Programming language: Kotlin
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Kotlin∇: Type-safe Symbolic Differentiation for Kotlin


Kotlin∇ is a type-safe automatic differentiation framework in Kotlin. It allows users to express differentiable programs with higher-dimensional data structures and operators. We attempt to restrict syntactically valid constructions to those which are algebraically valid and can be checked at compile-time. By enforcing these constraints in the type system, it eliminates certain classes of runtime errors that may occur during the execution of a differentiable program. Due to type-inference in the language, most types may be safely omitted by the end user. Kotlin∇ strives to be expressive, safe, and notationally similar to mathematics. It is currently pre-release and offers no stability guarantees at this time.

Table of contents


Inspired by Stalin∇, Autograd, DiffSharp, Myia, Nexus, Tangent, Lantern et al., Kotlin∇ attempts to port recent advancements in automatic differentiation (AD) to the Kotlin language. AD is useful for gradient descent and has a variety of applications in numerical optimization and machine learning. Our implementation adds a number of experimental ideas, including compile-time shape-safety, algebraic simplification and numerical stability checking with property-based testing. We aim to provide an algebraically-grounded implementation of AD for shape-safe tensor operations. Tensors in Kotlin∇ are represented as multidimensional arrays.


Kotlin∇ currently supports the following features:

  • Arithmetical operations on scalars, vectors and matrices
  • Shape-safe vector and matrix algebra
  • Partial and higher-order differentiation on scalars
  • Property-based testing for numerical gradient checking
  • Recovery of symbolic derivatives from AD

Additionally, it aims to support:

All of these features are implemented without access to bytecode or special compiler tricks - just using higher-order functions and lambdas as shown in Lambda the Ultimate Backpropogator, embedded DSLs a la Lightweight Modular Staging, and ordinary generics. Please see below for a more detailed feature comparison.



Kotlin∇ is hosted on JitPack.


repositories {

dependencies {





Kotlin∇ operators are higher-order functions, which take at most two inputs and return a single output, all of which are functions with the same numerical type, and whose shape is denoted using superscript in the rightmost column below.

Math Infix † Prefix Postfix‡ Operator Type Signature
a(b)a of b (a: ℝτ→ℝπ, b: ℝλ → ℝτ) → (ℝλ→ℝπ)
a + ba - b plus(a, b)minus(a, b) (a: ℝτ→ℝπ, b: ℝλ → ℝπ) → (ℝ?→ℝπ)
a * ba.times(b) times(a, b) (a: ℝτ→ℝm×n, b: ℝλ→ℝn×p) → (ℝ?→ℝm×p)
a / ba.div(b) div(a, b) (a: ℝτ→ℝm×n, b: ℝλ→ℝp×n) → (ℝ?→ℝm×p)
-a+a a.unaryMinus()a.unaryPlus() (a: ℝτ→ℝπ) → (ℝτ→ℝπ)
sin(a)cos(a)tan(a) a.sin()a.cos()a.tan() (a: ℝ→ℝ) → (ℝ→ℝ)
ln(a)log(a) a.ln()a.log() (a: ℝτ→ℝm×m) → (ℝτ→ℝm×m)
a.log(b) log(a, b) (a: ℝτ→ℝm×m, b: ℝλ→ℝm×m) → (ℝ?→ℝ)
a.pow(b) pow(a, b) (a: ℝτ→ℝm×m, b: ℝλ→ℝ) → (ℝ?→ℝm×m)
a.pow(1.0/2)a.root(3) sqrt(a)cbrt(a) a.sqrt()a.cbrt() (a: ℝτ→ℝm×m) → (ℝτ→ℝm×m)
a.d(b)d(a) / d(b) grad(a)[b] (a: C(ℝτ→ℝ)*, b: C(ℝλ→ℝ)) → (ℝ?→ℝ)
grad(a) a.grad() (a: C(ℝτ→ℝ)) → (ℝτ→ℝτ)
a.d(b)a.grad(b) grad(a, b)grad(a)[b] (a: C(ℝτ→ℝπ), b: C(ℝλ→ℝω)) → (ℝ?→ℝπ×ω)
divg(a) a.divg() (a: C(ℝτ→ℝm)) → (ℝτ→ℝ)
curl(a) a.curl() (a: C(ℝ3→ℝ3)) → (ℝ3→ℝ3)
grad(a) a.grad() (a: C(ℝτ→ℝm)) → (ℝτ→ℝm×τ)
hess(a) a.hess() (a: C(ℝτ→ℝ)) → (ℝτ→ℝτ×τ)
lapl(a) a.lapl() (a: C(ℝτ→ℝ)) → (ℝτ→ℝτ)

<!-- Equations -->

ℝ can be a Double, Float or BigDecimal. Specialized operators are defined for subsets of ℝ, e.g. Int, Short or BigInteger for subsets of ℤ, however differentiation is only defined for continuous functions on ℝ.

a and b are higher-order functions. These may be constants (e.g. 0, 1.0), variables (e.g. Var()) or expressions (e.g. x + 1, 2 * x + y).

