets a unique URL, and we can embed any expression in our code by referencing the appropriate URL.

#### Remote imports

We can use the `morte`

compiler to download, parse, and super-optimize programs written in the calculus of constructions. The morte compiler reads in a program from standard input, outputs the program's type to standard error, then super-optimizes the program and outputs the optimized program to standard output.

For example, we can compute `and True False`

at compile time by just replacing `and`

, `True`

, and `False`

by their appropriate URLs:

```
$ cabal install 'morte >= 1.2'
$ morte
#http://sigil.place/post/0/and
#http://sigil.place/post/0/True
#http://sigil.place/post/0/False
```

When we hit `<Ctrl-D>`

to signal end of standard input then `morte`

will compile the program:

```
$ morte
#http://sigil.place/post/0/and
#http://sigil.place/post/0/True
#http://sigil.place/post/0/False
<Ctrl-D>
∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False
```

The program's type is `Bool`

and `morte`

optimizes away the program to `False`

at compile time. Both the type (`Bool`

) and the value (`False`

) are encoded in the calculus of constructions.

Here we are using `morte`

as a compile-time calculator, mainly because Morte does not yet compile to a backend language. When I release a backend language I will go into more detail about how to embed expressions to evaluate at runtime instead of compile time.

#### Local imports

We can shorten this example further because `morte`

also lets you import expressions from local files using the same hashtag syntax. For example, we can create local files that wrap remote URLs like this:

```
$ echo "#http://sigil.place/post/0/Bool" > Bool
$ echo "#http://sigil.place/post/0/True" > True
$ echo "#http://sigil.place/post/0/False" > False
$ echo "#http://sigil.place/post/0/or" > or
```

We can then use these local files as convenient short-hand aliases for remote imports:

```
$ morte
#or #True #False
<Ctrl-D>
∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True
```

We can use imports anywhere in our program, even as types! For example, in the calculus of constructions you encode `if`

as the identity function on `#Bool`

s:

`λ(b : #Bool ) → b # Note: Imports must end with whitespace`

We can then save our implementation of `if`

to a file named `if`

, except using the ASCII symbols `\`

and `->`

instead of `λ`

and `→`

:

`$ echo "\(b : #Bool ) -> b" > if`

Now we can define our own `and`

function in terms of `if`

. Remember that the Haskell definition of `and`

is:

`and b1 b2 = if b1 then b2 else False`

Our definition won't be much different:

`$ echo "\(b1 : #Bool ) -> \(b2 : #Bool ) -> #if b1 #Bool b2 #False" > and`

Let's confirm that our new `and`

function works:

```
$ echo "#and #True #False" | morte
∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False
```

We can also ask `morte`

to resolve all imports for our `and`

function and optimize the result:

```
$ morte < and
∀(b1 : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
→ ∀(b2 : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
→ ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
λ(b1 : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
→ λ(b2 : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
→ b1 (∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
b2
(λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False)
```

We can then compare our version with the `and`

expression hosted online, which is identical:

```
$ curl sigil.place/post/0/and
λ(b1 : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
→ λ(b2 : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
→ b1 (∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
b2
(λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False)
```

#### Reduction

When we write an expression like this:

`#or #True #False`

The compiler resolves all imports transitively until all that is left is an expression in the calculus of constructions, like this one:

```
-- or
( λ(b1 : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
→ b1 (∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
(λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True)
)
-- True
(λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True )
-- False
(λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False)
```

Then the compiler reduces this expression using β-reduction and ε-reduction. We can safely reduce these expressions at compile time because these reductions always terminate in the calculus of constructions, which is a total and non-recursive language.

