ChapelCon2025-Slides/type-level/slides.md

theme	title	backgroundColor
gaia	Type-Level Programming in Chapel for Compile-Time Specialization

Type-Level Programming in Chapel for Compile-Time Specialization

Daniel Fedorin, HPE

---

Compile-Time Programming in Chapel

• Type variables, as their name suggests, store types instead of values.

  type myArgs = (int, real);
  

• Procedures with type return intent can construct new types.

  proc toNilableIfClassType(type arg) type do
    if isNonNilableClassType(arg) then return arg?; else return arg;
  

• param variables store values that are known at compile-time.

  param numberOfElements = 3;
  var threeInts: numberOfElements * int;
  

• Compile-time conditionals are inlined at compile-time.

  if false then somethingThatWontCompile();
  

---

Restrictions on Compile-Time Programming

• Compile-time operations do not have mutable state.
  • Cannot change values of param or type variables.
• Chapel's compile-time programming does not support loops.
  • param loops are kind of an exception, but are simply unrolled.
  • Without mutability, this unrolling doesn't give us much.
• Without state, our type and param functions are pure.

---

Did someone say "pure"?

I can think of another language that has pure functions...

• ✅ Haskell doesn't have mutable state by default.
• ✅ Haskell doesn't have imperative loops.
• ✅ Haskell functions are pure.

---

Programming in Haskell

Without mutability and loops, (purely) functional programmers use pattern-matching and recursion to express their algorithms.

• Data structures are defined by enumerating their possible cases. A list is either empty, or a head element followed by a tail list.

  data ListOfInts = Nil | Cons Int ListOfInts
  
  -- [] = Nil
  -- [1] = Cons 1 Nil
  -- [1,2,3] = Cons 1 (Cons 2 (Cons 3 Nil))
  

• Pattern-matching is used to examine the cases of a data structure and act accordingly.

  sum :: ListOfInts -> Int
  sum Nil = 0
  sum (Cons i tail) = i + sum tail
  

---

Evaluating Haskell

Haskell simplifies calls to functions by picking the case based on the arguments.

sum (Cons 1 (Cons 2 (Cons 3 Nil)))

-- case: sum (Cons i tail) = i + sum tail
= 1 + sum (Cons 2 (Cons 3 Nil))

-- case: sum (Cons i tail) = i + sum tail
= 1 + (2 + sum (Cons 3 Nil))

-- case: sum (Cons i tail) = i + sum tail
= 1 + (2 + (3 + sum Nil))

-- case: sum Nil = 0
= 1 + (2 + (3 + 0))

= 6

---

A Familiar Pattern

Picking a case based on the arguments is very similar to Chapel's function overloading.

• A very familiar example:

  proc foo(x: int) { writeln("int"); }
  proc foo(x: real) { writeln("real"); }
  foo(1);   // prints "int"
  

• A slightly less familiar example:

  proc foo(type x: int) { compilerWarning("int"); }
  proc foo(type x: real) { compilerWarning("real"); }
  foo(int);   // compiler prints "int"
  

---

A Type-Level List

Hypothesis: we can use Chapel's function overloading and types to write functional-ish programs.

record Nil {}
record Cons { param head: int; type tail; }


type myList = Cons(1, Cons(2, Cons(3, Nil)));



proc sum(type x: Nil) param do return 0;
proc sum(type x: Cons(?i, ?tail)) param do return i + sum(tail);

compilerWarning(sum(myList) : string); // compiler prints 6

data ListOfInts = Nil
                | Cons Int ListOfInts


myList = Cons 1 (Cons 2 (Cons 3 Nil))


sum :: ListOfInts -> Int
sum Nil = 0
sum (Cons i tail) = i + sum tail

                                                                

---

Type-Level Programming at Compile-Time

After resolution, our original program:

record Nil {}
record Cons { param head: int; type tail; }

type myList = Cons(1, Cons(2, Cons(3, Nil)));

proc sum(type x: Nil) param do return 0;
proc sum(type x: Cons(?i, ?tail)) param do return i + sum(tail);

writeln(sum(myList) : string); // compiler prints 6

Becomes:

writeln("6");

There is no runtime overhead!

---

Type-Level Programming at Compile-Time

---

Type-Level Programming at Compile-Time

Why would I want to do this?!

- You, probably

• Do you want to write parameterized code, without paying runtime overhead for the runtime parameters?
  • Worked example: linear multi-step method approximator
• Do you want to have powerful compile-time checks and constraints on your function types?
  • Worked example: type-safe printf function

---

Linear Multi-Step Method Approximator

---

Type-Safe printf

---

The printf Function

The printf function accepts a format string, followed by a variable number of arguments that should match:

// totally fine:
printf("Hello, %s! Your ChapelCon submission is #%d\n", "Daniel", 18);

// not good:
printf("Hello, %s! Your ChapelCon submission is #%d\n", 18, "Daniel");

Can we define a printf function in Chapel that is type-safe?

---

Yet Another Type-Level List

• The general idea for type-safe printf: take the format string, and extract a list of the expected argument types.

• To make for nicer error messages, include a human-readable description of each type in the list.

• I've found it more convenient to re-define lists for various problems when needed, rather than having a single canonical list definition.

record _nil {
  proc type length param do return 0;
}
record _cons {
  type expectedType;  // type of the argument to printf
  param name: string; // human-readable name of the type
  type rest;

  proc type length param do return 1 + rest.length();
}

---

Extracting Types from Format Strings

proc specifiers(param s: string, param i: int = 0) type {
  if i >= s.size then return _nil;

  if s[i] == "%" {
    if i + 1 >= s.size then
        compilerError("Invalid format string: unterminted %");

    select s[i + 1] {
      when "%" do return specifiers(s, i + 2);
      when "s" do return _cons(string, "a string", specifiers(s, i + 2));
      when "i" do return _cons(int, "a signed integer", specifiers(s, i + 2));
      when "u" do return _cons(uint, "an unsigned integer", specifiers(s, i + 2));
      when "n" do return _cons(numeric, "a numeric value", specifiers(s, i + 2));
      otherwise do compilerError("Invalid format string: unknown format type");
    }
  } else {
    return specifiers(s, i + 1);
  }
}

---

Extracting Types from Format Strings

Let's give it a quick try:

writeln(specifiers("Hello, %s! Your ChapelCon submission is #%i\n") : string);

The above prints:

_cons(string,"a string",_cons(int(64),"a signed integer",_nil))

---

Validating Argument Types

• The Chapel standard library has a nice isSubtype function that we can use to check if an argument matches the expected type.

• Suppose the .length of our type specifiers matches the number of arguments to printf

• Chapel doesn't currently support empty tuples, so if the lengths match, we know that specifiers is non-empty.

• Then, we can validate the types as follows:

  proc validate(type specifiers: _cons(?t, ?s, ?rest), type argTup, param idx) {
    if !isSubtype(argTup[idx], t) then
      compilerError("Argument " + (idx + 1) : string + " should be " + s + " but got " + argTup[idx]:string, idx+2);
  
    if idx + 1 < argTup.size then
      validate(rest, argTup, idx + 1);
  }
  

• The idx+2 argument to compilerError avoids printing the recursive validate calls in the error message.

---

The fprintln overloads

• I named it fprintln for "formatted print line".

• To support the empty-specifier case (Chapel varargs don't allow zero arguments):

  proc fprintln(param format: string) where specifiers(format).length == 0 {
    writeln(format);
  }
  

• If we do have type specifiers, to ensure our earlier assumption of size matching:

  proc fprintln(param format: string, args...)
    where specifiers(format).length != args.size {
    compilerError("'fprintln' with this format string expects " +
                  specifiers(format).length : string +
                  " argument(s) but got " + args.size : string);
  }
  

---

The fprintln overloads

• All that's left is the main fprintln implementation:

  proc fprintln(param format: string, args...) {
    validate(specifiers(format), args.type, 0);
  
    writef(format + "\n", (...args));
  }
  

---

Using fprintln

fprintln("Hello, world!");        // fine, prints "Hello, world!"
fprintln("The answer is %i", 42); // fine, prints "The answer is 42"

// compiler error: Argument 3 should be a string but got int(64)
fprintln("The answer is %i %i %s", 1, 2, 3);

More work could be done to support more format specifiers, escapes, etc., but the basic idea is there.

---

Beyond Lists

---

Beyond Lists

• I made grand claims earlier
  • "Write functional-ish program at the type level!"
• So far, we've just used lists and some recursion.
• Is that all there is?

---

Algebraic Data Types

• The kinds of data types that Haskell supports are called algebraic data types.
• At a fundamental level, they can be built up from two operations: Cartesian product and disjoint union.
• There are other concepts to build recursive data types, but we won't need them in Chapel.
  • To prove to you I know what I'm talking about, some jargon: initial algebras, the fixedpoint functor, catamorphisms...
  • Check out Bananas, Lenses, Envelopes and Barbed Wire by Meijer et al. for more.
• Claim: Chapel supports disjoint union and Cartesian product, so we can build any data type that Haskell can.

---

Algebraic Data Types

• The kinds of data types that Haskell supports are called algebraic data types.
• At a fundamental level, they can be built up from two operations: Cartesian product and disjoint union.
• There are other concepts to build recursive data types, but we won't need them in Chapel.
  • To prove to you I know what I'm talking about, some jargon: initial algebras, the fixedpoint functor, catamorphisms...
  • Check out Bananas, Lenses, Envelopes and Barbed Wire by Meijer et al. for more.
• Claim: Haskell supports disjoint union and Cartesian product, so we can build any data type that Haskell can.

---

A General Recipe

To translate a Haskell data type definition to Chapel:

• For each constructor, define a record with that constructor's name

• The fields of that record are type fields for each argument of the constructor

  • If the argument is a value (like Int), you can make it a param field instead

• A visual example, again:

  record C1 { type arg1; /* ... */ type argi; }
  // ...
  record Cn { type arg1; /* ... */ type argj; }
  

  data T = C1 arg1 ... argi
         | ...
         | Cn arg1 ... argj
  

---

Inserting and Looking Up in a BST

proc insert(type t: Empty, param x: int) type do return Node(x, Empty, Empty);
proc insert(type t: Node(?v, ?left, ?right), param x: int) type do
    select true {
      when x < v do return Node(v, insert(left, x), right);
      otherwise do return Node(v, left, insert(right, x));
    }

type test = insert(insert(insert(Empty, 2), 1), 3);



proc lookup(type t: Empty, param x: int) param do return false;
proc lookup(type t: Node(?v, ?left, ?right), param x: int) param do
    select true {
      when x == v do return true;
      when x < v do return lookup(left, x);
      otherwise do return lookup(right, x);
    }

insert :: Int -> BSTree -> BSTree
insert x Empty = Node x Empty Empty
insert x (Node v left right)

    | x < v     = Node v (insert x left) right
    | otherwise = Node v left (insert x right)
                                                                              

test = insert 3 (insert 1 (insert 2 Empty))


lookup :: Int -> BSTree -> Bool
lookup x Empty = False
lookup x (Node v left right)

    | x == v    = True
    | x < v     = lookup x left
    | otherwise = lookup x right

It really works!

writeln(test : string);
// prints Node(2,Node(1,Empty,Empty),Node(3,Empty,Empty))

writeln(lookup(test, 1));
// prints true for this one, but false for '4'

---

A Key-Value Map

record Empty {}
record Node { param key: int; param value; type left; type right; }

proc insert(type t: Empty, param k: int, param v) type do return Node(k, v, Empty, Empty);
proc insert(type t: Node(?k, ?v, ?left, ?right), param nk: int, param nv) type do
    select true {
      when nk < k do return Node(k, v, insert(left, nk, nv), right);
      otherwise do return Node(k, v, left, insert(right, nk, nv));
    }

proc lookup(type t: Empty, param k: int) param do return "not found";
proc lookup(type t: Node(?k, ?v, ?left, ?right), param x: int) param do
    select true {
      when x == k do return v;
      when x < k do return lookup(left, x);
      otherwise do return lookup(right, x);
    }

type test = insert(insert(insert(Empty, 2, "two"), 1, "one"), 3, "three");
writeln(lookup(test, 1)); // prints "one"
writeln(lookup(test, 3)); // prints "three"
writeln(lookup(test, 4)); // prints "not found"

---

Conclusion

• Chapel's type-level programming is surprisingly powerful.
• We can write compile-time programs that are very similar to Haskell programs.
• This allows us to write highly parameterized code without paying runtime overhead.
• This also allows us to devise powerful compile-time checks and constraints on our code.
• This approach allows for general-purpose programming, which can be applied to your use-case

---

Extra Slides

---

Linear Multi-Step Method Approximator

---

Euler's Method

A first-order differential equation can be written in the following form:

y' = f(t, y)

In other words, the derivative of of y depends on t and y itself. There is no solution to this equation in general; we have to approximate.

If we know an initial point (t_0, y_0), we can approximate other points. To get the point at t_1 = t_0 + h, we can use the formula:

\begin{align*}
y'(t_0) & = f(t_0, y_0) \\
y(t_0+h) & \approx y_0 + h \times y'(t_0) \\
         & \approx y_0 + h \times f(t_0, y_0)
\end{align*}

We can name the first approximated $y$-value y_1, and set it:

y_1 = y_0 + h \times f(t_0, y_0)

---

Euler's Method

On the previous slide, we got a new point (t_1, y_1). We can repeat the process to get y_2:

\begin{array}{c}
y_2 = y_1 + h \times f(t_1, y_1) \\
y_3 = y_2 + h \times f(t_2, y_2) \\
y_4 = y_3 + h \times f(t_3, y_3) \\
\cdots \\
y_{n+1} = y_n + h \times f(t_n, y_n) \\
\end{array}

---

Euler's Method in Chapel

This can be captured in a simple Chapel procedure:

proc runEulerMethod(step: real, count: int, t0: real, y0: real) {
    var y = y0;
    var t = t0;
    for i in 1..count {
        y += step*f(t,y);
        t += step;
    }
    return y;
}

---

Other Methods

• In Euler's method, we look at the slope of a function at a particular point, and use it to extrapolate the next point.

• Once we've computed a few points, we have more information we can incorporate.

  • When computing y_2, we can use both y_0 and y_1.
  • To get a good approximation, we have to weight the points differently.

    
    y_{n+2} = y_{n+1} + h \left(\frac{3}{2}f(t_{n+1}, y_{n+1}) - \frac{1}{2}f(t_{n}, y_{n})\right)
    

  • More points means better accuracy, but more computation.

• There are other methods that use more points and different weights.

  • Another method is as follows:

    
    y_{n+3} = y_{n+2} + h \left(\frac{23}{12}f(t_{n+2}, y_{n+2}) - \frac{16}{12}f(t_{n+1}, y_{n+1}) + \frac{5}{12}f(t_{n}, y_{n})\right)
    

---

Generalizing Multi-Step Methods

Explicit Adams-Bashforth methods in general can be encoded as the coefficients used to weight the previous points.

Method	Equation	Coefficient List
Euler's method	y_{n+1} = y_n + h \times f(t_n, y_n)	1
Two-step A.B.	y_{n+2} = y_{n+1} + h \left(\frac{3}{2}f(t_{n+1}, y_{n+1}) - \frac{1}{2}f(t_{n}, y_{n})\right)	\frac{3}{2},-\frac{1}{2}

---

Generalizing Multi-Step Methods

Explicit Adams-Bashforth methods in general can be encoded as the coefficients used to weight the previous points.