‡ For infix notation, . is optional. Parentheses are also optional depending on precedence.

§ Matrix division is defined iff B is invertible, although it could be possible to redefine this operator using the Moore-Penrose inverse.

∗ Where C(ℝm) is the space of all continuous functions over ℝ. If the function is not over ℝ, it will fail at compile-time. If the function is over ℝ but not continuous differentiable at the point under consideration, it will fail at runtime.

? The input shape is tracked at runtime, but not at the type level. While it would be nice to infer a union type bound over the inputs of binary functions, it is likely impossible using the Kotlin type system [without great effort](core/src/main/kotlin/edu/umontreal/kotlingrad/experimental/VariableCapture.kt). If the user desires type checking when invoking higher order functions with literal values, they will need to specify the combined input type explicitly, or do so at runtime.

τ, λ, π, ω Arbitrary products.

Higher Rank Derivatives

Kotlin∇ supports derivatives between tensors of up to rank 2. The shape of a tensor derivative depends on (1) the shape of the function under differentiation and (2) the shape of the variable with respect to which we are differentiating.

I/O Shape ℝ?→ℝ ℝ?→ℝm ℝ?→ℝj×k
ℝ?→ℝ ℝ?→ℝ ℝ?→ℝm ℝ?→ℝj×k
ℝ?→ℝn ℝ?→ℝn ℝ?→ℝm×n :x:
ℝ?→ℝh×i ℝ?→ℝh×i :x: :x:

Matrix-by-vector, vector-by-matrix, and matrix-by-matrix derivatives require rank 3+ tensors and are currently unsupported.

Higher Order Derivatives

Kotlin∇ supports arbitrary order derivatives on scalar functions, and up to 2nd order derivatives on vector functions. Higher order derivatives on matrix functions are unsupported.

Shape Safety

Shape safety is an important concept in Kotlin∇. There are three broad strategies for handling shape errors:

  • Hide the error somehow by implicitly reshaping or broadcasting arrays
  • Announce the error at runtime, with a relevant message, e.g. InvalidArgumentError
  • Do not allow programs which can result in a shape error to compile

In Kotlin∇, we use the last strategy to check the shape of tensor operations. Consider the following program:

// Inferred type: Vec<Double, D2>
val a = Vec(1.0, 2.0)
// Inferred type: Vec<Double, D3>
val b = Vec(1.0, 2.0, 3.0)

val c = b + b

// Does not compile, shape mismatch
// a + b

Attempting to sum two vectors whose shapes do not match will fail to compile, and they must be explicitly resized.

// Inferred type: Mat<Double, D1, D4>
val a = Mat1x4(1.0, 2.0, 3.0, 4.0)
// Inferred type: Mat<Double, D4, D1>
val b = Mat4x1(1.0, 2.0, 3.0, 4.0)

val c = a * b

// Does not compile, inner dimension mismatch
// a * a
// b * b

Similarly, attempting to multiply two matrices whose inner dimensions do not match will fail to compile.

val a = Mat2x4( 
  1.0, 2.0, 3.0, 4.0,
  5.0, 6.0, 7.0, 8.0

val b = Mat4x2( 
  1.0, 2.0,
  3.0, 4.0,
  5.0, 6.0,
  7.0, 8.0

// Types are optional, but encouraged
val c: Mat<Double, D2, D2> = a * b 

val d = Mat2x1(1.0, 2.0)

val e = c * d

val f = Mat3x1(1.0, 2.0, 3.0)

// Does not compile, inner dimension mismatch
// e * f

Explicit types are optional but encouraged. Type inference helps preserve shape information over long programs.

fun someMatFun(m: Mat<Double, D3, D1>): Mat<Double, D3, D3> = ...
fun someMatFun(m: Mat<Double, D2, D2>) = ...

When writing a function, it is mandatory to declare the input type(s), but the return type may be omitted. Shape-safety is currently supported up to rank-2 tensors, i.e. matrices.

Variable Capture

Kotlin∇ provides a DSL with support for type-safe variable capture with variadic currying. Consider the following example:

val q = X + Y * Z + Y + 0.0
val p0 = q(X to 1.0, Y to 2.0, Z to 3.0) // Name resolution
val p1 = q(X to 1.0, Y to 1.0)(Z to 1.0) // Variadic currying
val p3 = q(Z to 1.0)(X to 1.0, Y to 1.0) // Any order is possible
val p4 = q(Z to 1.0)(X to 1.0)(Y to 1.0) // Proper currying
val p5 = q(Z to 1.0)(X to 1.0) // Returns a partially applied function
val p6 = (X + Z + 0)(Y to 1.0) // Does not compile

For further details, please refer to [the implementation](core/src/main/kotlin/edu/umontreal/kotlingrad/experimental/VariableCapture.kt).