For example, the above expression reduces to:

```
( λ(b1 : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
→ b1 (∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
(λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True)
)
(λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True )
(λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False)
-- β-reduce
= (λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True )
(∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
(λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True)
(λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False)
-- β-reduce
= ( λ(True : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
→ λ(False : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
→ True
)
(λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True)
(λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False)
-- β-reduce
= ( λ(False : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
→ λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True
)
(λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False)
-- β-reduce
= λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True
```

#### Linking

The `and`

we defined is "dynamically linked", meaning that the file we saved has not yet resolved all imports:

```
$ cat and
\(b1 : #Bool ) -> \(b2 : #Bool ) -> #if b1 #Bool b2 #False
```

The `morte`

compiler will resolve these imports every time we import this expression within a program. To be precise, each import is resolved once per program and then cached and reused for subsequent duplicate imports. That means that the compiler only imports `#Bool`

once for the above program and not three times. Also, we can transparently cache these expressions just like any other web resource by providing the appropriate `Cache-Control`

HTTP header. For example, my static web server sets `max-age`

to a day so that expressions can be cached for up to one day.

If our imported expressions change then our program will reflect those changes, which may or may not be desirable. For the above program dynamic linking is undesirable because if we change the file `#False`

to point to sigil.place/post/0/True then we would break the behavior of the `and`

function.

Alternatively, we can "statically link" the `and`

function by resolving all imports using the `morte`

compiler. For example, I statically linked my remote `and`

expression because the behavior should never change:

```
$ curl sigil.place/post/0/and
λ(b1 : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
→ λ(b2 : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
→ b1 (∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
b2
(λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False)
```

In other scenarios you might want to dynamically link expressions if you want to automatically pull in upgrades from trusted upstream sources. This is the same rationale behind service-oriented architectures which optimize for transparent system-wide updates, except that instead of updating a service we update an expression.

#### Partial application

We can store partially applied functions in files, too. For example, we could store `and True`

in a statically linked file named `example`

using `morte`

:

```
$ echo "#and #True" | morte > example
∀(b2 : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool)
→ ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
```

The type still goes to standard error, but the partially applied function goes to the `example`

file. We can use the partially applied function just by referencing our new file:

```
$ morte
#example #False -- Same as: #and #True #False
<Ctrl-D>
∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → False
```

We can even view `example`

and see that it's still just an ordinary text source file:

```
$ cat example
λ(b2 : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) → b2
```

We can also see that `morte`

was clever and optimized `#and #True`

to the identity function on `#Bool`

s.

If we wanted to share our `example`

code with our friends, we'd just host the file using any static web server. I like to use Haskell's `warp`

server (from the `wai-app-static`

package) for static hosting, but even something like `python -m SimpleHTTPServer`

would work just as well:

```
$ cabal install wai-app-static
$ warp
Serving directory /tmp/code on port 3000 with ["index.html","index.htm"] index files.
```

Then we could provide our friends with a URL pointing to the `example`

file and they could embed our code within their program by pasting in our URL.

#### Types

The calculus of constructions is typed, so if you make a mistake, you'll know immediately:

```
$ morte
#True #True
<Ctrl-D>
Expression:
(λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True)
(λ(Bool : *) → λ(True : Bool) → λ(False : Bool) → True)
Error: Function applied to argument of the wrong type
Expected type: *
Argument type: ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
```

Types are what differentiates `morte`

from a curl into sh. You can use the type system to whitelist the set of permissible values to import.

For example, in this code, there are only two values of `#x`

that will type-check (up to α-conversion):

`(λ(b : ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool) → b) #x`

Therefore, we can safely import a remote value knowing that the type-checker will reject attempts to inject arbitrary code.

When building a program with effects, we can similarly refine the set of permissible actions using the types. I introduced one such example in my previous post on `morte`

, where the `recursive.mt`

program restricts the effects to reading and printing lines of text and nothing else. You could then import a remote expression of type:

```
∀(String : *)
→ ∀(U : *)
→ ∀(Unit : U)
→ ∀(IO : *)
→ ∀(GetLine : String → IO → IO)
→ ∀(PutStrLn : (String → IO) → IO)
→ ∀(Return : U → IO)
→ IO
```

... which is the type of an effect syntax tree built from `GetLine`

/`PutStrLn`

/`Return`

constructors. The type-checker will then enforce that the imported syntax tree cannot contain any other constructors and therefore cannot be interpreted to produce any other effects.