Method	Equation	Chapel Type Expression
Euler's method	y_{n+1} = y_n + h \times f(t_n, y_n)	Cons(1,Nil)
Two-step A.B.	y_{n+2} = y_{n+1} + h \left(\frac{3}{2}f(t_{n+1}, y_{n+1}) - \frac{1}{2}f(t_{n}, y_{n})\right)	Cons(3/2,Cons(-1/2, Nil))

---

Supporting Functions for Coefficient Lists

proc length(type x: Cons(?w, ?t)) param do return 1 + length(t);
proc length(type x: Nil)          param do return 0;

proc coeff(param x: int, type lst: Cons(?w, ?t)) param where x == 0 do return w;
proc coeff(param x: int, type lst: Cons(?w, ?t)) param where x  > 0 do return coeff(x-1, t);

---

A General Solver

proc runMethod(type method, h: real, count: int, start: real,
               in ys: real ... length(method)): real {

• type method accepts a type-level list of coefficients.
• h encodes the step size.
• start is t_0, the initial time.
• count is the number of steps to take.
• in ys makes the function accept as many real values (for y_0, y_1, \ldots) as there are weights

---

A General Solver

param coeffCount = length(method);
// Repeat the methods as many times as requested
for i in 1..count {
    // We're computing by adding h*b_j*f(...) to y_n.
    // Set total to y_n.
    var total = ys(coeffCount - 1);
    // 'for param' loops are unrolled at compile-time -- this is just
    // like writing out each iteration by hand.
    for param j in 1..coeffCount do
        // For each coefficient b_j given by coeff(j, method),
        // increment the total by h*bj*f(...)
        total += step * coeff(j, method) *
            f(start + step*(i-1+coeffCount-j), ys(coeffCount-j));
    // Shift each y_i over by one, and set y_{n+s} to the
    // newly computed total.
    for param j in 0..< coeffCount - 1 do
        ys(j) = ys(j+1);
    ys(coeffCount - 1) = total;
}
// return final y_{n+s}
return ys(coeffCount - 1);

---

Using the General Solver

type euler = cons(1.0, empty);
type adamsBashforth = cons(3.0/2.0, cons(-0.5, empty));
type someThirdMethod = cons(23.0/12.0, cons(-16.0/12.0, cons(5.0/12.0, empty)));

Take a simple differential equation y' = y. For this, define f as follows:

proc f(t: real, y: real) do return y;

Now, we can run Euler's method like so:

writeln(runMethod(euler, step=0.5, count=4, start=0, 1)); // 5.0625

To run the 2-step Adams-Bashforth method, we need two initial values:

var y0 = 1.0;
var y1 = runMethod(euler, step=0.5, count=1, start=0, 1);
writeln(runMethod(adamsBashforth, step=0.5, count=3, start=0.5, y0, y1)); // 6.02344

---

The General Solver

We can now construct solvers for any explicit Adams-Bashforth method, without writing any new code.

---

Cartesian Product

For any two types, the Cartesian product of these two types defines all pairs of values from these types.

• This is like a two-element tuple at the value level in Chapel.

• We write this as A \times B for two types A and B.

• In (type-level) Chapel and Haskell:

  record Pair {
      type fst;
      type snd;
  }
  
  type myPair = Pair(myVal1, myVal2);
  

  data Pair = MkPair
      { fst :: A
      , snd :: B
      }
  
  myPair = MkPair myVal1 myVal2
  

---

Disjoint Union

For any two types, the disjoint union of these two types defines values that are either from one type or the other.

• This is almost like a union in Chapel or C...

• But there's extra information to tell us which of the two types the value is from.

• We write this as A + B for two types A and B.

• In Chapel and Haskell:

  
  record InL { type value; }
  record InR { type value; }
  
  type myFirstCase = InL(myVal1);
  type mySecondCase = InR(myVal2);
  

  data Sum
    = InL A
    | InR B
  
  myFirstCase = InL myVal1
  mySecondCase = InR myVal2
  

---

Algebraic Data Types

• We can build up more complex types by combining these two operations.

  • Need a triple of types A, B, and C? Use A \times (B \times C).
  • Similarly, "any one of three types" can be expressed as A + (B + C).
  • A Option<T> type (in Rust, or optional<T> in C++) is T + \text{Unit}.
    • Unit is a type with a single value (there's only one None / std::nullopt).

• Notice that in Chapel, we moved up one level

  Thing	Chapel	Haskell
  Nil	type	value
  Cons	type constructor	value constructor
  List	???	type

---

Algebraic Data Types

• Since Chapel has no notion of a type-of-types, we can't enforce that our values are only InL or InR (in the case of Sum).

• This is why, in Chapel versions, type annotations like A and B are missing.

  record Pair {
      type fst; /* : A */
      type snd; /* : B */
  }
  

  data Pair = MkPair
      { fst :: A
      , snd :: B
      }
  

• So, we can't enforce that the user doesn't pass int to our length function defined on lists.

• We also can't enforce that InL is instantiated with the right type.

• So, we lose some safety compared to Haskell...

• ...but we're getting the compiler to do arbitrary computations for us at compile-time.

---

Worked Example: Binary Search Tree

In Haskell, binary search trees can be defined as follows:

data BSTree = Empty
            | Node Int BSTree BSTree

balancedOneTwoThree = Node 2 (Node 1 Empty Empty) (Node 3 Empty Empty)

Written using Algebraic Data Types, this is:

\text{BSTree} = \text{Unit} + (\text{Int} \times \text{BSTree} \times \text{BSTree})

In Haskell (using sums and products):

type BSTree' = Unit `Sum` (Int `Pair` (BSTree' `Pair` BSTree'))

balancedOneTwoThree' = InR (2 `MkPair` (InR (1 `MkPair` (InL MkUnit `MkPair` InL Unit)) `MkPair`
                                        InR (3 `MkPair` (InL MkUnit `MkPair` InL Unit))))

---

Worked Example: Binary Search Tree

• Recalling the Haskell version:

  type BSTree' = Unit `Sum` (Int `Pair` (BSTree' `Pair` BSTree'))
  
  balancedOneTwoThree' = InR (2 `MkPair` (InR (1 `MkPair` (InL MkUnit `MkPair` InL Unit)) `MkPair`
                                          InR (3 `MkPair` (InL MkUnit `MkPair` InL Unit))))
  

• We can't define BSTree' in Chapel (no type-of-types), but we can define balancedOneTwoThree':

  type balancedOneTwoThree =
    InR(Pair(2, Pair(InR(Pair(1, Pair(InL(), InL()))),
                     InR(Pair(3, Pair(InL(), InL()))))));
  

• ✅ We can use algebraic data types to build arbitrarily complex data structures ◼.

---

Returning to Pragmatism

• We could've defined our list type in terms of InL, InR, and Pair.

• However, it was cleaner to make it look more like the non-ADT Haskell version.

• Recall that it looked like this:

  record Nil {}
  record Cons { param head: int; type tail; }
  
  type myList = Cons(1, Cons(2, Cons(3, Nil)));
  

  data ListOfInts = Nil
                  | Cons Int ListOfInts
  
  myList = Cons 1 (Cons 2 (Cons 3 Nil))
  

• We can do the same thing for our binary search tree:

  record Empty {}
  record Node { param value: int; type left; type right; }
  
  type balancedOneTwoThree = Node(2, Node(1, Empty, Empty),
                                     Node(3, Empty, Empty));
  

  data BSTree = Empty
              | Node Int BSTree BSTree
  
  balancedOneTwoThree = Node 2 (Node 1 Empty Empty)
                               (Node 3 Empty Empty)
  

---