The following example shows how to derive higher-order partials of a function z of type ℝ²→ℝ:

with(DoublePrecision) {
  val x by Var()
  val y by Var()

  val z = x * (-sin(x * y) + y) * 4  // Infix notation
  val `∂z∕∂x` = d(z) / d(x)          // Leibniz notation [Christianson, 2012]
  val `∂z∕∂y` = d(z) / d(y)          // Partial derivatives
  val `∂²z∕∂x²` = d(`∂z∕∂x`) / d(x)  // Higher order derivatives
  val `∂²z∕∂x∂y` = d(`∂z∕∂x`) / d(y) // Higher order partials
  val `∇z` = z.grad()                // Gradient operator

  val values = arrayOf(x to 0, y to 1)

  println("z(x, y) \t= $z\n" +
    "z(${values.map { it.second }.joinToString(", ")}) \t\t= ${z(*values)}\n" +
    "∂z/∂x \t\t= $`∂z∕∂x` \n\t\t= " + `∂z∕∂x`(*values) + "\n" +
    "∂z/∂y \t\t= $`∂z∕∂y` \n\t\t= " + `∂z∕∂y`(*values) + "\n" +
    "∂²z/∂x² \t= $`∂z∕∂y` \n\t\t= " + `∂²z∕∂x²`(*values) + "\n" +
    "∂²z/∂x∂y \t= $`∂²z∕∂x∂y` \n\t\t= " + `∂²z∕∂x∂y`(*values) + "\n" +
    "∇z \t\t= $`∇z` \n\t\t= [${`∇z`[x]!!(*values)}, ${`∇z`[y]!!(*values)}]ᵀ")

Any backticks and unicode characters above are simply for readability and have no effect on the behavior. Running [this program](samples/src/main/kotlin/edu/umontreal/kotlingrad/samples/HelloKotlingrad.kt) via ./gradlew HelloKotlingrad should produce the following output:

z(x, y)         = ((x) * ((- (sin((x) * (y)))) + (y))) * (4.0)
z(0, 1)         = 0.0
∂z/∂x           = d(((x) * ((- (sin((x) * (y)))) + (y))) * (4.0)) / d(x) 
                = 4.0
∂z/∂y           = d(((x) * ((- (sin((x) * (y)))) + (y))) * (4.0)) / d(y) 
                = 0.0
∂²z/∂x²         = d(((x) * ((- (sin((x) * (y)))) + (y))) * (4.0)) / d(y) 
                = 4.0
∂²z/∂x∂y        = d(d(((x) * ((- (sin((x) * (y)))) + (y))) * (4.0)) / d(x)) / d(y) 
                = 4.0
∇z              = {y=d(((x) * ((- (sin((x) * (y)))) + (y))) * (4.0)) / d(y), x=d(((x) * ((- (sin((x) * (y)))) + (y))) * (4.0)) / d(x)} 
                = [4.0, 0.0]ᵀ

Visualization tools

Kotlin∇ provides various graphical tools that can be used for visual debugging.

Dataflow graphs

Kotlin∇ functions are a type of directed acyclic graph, called dataflow graphs (DFGs). For example, running the expression ((1 + x * 2 - 3 + y + z / y).d(y).d(x) + z / y * 3 - 2).render() will display the following DFG:


Red and blue edges indicate the right and left inputs to a binary operator, respectively. Consider the DFG for a batch of stochastic gradients on [linear regression](samples/src/main/kotlin/edu/umontreal/kotlingrad/samples/LinearRegression.kt), which can be written in matrix form as :


Thetas represent the hidden parameters under differentiation and the constants are the batch inputs (X) and targets (Y). When all the free variables are bound to numerical values, the graph collapses into a single node, which can be unwrapped into a Kotlin Number.


To generate the [sample 2D plots](samples/src/main/kotlin/edu/umontreal/kotlingrad/samples/Plot2D.kt) below, run ./gradlew Plot2D.

Plotting is also possible in higher dimensions, [for example](samples/src/main/kotlin/edu/umontreal/kotlingrad/samples/Plot3D.kt) in 3D via ./gradlew Plot3D:

[](samples/src/main/resources/ripple.png) [](samples/src/main/resources/pulsar.png) [](samples/src/main/resources/starquake.png) [](samples/src/main/resources/novaflux.png)

Loss curves

Gradient descent is one application for Kotlin∇. Below, is a typical loss curve of SGD on [a multilayer perceptron](samples/src/main/kotlin/edu/umontreal/kotlingrad/samples/MLP.kt):


To train the model, execute ./gradlew MLP from within the parent directory.


To run [the tests](core/src/test/kotlin/edu/umontreal/kotlingrad), execute: ./gradlew test

Kotlin∇ claims to eliminate certain runtime errors, but how do we know the proposed implementation is not incorrect? One method, borrowed from the Haskell community, is called property-based testing (PBT), closely related to metamorphic testing. Notable implementations include QuickCheck, Hypothesis and ScalaTest (ported to Kotlin in KotlinTest). PBT uses algebraic properties to verify the result of an operation by constructing semantically equivalent but syntactically distinct expressions, which should produce the same answer. Kotlin∇ uses two such equivalences to validate its AD implementation:

For example, consider the following test, which checks whether the analytical derivative and the automatic derivative, when evaluated at a given point, are equal to each other within the limits of numerical precision:

val x by Var()
val y by Var()

val z = y * (sin(x * y) - x)            // Function under test
val `∂z∕∂x` = d(z) / d(x)               // Automatic derivative
val manualDx = y * (cos(x * y) * y - 1) // Analytical derivative 

"∂z/∂x should be y * (cos(x * y) * y - 1)" {
  NumericalGenerator.assertAll { ẋ, ẏ ->
    // Evaluate the results at a given seed
    val autoEval = `∂z∕∂x`(x to ẋ, y to ẏ) 
    val manualEval = manualDx(x to ẋ, y to ẏ)
    // Should pass iff Δ(adEval, manualEval) < Ɛ
    autoEval shouldBeApproximately manualEval

PBT will search the input space for two numerical values and , which violate the specification, then "shrink" them to discover pass-fail boundary values. We can construct a similar test using finite differences:

"d(sin x)/dx should be equal to (sin(x + dx) - sin(x)) / dx" {
  NumericalGenerator.assertAll { ẋ ->
    val f = sin(x)

    val `df∕dx` = d(f) / d(x)
    val adEval = `df∕dx`(ẋ) 

    val dx = 1E-8
    // Since ẋ is a raw numeric type, sin => kotlin.math.sin
    val fdEval = (sin(ẋ + dx) - sin(ẋ)) / dx
    adEval shouldBeApproximately fdEval


Above, we [compare numerical errors](samples/src/main/kotlin/edu/umontreal/kotlingrad/samples/ADSDComparison.kt) for three types of computational differentiation: (1) finite precision automatic differentiation (AD), (2) finite precision symbolic differentiation (SD) and (3) finite precision finite differences (FD) against infinite precision symbolic differentiation (IP). AD and SD both exhibit relative errors (i.e. with respect to each other) several orders of magnitude lower than their absolute errors (i.e. with respect to IP), which roughly agree to within numerical precision. As expected, FD exhibits numerical error significantly higher than AD and SD due to the inaccuracy of floating point division.

There are many other ways to independently verify the numerical gradient, such as dual numbers or the complex step derivative. Another method is to compare the numerical output against a well-known implementation, such as TensorFlow. We plan to conduct a more thorough comparison of numerical accuracy and performance.


To understand the core of Kotlin∇'s AD implementation, please refer to the [scalar example](core/src/main/kotlin/edu/umontreal/kotlingrad/experimental/Scalar.kt).

This project relies on a few Kotlin-native language features, which together enable a concise, flexible and type-safe user interface. The following features have proven beneficial to the development of Kotlin∇:

Operator overloading

Operator overloading enables concise notation for arithmetic on abstract types, where the types encode algebraic structures, e.g. [Group](core/src/main/kotlin/edu/umontreal/kotlingrad/algebra/Group.kt), [Ring](core/src/main/kotlin/edu/umontreal/kotlingrad/algebra/Ring.kt), and [Field](core/src/main/kotlin/edu/umontreal/kotlingrad/algebra/Field.kt). These abstractions are extensible to other kinds of mathematical structures, such as complex numbers and quaternions.

For example, suppose we have an interface Group, which overloads the operators + and *, and is defined like so:

interface Group<T: Group<T>> {
  operator fun plus(addend: T): T

  operator fun times(multiplicand: T): T

Here, we specify a recursive type bound using a method known as F-bounded quantification to ensure that operations return the concrete type variable T, rather than something more abstract like Group. Imagine a class Expr which has implemented Group. It can be used as follows:

fun <T: Group<T>> cubed(t: T): T = t * t * t

fun <E: Expr<E>> twiceExprCubed(e: E): E = cubed(e) + cubed(e)

Like Python, Kotlin supports overloading a limited set of operators, which are evaluated using a fixed precedence. In the current version of Kotlin∇, operators do not perform any computation, they simply construct a directed acyclic graph representing the symbolic expression. Expressions are only evaluated when invoked as a function.

First-class functions

With higher-order functions and lambdas, Kotlin treats functions as first-class citizens. This allows us to represent mathematical functions and programming functions with the same underlying abstractions (typed FP). A number of recent papers have demonstrated the expressiveness of this paradigm for automatic differentiation.

In Kotlin∇, all expressions can be treated as functions. For example:

fun <T: Group<T>> makePoly(x: Var<T>, y: Var<T>) = x * y + y * y + x * x
val x by Var()
val y by Var()
val f = makePoly(x, y)
val z = f(1.0, 2.0) // Returns a value
println(z) // Prints: 7

Additionally, it is possible to build functions consisting of varying dimensional inputs:

fun <T: Fun<T>> mlp(p1: VFun<T, D3>, p2: MFun<T, D3, D3>, p3: T) =
  ((p1 * p2 + p1 * p2 * p2 dot p1 + p1) - p3) pow p3


Coroutines are a generalization of subroutines for non-preemptive multitasking, typically implemented using continuations. One form of continuation, known as shift-reset a.k.a. delimited continuations, are sufficient for implementing reverse mode AD with operator overloading alone (without any additional data structures) as described by Wang et al. in Shift/Reset the Penultimate Backpropagator and later in Backpropagation with Continuation Callbacks. While Kotlin callbacks are single-shot by default, delimited continuations and reentrant or "multi-shot" delimited continuations can also be implemented using Kotlin coroutines and would be an interesting extension to this work. Please stay tuned!