#### Recursive data types

You can encode recursive data types and functions in the calculus of constructions. This is all the more amazing when you realize that the calculus of constructions does not permit recursion! Also, `morte`

's import system forbids recursion as well. If you try to recurse using imports you will get an error:

```
$ echo "#foo" > bar
$ echo "#bar" > foo
$ morte < foo
morte:
⤷ #bar
⤷ #foo
Cyclic import: #bar
```

Joe Armstrong once proposed that the core language for an internet of code would require built-in support for recursion (via `letrec`

or something similar), but that's actually not true! The paper "Automatic synthesis of typed Λ-programs on term algebras" spells out how to encode recursive data types in the non-recursive System F language. What's amazing is that the algorithm works even for mutually recursive data types like `Even`

and `Odd`

from our original Haskell example.

You don't have to take my word for it! You can verify for yourself that the `Even`

and `Odd`

types and the `Zero`

/`SuccE`

/`SuccO`

constructors that I hosted online are not recursive:

- sigil.place/post/0/Even
- sigil.place/post/0/Odd
- sigil.place/post/0/Zero
- sigil.place/post/0/SuccE
- sigil.place/post/0/SuccO

Let's create local aliases for the constructors so we can built our own `Even`

or `Odd`

values:

```
$ echo "#http://sigil.place/post/0/Zero" > Zero
$ echo "#http://sigil.place/post/0/SuccE" > SuccE
$ echo "#http://sigil.place/post/0/SuccO" > SuccO
```

We can then assemble the number `four`

using these constructors:

```
$ morte
#SuccE (#SuccO (#SuccE (#SuccO #Zero )))
<Ctrl-D>
∀(Even : *)
→ ∀(Odd : *)
→ ∀(Zero : Even)
→ ∀(SuccE : ∀(pred : Odd) → Even)
→ ∀(SuccO : ∀(pred : Even) → Odd)
→ Even
λ(Even : *)
→ λ(Odd : *)
→ λ(Zero : Even)
→ λ(SuccE : ∀(pred : Odd) → Even)
→ λ(SuccO : ∀(pred : Even) → Odd)
→ SuccE (SuccO (SuccE (SuccO Zero)))
```

The result is identical to the `four`

that I hosted:

```
$ curl sigil.place/post/0/four
λ(Even : *)
→ λ(Odd : *)
→ λ(Zero : Even)
→ λ(SuccE : ∀(pred : Odd) → Even)
→ λ(SuccO : ∀(pred : Even) → Odd)
→ SuccE (SuccO (SuccE (SuccO Zero)))
```

We can even encode functions over mutually recursive types like `doubleEven`

and `doubleOdd`

. You can verify that the ones I wrote are not recursive:

... and then we can test that they work by doubling the number `four`

:

```
$ morte
#http://sigil.place/post/0/doubleEven
#http://sigil.place/post/0/four
<Ctrl-D>
∀(Even : *)
→ ∀(Odd : *)
→ ∀(Zero : Even)
→ ∀(SuccE : ∀(pred : Odd) → Even)
→ ∀(SuccO : ∀(pred : Even) → Odd)
→ Even
λ(Even : *)
→ λ(Odd : *)
→ λ(Zero : Even)
→ λ(SuccE : ∀(pred : Odd) → Even)
→ λ(SuccO : ∀(pred : Even) → Odd)
→ SuccE (SuccO (SuccE (SuccO (SuccE (SuccO (SuccE (SuccO Zero)))))))
```

We get back the `Even`

number eight encoded in the calculus of constructions.

#### Stack traces

`morte`

will provide a "stack trace" if there is a type error or parse error:

```
$ echo "\(a : *) ->" > foo # This is a malformed program
$ echo "#foo" > bar
$ echo "#bar" > baz
$ echo "#baz" > qux
$ morte < qux
morte:
⤷ #qux
⤷ #baz
⤷ #bar
⤷ #foo
Line: 2
Column: 1
Parsing: EOF
Error: Parsing failed
```

You can learn more about how `morte`

's import system works by reading the newly add "Imports" section of the `morte`

tutorial.