Extension Functions

Extension functions augment external classes with new fields and methods. Via context oriented programming, Kotlin∇ can expose its custom extensions (e.g. in [DoublePrecision](core/src/main/kotlin/edu/umontreal/kotlingrad/numerical/Protocol.kt)) to [consumers](samples/src/main/kotlin/edu/umontreal/kotlingrad/samples/HelloKotlingrad.kt) without requiring subclasses or inheritance.

data class Const<T: Group<T>>(val number: Double) : Expr()
data class Sum<T: Group<T>>(val e1: Expr, val e2: Expr) : Expr()
data class Prod<T: Group<T>>(val e1: Expr, val e2: Expr) : Expr()

class Expr<T: Group<T>>: Group<Expr<T>> {
  operator fun plus(addend: Expr<T>) = Sum(this, addend)

  operator fun times(multiplicand: Expr<T>) = Prod(this, multiplicand)

object DoubleContext {
  operator fun Number.times(expr: Expr<Double>) = Const(toDouble()) * expr

Now, we can use the context to define another extension, Expr.multiplyByTwo, which computes the product inside a DoubleContext, using the operator overload we defined above:

fun Expr<Double>.multiplyByTwo() = with(DoubleContext) { 2 * this } // Uses `*` operator in DoubleContext

Extensions can also be defined in another file or context and imported on demand.

Algebraic data types

Algebraic data types (ADTs) in the form of sealed classes (a.k.a. sum types) facilitate a limited form of pattern matching over a closed set of subclasses. When matching against subclasses of a sealed class, the compiler forces the author to provide an exhaustive control flow over all concrete subtypes of an abstract class. Consider the following classes:

class Const<T: Fun<T>>(val number: Number) : Fun<T>()
class Sum<T: Fun<T>>(val left: Fun<T>, val right: Fun<T>) : Fun<T>()
class Prod<T: Fun<T>>(val left: Fun<T>, val right: Fun<T>) : Fun<T>()
class Var<T: Fun<T>>: Fun<T>() { override val variables: Set<Var<X>> = setOf(this) }
class Zero<T: Fun<T>>: Const<T>(0.0)
class One<T: Fun<T>>: Const<T>(1.0)

When branching on the type of a sealed class, consumers must explicitly handle every case, since incomplete control flow will not compile rather than fail silently at runtime. Let us now consider a simplified definition of Fun, a sealed class which defines the behavior of function invocation and differentiation, using a restricted form of pattern matching. It can be constructed with a set of Vars, and can be invoked with a numerical value:

sealed class Fun<X: Fun<X>>(open val variables: Set<Var<X>> = emptySet()): Group<Fun<X>> {
    constructor(vararg fns: Fun<X>): this(fns.flatMap { it.variables }.toSet())

    // Since the subclasses of Fun are a closed set, no `else  ...` is required.
    operator fun invoke(map: Map<Var<X>, X>): Fun<X> = when (this) {
        is Const -> this
        is Var -> map.getOrElse(this) { this } // Partial application is permitted
        is Prod -> left(map) * right(map) // Smart casting implicitly casts after checking
        is Sum -> left(map) + right(map)

    fun d(variable: Var<X>): Fun<X> = when(this) {
       is Const -> Zero
       is Var -> if (variable == this) One else Zero
       // Product rule: d(u*v)/dx = du/dx * v + u * dv/dx
       is Prod -> left.d(variable) * right + left * right.d(variable)
       is Sum -> left.d(variable) + right.d(variable)

    operator fun plus(addend: Fun<T>) = Sum(this, addend)

    operator fun times(multiplicand: Fun<T>) = Prod(this, multiplicand)

Kotlin's smart casting is an example of flow-sensitive type analysis where the abstract type Fun can be treated as Sum after performing an is Sum check. Without smart casting, we would need to write (this as Sum).left to access the member, left, causing a potential ClassCastException if the cast were mistaken.

Multiple Dispatch

In conjunction with ADTs, Kotlin∇ also uses multiple dispatch to instantiate the most specific result type of applying an operator based on the type of its operands. While multiple dispatch is not an explicit language feature, it can be emulated using inheritance.

Building on the previous example, a common task in AD is to simplify a graph. This is useful in order to minimize the total number of calculations required, improving numerical stability. We can eagerly simplify expressions based on algebraic rules of replacement. Smart casting allows us to access members of a class after checking its type, without explicitly casting it:

override fun times(multiplicand: Function<X>): Function<X> = when {
  this == zero -> this
  this == one -> multiplicand
  multiplicand == one -> this
  multiplicand == zero -> multiplicand
  this == multiplicand -> pow(two)
  this is Const && multiplicand is Const -> const(value * multiplicand.value)
  // Further simplification is possible using rules of replacement
  else -> Prod(this, multiplicand)

val result = Const(2.0) * Sum(Var(2.0), Const(3.0)) // Sum(Prod(Const(2.0), Var(2.0)), Const(6.0))

This allows us to put all related control flow on a single abstract class which is inherited by subclasses, simplifying readability, debugging and refactoring.

Shape-safe Tensor Operations

While first-class dependent types are useful for ensuring arbitrary shape safety (e.g. when concatenating and reshaping matrices), they are unnecessary for simple equality checking (such as when multiplying two matrices).* When the shape of a tensor is known at compile-time, it is possible to encode this information using a less powerful type system, as long as it supports subtyping and parametric polymorphism (a.k.a. generics). In practice, we can implement a shape-checked tensor arithmetic in languages like Java, Kotlin, C++, C# or Typescript, which accept generic type parameters. In Kotlin, whose type system is less expressive than Java, we use the following strategy.

Shape safety is currently supported up to rank-2 tensors, i.e. matrices. To perform dimension checking in our type system, first we enumerate a list of integer type literals as a chain of subtypes, C <: C - 1 <: C - 2 <: ... <: 1 <: 0, where C is the largest fixed-length dimension we wish to represent, which can be specified by the user prior to compilation. This guarantees linear space and time complexity for subtype checking, with a constant upper bound.

interface Nat<T: D0> { val i: Int } // Used for certain type bounds
sealed class D0(open val i: Int = 0) { companion object: D0(), Nat<D0> }
sealed class D1(override val i: Int = 1): D0(i) { companion object: D1(), Nat<D1> }
sealed class D2(override val i: Int = 2): D1(i) { companion object: D2(), Nat<D2> }
sealed class D3(override val i: Int = 3): D2(i) { companion object: D3(), Nat<D3> }

Next, we overload the call operator to emulate instantiating a collection literal, using arity to infer its dimensionality. Consider the rank-1 case for length inference on vector literals:

open class Vec<E, Len: D1> constructor(val contents: List<E>) {
    companion object {
        operator fun <T> invoke(t: T): Vec<T, D1> = Vec(listOf(t))
        operator fun <T> invoke(t0: T, t1: T): Vec<T, D2> = Vec(listOf(t0, t1))
        operator fun <T> invoke(t0: T, t1: T, t2: T): Vec<T, D3> = Vec(listOf(t0, t1, t2))

Finally, we encode length as a parameter of the operand type. Since integer literals are a chain of subtypes, we only need to define one operator using the highest literal, and can rely on Liskov substitution to preserve shape safety for all subtypes.

@JvmName("floatVecPlus") infix operator fun <C: D1, V: Vec<Float, C>> V.plus(v: V): Vec<Float, C> = 
  Vec(length, contents.zip(v.contents).map { it.first + it.second })

The operator + can now be used like so. Incompatible operands will cause a type error:

val one = Vec(1, 2, 3) + Vec(1, 2, 3)          // Always runs safely
val add = Vec(1, 2, 3) + Vec(D3, listOf(...))  // May fail at runtime
val sum = Vec(1, 2) + add                      // Does not compile

A similar syntax is available for [matrices](core/src/main/kotlin/edu/umontreal/kotlingrad/experimental/Matrix.kt) and higher-rank [tensors](core/src/main/kotlin/edu/umontreal/kotlingrad/experimental/Tensor.kt). For example, Kotlin∇ can infer the shape of multiplying two matrices, and will not compile if their inner dimensions do not match:

open class Mat<X, R: D1, C: D1>(vararg val rows: Vec<X, C>)
fun <X> Mat1x2(d0: X, d1: X): Mat<X, D1, D2> = Mat(Vec(d0, d1))
fun <X> Mat2x1(d0: X, d1: X): Mat<X, D2, D1> = Mat(Vec(d0), Vec(d1))
// ...
operator fun <Q: D1, R: D1, S: D1> Mat<Int, Q, R>.times(m: Mat<Int, R, S>): Mat<Int, Q, S> = TODO()

// Inferred type: Mat<Int, D4, D4>
val l = Mat4x4(
  1, 2, 3, 4,
  5, 6, 7, 8,
  9, 0, 0, 0,
  9, 0, 0, 0

// Inferred type: Mat<Int, D4, D3>
val m = Mat4x3(
  1, 1, 1,
  2, 2, 2,
  3, 3, 3,
  4, 4, 4

// Inferred type: Mat<Int, D4, D3>
val lm = l * m
// m * m // Compile error: Expected Mat<3, *>, found Mat<4, 3>

[Further examples](core/src/main/kotlin/edu/umontreal/kotlingrad/experimental/Matrix.kt) are provided for shape-safe matrix operations such as addition, subtraction and transposition.

A similar technique is possible in Haskell, which is capable of a more powerful form of type-level computation, type arithmetic. Type arithmetic makes it easy to express convolutional arithmetic and other arithmetic operations on shape variables (say, splitting a vector in half), which is currently not possible, or would require enumerating every possible combination of type literals.

∗ Many less powerful type systems are still capable of performing arbitrary computation in the type checker. As specified, Java's type system is known to be Turing Complete. It may be possible to emulate a limited form of dependent types in Java by exploiting this property, although this may not computationally tractable due to the practical limitations noted by Grigore.

Property Delegation

Property delegation is a reflection feature in the Kotlin language which lets us access properties to which an instance is bound. For example, we can read the property name like so:

class Var(val name: String?) {
  operator fun getValue(thisRef: Any?, property: KProperty<*>) = Var(name ?: property.name)

This feature allows consumers to instantiate variables e.g. in an embedded DSL without redeclaring their names:

val x by Var()   // With property delegation
val x = Var("x") // Without property delegation

Without property delegation, users would need to repeat the property name in the constructor.

Ideal API (WIP)

The current API is experimental, but can be improved in many ways. Currently, Kotlin∇ does not infer a function's input dimensionality (i.e. free variables and their corresponding shape). While it is possible to perform variable capture over a small alphabet using [type safe currying](core/src/main/kotlin/edu/umontreal/kotlingrad/experimental/VariableCapture.kt), this technique incurs a large source code overhead. It may be possible to reduce the footprint using phantom types or some form of union type bound (cf. Kotlin, Java).

When the shape of an N-dimensional array is known at compile-time, we can use [type-level integers](core/src/main/kotlin/edu/umontreal/kotlingrad/dependent) to ensure shape conforming tensor operations (inspired by Nexus and others).

Allowing users to specify a matrix's structure in its type signature, (e.g. Singular, Symmetric, Orthogonal, Unitary, Hermitian, Toeplitz) would allows us to specialize derivation over such matrices (cf. section 2.8 of The Matrix Cookbook).

Scalar functions

A function's type would ideally encode arity, based on the number of unique variables:

val f = x * y + sin(2 * x + 3 * y)              // f: BinaryFunction<Double> "
val g = f(x to -1.0)                            // g: UnaryFunction<Double> == -y + sin(-2 + 3 * y)
val h = f(x to 0.0, y to 0.0)                   // h: Const<Double> == 0 + sin(0 + 0) == 0

However inferring arity for arbitrary expressions at compile-time would be difficult in the Kotlin type system. Instead, we can have the user specify it directly.

val x by Var()                                  // x: Variable<Double> inferred type
val y by Var()                                  // x: Variable<Double> "
val f = Fun(D2) { x * y + sin(2 * x + 3 * y) }  // f: BinaryFunction<Double> "
val g = f(x to -1.0)                            // g: UnaryFunction<Double> == -y + sin(-2 + 3 * y)
val h = f(x to 0.0, y to 0.0)                   // h: Const<Double> == 0 + sin(0 + 0) == 0


Below is the approximate BNF grammar for Kotlin∇. This is incomplete and subject to change without notice.

       type = "Double" | "Float" | "Int" | "BigInteger" | "BigDouble";
        nat = "1" | ... | "99";
     output = "Fun<" type "Real>" | "VFun<" type "Real," nat ">" | "MFun<" type "Real," nat "," nat ">";
        int = "0" | nat int;
      float = int "." int;
        num = type "(" int ")" | type "(" float ")";
        var = "x" | "y" | "z" | "ONE" | "ZERO" | "E" | "Var()";
     signOp = "+" | "-";
      binOp = signOp | "*" | "/" | "pow";
     trigOp = "sin" | "cos" | "tan" | "asin" | "acos" | "atan" | "asinh" | "acosh" | "atanh";
    unaryOp = signOp | trigOp | "sqrt" | "log" | "ln" | "exp";
        exp = var | num | unaryOp exp | var binOp exp | "(" exp ")";
    expList = exp | exp "," expList;
      linOp = signOp | "*" | " dot ";
        vec = "Vec(" expList ")" | "Vec" nat "(" expList ")";
     vecExp = vec | signOp vecExp | exp "*" vecExp | vec linOp vecExp | vecExp ".norm(" int ")";
        mat = "Mat" nat "x" nat "(" expList ")";
     matExp = mat | signOp matExp | exp linOp matExp | vecExp linOp matExp | mat linOp matExp;
     anyExp = exp | vecExp | matExp | derivative | invocation;
   bindings = exp " to " exp | exp " to " exp "," bindings;
 invocation = anyExp "(" bindings ")";
 derivative = "d(" anyExp ") / d(" exp ")" | anyExp ".d(" exp ")" | anyExp ".d(" expList ")";
   gradient = exp ".grad()";

UML Diagram



Unlike certain frameworks which simply wrap an existing AD library in a type-safe DSL, Kotlin∇ contains a fully shape-safe implementation of algorithmic differentiation, written in pure Kotlin. By doing so, it can leverage Kotlin language features such as typed functional programming, as well as interoperability with other languages on the JVM platform. Furthermore, it implements symbolic differentiation, which unlike Wengert tape or dual-number based ADs, allows it to calculate derivatives of arbitrarily high order with zero extra engineering required. Further details can be found below.

Framework Language SD¹ AD² HD³ DP⁴ FP⁵ TS⁶ SS⁷ DT⁸ MP⁹
Kotlin∇ Kotlin :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :x: :construction:
DiffSharp F# :x: :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :x: :x: :x:
TensorFlow.FSharp F# :x: :x: :x: :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :x: :x:
Nexus Scala :x: :heavy_check_mark: :x: :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :x: :x:
Lantern Scala :x: :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :x: :x: :x:
Hipparchus Java :x: :heavy_check_mark: :x: :x: :x: :heavy_check_mark: :x: :x: :x:
JAutoDiff Java :heavy_check_mark: :heavy_check_mark: :x: :x: :x: :heavy_check_mark: :x: :x: :x:
Eclipse DL4J Java :x: :construction: :x: :x: :x: :heavy_check_mark: :x: :x: :x:
Halide C++ :x: :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :x: :heavy_check_mark: :x: :x: :x:
Tensor Safe Haskell :x: :x: :x: :x: :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :x:
HaskTorch Haskell :x: :x: :x: :x: :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :x: :x:
Grenade Haskell :x: :x: :x: :x: :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :x: :x:
Stalin∇ Scheme :x: :heavy_check_mark: :x: :x: :heavy_check_mark: :x: :x: :x: :x:
Myia Python :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :heavy_check_mark: :x: :x: :x: :construction:
Autograd Python :x: :heavy_check_mark: :x: :x: :x: :x: :x: :x: :x:
JAX Python :x: :heavy_check_mark: :x: :heavy_check_mark: :heavy_check_mark: :x: :x: :x: :construction:
Tangent Python :x: :heavy_check_mark: :x: :x: :x: :x: :x: :x: :x:
Analitik Analitik :heavy_check_mark: :x: :x: :x: :heavy_check_mark: :x: :x: :x: :x:

¹ Symbolic differentiation

² Automatic differentiation

³ Higher order differentiation

⁴ Differentiable programming

⁵ Functional programming

⁶ Compile-time type safety

⁷ Compile-time shape safety

⁸ Dependently Typed

⁹ Multiplatform


If you would like to cite Kotlin∇, please use the following bibtex entry:

  authors = {Considine, Breandan and Famelis, Michalis and Paull, Liam},
  title = {Kotlin{\nabla}: A Shape-Safe e{DSL} for Differentiable Programming},
  year = {2019},
  publisher = {GitHub},
  journal = {GitHub repository},
  howpublished = {\url{https://github.com/breandan/kotlingrad}},


To the author's knowledge, Kotlin∇ is the first AD implementation in native Kotlin. While the particular synthesis of these ideas (i.e. shape-safe, functional AD, using generic types) is unique, it has been influenced by a long list of prior work in AD. Below is a list of projects and publications that helped inspire this work.

Automatic Differentiation

Differentiable Programming


Computer Algebra

Symbolic Mathematics

  • KMath - Kotlin mathematics extensions library
  • Hipparchus - An efficient, general-purpose mathematics components library in the Java programming language
  • tensor - Linear algebra for tensors with symbolic and numeric scalars
  • SymJava - A Java library for fast symbolic-numeric computation
  • JAS - Java Algebra System
  • jalgebra - An abstract algebra library for Java
  • COJAC - Numerical sniffing tool and Enriching number wrapper for Java
  • chebfun - Allows representing functions as Chebyshev polynomials, for easy symbolic differentiation (or integration)
  • horeilly1101/deriv - Open source derivative calculator REST API (and Java library)

Neural Networks

Type Systems

Domain-Specific Languages

Automated Testing

AD Libraries

  • TensorFlow.FSharp: An eDSL for writing numerical models in F# with support for interactive tensor shape-checking
  • Stalin∇, a brutally optimizing compiler for the VLAD language, a pure dialect of Scheme with first-class automatic differentiation operators
  • Autograd - Efficiently computes derivatives of NumPy code
  • DiffSharp, a functional AD library implemented in the F# language
  • Myia - SCT based AD, adapted from Pearlmutter & Siskind's "Reverse Mode AD in a functional framework"
  • Nexus - Type-safe tensors, deep learning and probabilistic programming in Scala
  • Tangent - "Source-to-Source Debuggable Derivatives in Pure Python"
  • Grenade - composable, dependently typed, practical, and fast RNNs in Haskell
  • Lantern - a framework in Scala, based on delimited continuations and multi-stage programming
  • JAutoDiff - An Automatic Differentiation Library
  • Analitik - Algebraic language for the description of computing processes using analytical transformations

Special Thanks

The following individuals have helped shape this project through their enthusiasm and thoughtful feedback. Please check out their work.