#### Comparison to other software architectures

`morte`

's code distribution system most closely resembles the distribution model of Javascript, meaning that code can be downloaded from any URL and is compiled or interpreted client-side. The most important difference between the two is the granularity of imports and the import mechanism.

In `morte`

the unit of distribution is individual types, terms, and constructors and you can inject a remote expression anywhere in the syntax tree by referencing its URL. This is why we can do crazy things like use a URL for a type:

`λ(b : #http://sigil.place/post/0/Bool ) → ...`

The second difference is that `morte`

is designed from the ground up to be typed and highly optimizable (analogous to `asm.js`

, a restricted subset of Javascript designed for ease of optimization).

The third difference is that `morte`

lets you precisely delimit what remote code can do using the type system, unlike Javascript.

#### Future directions

This is just one piece of the puzzle in a long-term project of mine to build a typed and distributed intermediate language that we can use to share code across language boundaries. I want to give people the freedom to program in the language of their choice while still interoperating freely with other languages. In other words, I'm trying to build a `pandoc`

for programming languages.

However, this project is still not really usable, even in anger. There are several missing features to go, some of which will be provided by my upcoming `annah`

library:

**Requirement #1:** There needs to be a way to convert between restricted subsets of existing programming languages and the calculus of constructions

`annah`

currently provides logic to encode medium-level language abstractions to and from the calculus of constructions. In fact, that's how I converted the Haskell example at the beginning of this post into the calculus of constructions. For example, I used `annah`

to derive how to encode the `SuccE`

constructor in the calculus of constructions:

```
$ annah compile
type Even
data Zero
data SuccE (pred : Odd)
type Odd
data SuccO (pred : Even)
in SuccE
<Ctrl-D>
```

... and `annah`

correctly deduced the type and value in the calculus of constructions:

```
∀(pred : ∀(Even : *)
→ ∀(Odd : *)
→ ∀(Zero : Even)
→ ∀(SuccE : ∀(pred : Odd)
→ Even )
→ ∀(SuccO : ∀(pred : Even) → Odd) → Odd)
→ ∀(Even : *)
→ ∀(Odd : *)
→ ∀(Zero : Even)
→ ∀(SuccE : ∀(pred : Odd) → Even)
→ ∀(SuccO : ∀(pred : Even) → Odd)
→ Even
λ(pred : ∀(Even : *)
→ ∀(Odd : *)
→ ∀(Zero : Even)
→ ∀(SuccE : ∀(pred : Odd) → Even)
→ ∀(SuccO : ∀(pred : Even) → Odd)
→ Odd )
→ λ(Even : *)
→ λ(Odd : *)
→ λ(Zero : Even)
→ λ(SuccE : ∀(pred : Odd) → Even)
→ λ(SuccO : ∀(pred : Even) → Odd)
→ SuccE (pred Even Odd Zero SuccE SuccO)
```

Among other things, `annah`

automates the algorithm from "Automatic synthesis of typed Λ-programs on term algebras", which is known as "Böhm-Berarducci encoding".

**Requirement #2:** There must be an efficient way to transmit bytes, text, and numbers alongside code.

My plan is to transmit this information out-of-band as a separate file rather than embedding the data directly within the code and `annah`

will provide a systematic convention for distributing data and referencing that data within source code.

**Requirement #3:** There needs to be a standard library of types, data structures, functions, and side effects that all target languages must support.

In other words, there needs to be some sort of `thrift`

for code so that languages can maximize code sharing.

**Requirement #4:** There must be better tooling for mass installation and hosting of expressions.

For example, I'd like to be able to alias all imports within a remote directory to local files with a single command.

**Requirement #5:** I need to figure out a way to mesh type inference with an expression-level distribution system.

As far as I can tell this is still an open research problem and this is most likely going to be the greatest obstacle to making this usable in practice.

#### Resources

If you would like to learn more about Morte or contribute, then check out the following resources: