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<! @file:Suppress("ClassName") > <! @file:Suppress("PropertyName") >
Kotlin∇: Typesafe Symbolic Differentiation for the JVM
Kotlin∇ is a typesafe automatic differentiation framework written in Kotlin. It allows users to express differentiable programs with higherdimensional data structures and operators. We attempt to restrict syntactically valid constructions to those which are algebraically valid and can be checked at compiletime. 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 typeinference, most type declarations may be safely omitted by the enduser. Kotlin∇ strives to be expressive, safe, and notationally similar to mathematics.
Table of contents
 Introduction
 Supported features
 Usage
 Visualization
 Testing and gradient checking
 How does it work?
 Experimental ideas
 Formal grammar
 UML diagram
 Comparison to other frameworks
 References
 Acknowledgements
Introduction
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 compiletime shapesafety, algebraic simplification and numerical stability checking with propertybased testing. We aim to provide an algebraicallygrounded implementation of AD for shapesafe tensor operations. Tensors in Kotlin∇ are represented as multidimensional arrays.
Features
Kotlin∇ currently supports the following features:
 Arithmetical operations on scalars, vectors and matrices
 Shapesafe vector and matrix algebra
 Partial and higherorder differentiation on scalars
 Propertybased testing for numerical gradient checking
 Recovery of symbolic derivatives from AD
Additionally, it aims to support:
 PyTorchstyle definebyrun semantics
 Ndimensional tensors and higherorder tensor operators
 Fullygeneral AD over control flow, variable reassignment (via delegation), and array programming, possibly using a typed IR such as Myia
All of these features are implemented without access to bytecode or special compiler tricks  just using higherorder 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.
Usage
Installation
Kotlin∇ is hosted on Maven Central. An example project is provided here.
Gradle
dependencies {
implementation("ai.hypergraph:kotlingrad:0.4.7")
}
Maven
<dependency>
<groupId>ai.hypergraph</groupId>
<artifactId>kotlingrad</artifactId>
<version>0.4.7</version>
</dependency>
Jupyter Notebook
To access Kotlin∇'s notebook support, use the following line magic:
@file:DependsOn("ai.hypergraph:kotlingrad:0.4.7")
For more information, explore the [tutorial](samples/notebooks/hello_kotlingrad.ipynb).
Notation
Kotlin∇ operators are higherorder 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 + b a  b 
plus(a, b) minus(a, b) 
(a : ℝτ→ℝπ, b : ℝλ → ℝπ) → (ℝ?→ℝπ) 

a * b a.times(b) 
times(a, b) 
(a : ℝτ→ℝm×n, b : ℝλ→ℝn×p) → (ℝ?→ℝm×p) 

a / b a.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 continuously differentiable functions on ℝ.
† a
and b
are higherorder 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 MoorePenrose inverse.
∗ Where C(ℝm) is the space of all continuous functions over ℝ. If the function is not over ℝ, it will fail at compiletime. 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/commonMain/kotlin/ai/hypergraph/kotlingrad/typelevel/Variables.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.
HigherRank 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: 
Matrixbyvector, vectorbymatrix, and matrixbymatrix derivatives require rank 3+ tensors and are currently unsupported.
Higherorder derivatives
Kotlin∇ supports arbitrary order derivatives on scalar functions, and up to 2nd order derivatives on vector functions. Higherorder 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. Shapesafety is currently supported up to rank2 tensors, i.e. matrices.
Example
The following example shows how to derive higherorder partials of a function z
of type ℝ²→ℝ:
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) // Higherorder derivatives
val `∂²z∕∂x∂y` = d(`∂z∕∂x`) / d(y) // Higherorder 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/ai/hypergraph/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
Variable capture
Not only does Kotlin∇'s type system encode output shape, it is also capable of tracking free and bound variables, for orderindependent name binding and partial application. Expressions inhabited by free variables are typed as functions until fully bound, at which time they return a concrete value. 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 binding
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
This feature is made possible by encoding a typelevel Hasse diagram over a small set of predefined variable names, with skipconnections for variadic combination and partial application. Curious readers may glean further details by referring to [the implementation](core/src/commonMain/gen/ai/hypergraph/kotlingrad/typelevel/arity/Variables.kt) and [usage example](samples/src/main/kotlin/ai/hypergraph/kotlingrad/samples/VariableCapture.kt).
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:
[](samples/src/main/resources/dataflow.svg)
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/ai/hypergraph/kotlingrad/samples/LinearRegression.kt), which can be written in matrix form as :
[](samples/src/main/resources/lr_batch_loss_graph.svg)
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
.
Plotting
To generate the [sample 2D plots](samples/src/main/kotlin/ai/hypergraph/kotlingrad/samples/Plot2D.kt) below, run ./gradlew Plot2D
.
Plotting is also possible in higher dimensions, [for example](samples/src/main/kotlin/ai/hypergraph/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/ai/hypergraph/kotlingrad/samples/MLP.kt):
[](samples/src/main/resources/mlp_loss.svg)
To train the model, execute ./gradlew MLP
from within the parent directory.
Testing
To run [the tests](core/src/jvmTest/kotlin/ai/hypergraph/kotlingrad), execute ../gradlew allTests
from the core
directory.
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 propertybased testing (PBT), closely related to metamorphic testing. Notable implementations include QuickCheck, Hypothesis and ScalaTest (ported to Kotlin in Kotest). 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:
 Analytic differentiation: manually differentiate and compare the values returned on a subset of the domain with AD.
 Finite difference approximation: sample space of symbolic (differentiable) functions, comparing results of AD to FD.
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 ẋ
and ẏ
, which violate the specification, then "shrink" them to discover passfail 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 = 1E8
// Since ẋ is a raw numeric type, sin => kotlin.math.sin
val fdEval = (sin(ẋ + dx)  sin(ẋ)) / dx
adEval shouldBeApproximately fdEval
}
}
[](samples/src/main/resources/comparison.svg)
Above, we [compare numerical errors](samples/src/main/kotlin/ai/hypergraph/kotlingrad/samples/ADSDComparison.kt) for three types of computational differentiation against infinite precision symbolic differentiation (IP):
 Finite precision automatic differentiation (AD)
 Finite precision symbolic differentiation (SD)
 Finite precision finite differences (FD)
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 floatingpoint 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 wellknown implementation, such as TensorFlow. We plan to conduct a more thorough comparison of numerical accuracy and performance.
How?
To understand the core of Kotlin∇'s AD implementation, please refer to the [scalar example](core/src/commonMain/kotlin/ai/hypergraph/kotlingrad/api/Scalar.kt).
This project relies on a few Kotlinspecific language features, which together enable a concise, flexible and typesafe 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
, Ring
, and Field
. 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 Fbounded quantification to ensure that operations return the concrete type variable T
, rather than something more abstract like Group
. Imagine a class Fun
that has implemented Group
. It can be used as follows:
fun <T: Group<T>> cubed(t: T): T = t * t * t
fun <T: Group<T>> twiceCubed(t: T): T = cubed(t) + cubed(t)
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.
Firstclass functions
With higherorder functions and lambdas, Kotlin treats functions as firstclass citizens. This allows us to represent mathematical functions and programming functions with the same underlying abstractions (typed FP). Several 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
Multistage programming
Kotlin∇ uses operator overloading in the host language to first construct a dataflow graph, but evaluates the graph lazily. Called "multistage programming", or staging, this is a metaprogramming technique from the ML community which enables typesafe runtime code translation and compilation. More recently, staging has been put to effective use for compiling embedded DSLs similar to Kotlin∇.
In its current form, Kotlin∇ takes a "shallow embedding" approach. Similar to an interpreter, it adheres closely to the userdefined program and does not perform much code specialization or rewriting for optimization purposes. Unlike an interpreter, it postpones evaluation until all free variables in an expression have been bound. Consider the following snippet, which decides when to evaluate an expression:
var EAGER = false
operator fun invoke(newBindings: Bindings<X>): Fun<X> =
Composition(this, newBindings).run { if (bindings.complete  EAGER) evaluate() else this }
If bindings
are complete
, this means there are no unbound variables remaining (implementation omitted for brevity), and we can evaluate the expression to obtain a numerical result. Suppose we have the following user code:
val x = Var()
val y = Var()
val z = Var()
val f0 = x + y * z
var f1 = f0(x to 1).also { println(it) } // Prints: (x + y * z)(x=1)
var f2 = f1(y to 2).also { println(it) } // Prints: (x + y * z)(x=1)(y=2)
var f3 = f2(z to 3).also { println(it) } // Prints: 7
Once the last line is reached, all variables are bound, and instead of returning a Composition
, Kotlin∇ evaluates the function, returning a constant. Alternatively, if EAGER
mode is enabled, each invocation is applied as early as possible:
EAGER = true
f1 = f0(x to 1).also { println(it) } // Prints: 1 + y * z
f2 = f1(y to 2).also { println(it) } // Prints: 1 + 2 * z
f3 = f2(z to 3).also { println(it) } // Prints: 7
In the following section, we describe how evaluation works.
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. By using these, the compiler forces us to provide an exhaustive control flow when type checking a sealed 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 checking the type of a sealed class, consumers must explicitly handle every case, as incomplete control flow will produce a compiler error rather than fail at runtime. Consider a simplified definition of the superclass Fun
, which defines invocation and differentiation using a restricted form of pattern matching:
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: Bindings<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)
}
Symbolic differentiation as implemented by Kotlin∇ has two distinct passes, one for differentiation and one for evaluation. Differentiation constitutes a topdown substitution process on the computation graph and evaluation propagates the values from the bottom, up. This reduction semantics for this procedure are described more precisely in the specification.
[](latex/figures/kotlingrad_diagram.png)
Kotlin∇ functions are not only data structures, but Kotlin functions which can be invoked by passing a Bindings
instance (effectively, a Map<Fun<X>, Fun<X>>
). To enable this functionality, we overload the invoke
operator, then recurse over the graph, using Bindings
as a lookup table. If a matching subexpression is found, we propagate the bound value instead of the matching function. This is known as the interpreter pattern.
Kotlin's smart casting is an example of flowsensitive 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.
Extension functions
By using extension functions, users can convert between numerical types in the host language and our eDSL, by augmenting classes with additional operators. Contextoriented programming, allows users to define custom extensions without requiring subclasses or inheritance.
data class Const<T: Group<T>>(val number: Double) : Fun()
data class Sum<T: Group<T>>(val e1: Fun, val e2: Fun) : Fun()
data class Prod<T: Group<T>>(val e1: Fun, val e2: Fun) : Fun()
class Fun<T: Group<T>>: Group<Fun<T>> {
operator fun plus(addend: Fun<T>) = Sum(this, addend)
operator fun times(multiplicand: Fun<T>) = Prod(this, multiplicand)
}
object DoubleContext {
operator fun Number.times(expr: Fun<Double>) = Const(toDouble()) * expr
}
Now, we can use the context to define another extension, Fun.multiplyByTwo
, which computes the product inside a DoubleContext
, using the operator overload we defined above:
fun Fun<Double>.multiplyByTwo() = with(DoubleContext) { 2 * this } // Uses `*` operator in DoubleContext
Extensions can also be defined in another file or context and imported on demand. For example, Kotlin∇ also uses extensions to define shapesafe constructors and operators for vector and matrix arithmetic.
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.
Shapesafe tensor operations
While firstclass 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 compiletime, 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 shapechecked 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 rank2 tensors, i.e. matrices. To perform dimension checking in our type system, we first enumerate a list of integer type literals as a chain of subtypes, C <: C  1 <: C  2 <: ... <: 1 <: 0
, where C
is the largest fixedlength 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.
@file:Suppress("ClassName")
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> }
//... † Automatically generated
Next, we overload the call operator to emulate instantiating a collection literal, using arity to infer its dimensionality. Consider the rank1 case for length inference on vector literals:
open class Vec<E, Len: D1>(val contents: List<E>)
fun <T> Vec(t1: T): Vec<T, D1> = Vec(listOf(t1))
fun <T> Vec(t1: T, t2: T): Vec<T, D2> = Vec(listOf(t1, t2))
fun <T> Vec(t1: T, t2: T, t3: T): Vec<T, D3> = Vec(listOf(t1, t2, t3))
//... † Automatically generated
Finally, we encode length as a parameter of the operand type. Since integer literals are a chain of subtypes, we need only define one operator using the highest literal, and can rely on Liskov substitution to preserve shape safety for all subtypes.
infix operator fun <C: D1, V: Vec<Int, C>> V.plus(v: V): Vec<Int, C> =
Vec(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(listOf(/*...*/)) // May fail at runtime
val sum = Vec(1, 2) + add // Does not compile
A similar syntax is available for [matrices](core/src/commonMain/kotlin/ai/hypergraph/kotlingrad/api/Matrix.kt) and higherrank [tensors](core/src/commonMain/kotlin/ai/hypergraph/kotlingrad/api/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))
//... † Automatically generated
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](samples/src/main/kotlin/ai/hypergraph/kotlingrad/samples/MatrixDemo.kt) are provided for shapesafe matrix operations such as addition, subtraction and transposition.
A similar technique is possible in Haskell, which is capable of a more powerful form of typelevel 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 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 be computationally tractable due to the practical limitations noted by Grigore.
† Statically generated code, shipped within the library. To regenerate these methods (e.g., using larger dimensions), a code generator is [provided](shipshape/src/main/kotlin/ai/hypergraph/shipshape).
Intermediate representation
Kotlin∇ programs are staged into Kaliningraph, an experimental IR for graph computation. As written by the user, many graphs are computationally suboptimal due to expression swell and parameter sharing. To accelerate forward and backpropagation, it is often advantageous to simplify the graph by applying the reduction semantics in a process known as graph canonicalization. Kaliningraph enables compilerlike optimizations over the graph such as expression simplification and analytic rootfinding, and supports features for visualization and debugging, e.g., in computational notebooks.
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.
Experimental ideas
The current API is stable 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](samples/src/main/kotlin/ai/hypergraph/kotlingrad/samples/VariableCapture.kt), this technique incurs a large source code [overhead](core/src/commonMain/kotlin/ai/hypergraph/kotlingrad/typelevel/VariableCapture.kt). 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 Ndimensional array is known at compiletime, we can use [typelevel integers](shipshape/src/main/kotlin/ai/hypergraph/shipshape/DimGen.kt) 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 allow us to specialize derivation over such matrices (cf. section 2.8 of The Matrix Cookbook).
Church encoding
Computers appear to be very complicated machines. Beneath this complexity lies a remarkably simple idea: many apparently complex routines can be rewritten in terms of function composition. Consider the binary operator ^
, which can be lowered as follows:
a ^ b := a * ... * a
\_________/
b times
a * b := a + ... + a
\_________/
b times
a + b := a + 1 + ... + 1
\_________/
b times
a := next*(next(...next(1)...))
\________________/
a times
∗ next
is also called S
in Peano arithmetic.
By using the λcalculus, Church tells us, we can lower a large portion of mathematics onto a single operator: function application. Curry, by way of Schönfinkel, gives us combinatory logic, a kind of Rosetta stone for deciphering and translating between a host of cryptic languages. These two ideas, λcalculus and combinators, are keys to unlocking many puzzles in computer science and mathematics.
Though mathematically elegant, Church numerals are not particularly efficient or pleasant to read. One discovers that trying to encode Church arithmetic in a language without dependent types grows quickly impractical. By selecting a higher radix, however, it is possible to reduce spatial complexity and improve readability, albeit at the cost of increased temporal complexity on certain operations (e.g., +
and 
). Kotlin∇ uses a binary encoding by default, however generators for other bases are also provided for convenience.
Type classes
The trouble with numerical towers is that they assume all inheritors are aware of the tower. In practice, many types we would like to reuse are entirely oblivious to our DSL. How do we allow users to bring in existing types without needing to modify their source code? This kind of ad hoc polymorphism can be achieved using a pattern called the type class. While the JVM does not allow multiple inheritance on classes, it does support multiple inheritance and default methods on interfaces, allowing users to implement an interface via delegation rather than inheritance.
Suppose we have a base type, Nat
defined as an interface with a unitary member, nil
, and its successor function, next
, representing the Church encoding for natural numbers. To emulate instantiation, we can provide a nested class equipped with a constructor overriding nil
and next
as follows:
interface Nat<T> {
val nil: T
val one: T get() = nil.next()
fun T.next(): T
class of<T>(
override val nil: T,
val vnext: T.() > T
): Nat<T> {
override fun T.next(): T = vnext()
}
}
Now, if we wanted to wrap an external type, such as Double
, inside our tower, we could do so as follows:
val doubleNat = Nat.of(nil = 0.0) { this + 1.0 }
Although the Nat
interface is very expressive, evaluating arithmetic expressions on Nat
s can be computationally expensive. For instance, we could define the first three hyperoperations naïvely as follows:
tailrec fun <T> Nat<T>.plus(l: T, r: T, acc: T = l, i: T = nil): T =
if (i == r) acc else plus(l, r, acc.next(), i.next())
tailrec fun <T> Nat<T>.times(l: T, r: T, acc: T = nil, i: T = nil): T =
if (i == r) acc else times(l, r, acc + l, i.next())
tailrec fun <T> Nat<T>.pow(base: T, exp: T, acc: T = one, i: T = one): T =
if (i == exp) acc else pow(base, exp, acc * base, i.next())
However, we note that computing pow(a, b)
using this representation requires 𝓞(a↑b) operations using Knuth notation. Clearly, we must do better if this encoding is to be usable. We can make Nat
more efficient by introducing a subtype, Group
, which forces implementors to define a native addition operator:
interface Group<T>: Nat<T> {
override fun T.next(): T = this + one
override fun T.plus(t: T): T
class of<T>(
override val nil: T, override val one: T,
val plus: (T, T) > T
): Group<T> {
override fun T.plus(t: T) = plus(this, t)
}
}
Given a Group
, we can now define a more efficient implementation of Fibonacci. This will use the groupspecific addition operator:
tailrec fun <T> Nat<T>.fibonacci(
n: T,
seed: Pair<T, T> = nil to one,
fib: (Pair<T, T>) > Pair<T, T> = { (a, b) > b to a + b },
i: T = nil,
): T =
if (i == n) fib(seed).first
else fibonacci(n = n, seed = fib(seed), i = i.next())
val doubleGroup = Group.of(one = 1.0, plus = { a, b > a + b })
println(doubleGroup.fibonacci(10.0)) // Prints: 233.0
We could further extend this chain by introducing a subtype called Ring
, which overrides +
and requires implementors to define a native *
operator. Ring
s and their relatives are known to have many useful applications in graph theory and statistics:
interface Ring<T>: Group<T> {
override fun T.plus(t: T): T
override fun T.times(t: T): T
class of<T>(
override val nil: T, override val one: T,
val plus: (T, T) > T,
val times: (T, T) > T
): Ring<T> {
override fun T.plus(t: T) = plus(this, t)
override fun T.times(t: T) = times(this, t)
}
}
val doubleRing = Ring.of(one = 1.0, plus = { a, b > a + b }, times = { a, b > a * b })
Since differentiation is a linear map between function spaces, we now have the primitives necessary to build a fullygeneric AD system, and could easily implement the sum and product rules. To view the above example in full, see Types.kt
.
What benefit does this abstraction provide to the end user? By parameterizing over primitive operators, Kotlin∇ consumers can easily swap out a tensor backend without needing to alter or recompile any upstream dependencies. This feature makes multiplatform development a breeze: wherever a type class operator (e.g., +
or *
) with matching signature is encountered across a project, it will be dispatched to the usersupplied lambda delegate for specialized execution on custom hardware. Runtime indirection can be elided with proper compiler inlining for zerocost abstraction.
Type arithmetic
By default, Kotlin∇ supports compile time type arithmetic in the following domain:
 Fully symmetric arithmetic:
{ a ⍟ b ϵ [0..16){+,,*}[0..16)  0 ≤ a ⍟ b }
 Asymmetric arithmetic:
{ a ⍟ b ϵ [0..512){+,}[0..16)  0 ≤ a ⍟ b < 512 }
 Semisymmetric arithmetic:
{ a / b = c, a = b * c  a, b, c ϵ [0..128) & a % b = 0 }
Arithmetic outside this domain is checked at runtime, prior to evaluation.
Compile time type arithmetic is achieved by generating a typelevel representation of the Church encoding. A usage example is shown in ChurchArithmeticTest.kt
, which may be run with the following command:
./gradlew :kotlingrad:cleanJvmTest :kotlingrad:jvmTest tests "ai.hypergraph.kotlingrad.typelevel.church.ChurchArithmeticTest"
Extensions to other bases, including binary and decimal are also provided, which may be used as follows:
// Boolean arithmetic
val b32 = T.F
.let { it + T.F } // B_4<Ø>
.let { it + T.F.F } // B_8<Ø>
.let { it + T.T } // T<T<F<T<Ø>>>>
.let { it + T.F } // T<F<T<T<Ø>>>>
.let { it  T.F } // T<T<F<T<Ø>>>>
.let { it + T.F } // T<F<T<T<Ø>>>>
.let { it + T.F } // T<T<T<T<Ø>>>>
.let { it + T } // T<F<F<F<Ø>>>>
assertEquals(T.F.F.F.F, b32)
// Chinese arithmetic
val 四十二 = (十七 减 九)
.let { it 加 it } // 六<一<无>>
.let { (it 加 八) 加 六 } // 零<三<无>>
.let { (it 减 三) 加 九 } // 六<三<无>>
.let { (it 加 六) 除 六 } // 七<无>
.let { (it 乘 六) 加 五 } // 七<四<无>>
.let { (it 减 三) 减 九 } // 五<三<无>>
.let { (it 加 五) 加 二 } // 二<四<无>>
.also { assertEquals(六 乘 七, it) }
assertEquals(42, 四十二.toInt())
To alter the arithmetic domain, edit the file BinGen.kt
/算盘厂.kt
, then use the following command to regenerate Arithmetic.kt
/算盘.kt
:
./gradlew genShapes
In practice, compile time type arithmetic may struggle to compute numbers in excess of 4095
. The Kotlin team has been informed of these issues:
This API is experimental and subject to change without notice. In the future, it will be used to statically type check tensor functions whose output shape is an arithmetic function of the input shapes, e.g., concatenation, splitting and convolution.
Grammar
For a detailed grammar and semantics, please refer to [the Kotlin∇ specification](specification.md).
UML Diagram
The following graph depicts the subtyping relation between classes and interfaces in Kotlin∇.
[](samples/src/main/resources/uml_diagram.svg)
Comparison
Unlike certain frameworks which simply wrap an existing AD library in a typesafe DSL, Kotlin∇ contains a fully shapesafe 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 dualnumber 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:  :construction:  :heavy_check_mark: 
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: 
shapesafe  Scala  :construction:  :construction:  :construction:  :construction:  :heavy_check_mark:  :heavy_check_mark:  :heavy_check_mark:  :construction:  :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: 
SICMUtils  Clojure  :heavy_check_mark:  :heavy_check_mark:  :heavy_check_mark:  :heavy_check_mark:  :heavy_check_mark:  :x:  :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: 
Dex  Haskell  :x:  :heavy_check_mark:  :heavy_check_mark:  :heavy_check_mark:  :heavy_check_mark:  :heavy_check_mark:  :heavy_check_mark:  :construction:  :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:  :heavy_check_mark:  :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 Higherorder/rank differentiation, ⁴ Differentiable programming*, ⁵ Functional programming, ⁶ Compiletime type safety, ⁷ Compiletime shape safety, ⁸ Dependently Typed, ⁹ Multiplatform
∗ Although we do not distinguish between AD and SD, here we adopt the authors' preferred nomenclature. We do make a distinction between differentiable programming libraries and those which simply construct neural networks. The :construction: symbol indicates work in progress.
References
To the author's knowledge, Kotlin∇ is the first AD implementation in native Kotlin. While the particular synthesis of these ideas (i.e. shapesafe, 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
 The Simple Essence of Automatic Differentiation
 ReverseMode AD in a Functional Framework: Lambda the Ultimate Backpropagator
 Automatic differentiation in ML: Where we are and where we should be going
 A Leibniz Notation for Automatic Differentiation
 FirstClass Automatic Differentiation in Swift: A Manifesto
 The (JAX) Autodiff Cookbook
 Automatic Differentiation in PyTorch
 Automatic Differentiation in Machine Learning: a Survey
 Complexity of Derivatives Generated by Symbolic Differentiation
 EigenAD: Algorithmic Differentiation of the Eigen Library
Complexity
 Fast parallel computation of polynomials using few processors, Valiant and Skyum (1983)
 The complexity of partial derivatives, Baur and Strassen (1983)
 Lower Bounds on Arithmetic Circuits via Partial Derivatives
 Learning Restricted Models of Arithmetic Circuits
Differentiable programming
 Neural Networks, Types, and Functional Programming
 Backpropagation with Continuation Callbacks: Foundations for Efficient and Expressive Differentiable Programming
 Backprop as Functor: A compositional perspective on supervised learning
 Demystifying Differentiable Programming: Shift/Reset the Penultimate Backpropagator
 Efficient Differentiable Programming in a Functional ArrayProcessing Language
 Operational Calculus for Differentiable Programming
 Differentiable Functional Programming
 Differentiable Programming for Image Processing and Deep Learning in Halide
 Software 2.0
Calculus
 The Matrix Calculus You Need For Deep Learning, Parr and Howard (2018)
 Backpropagation in matrix notation, Mishachev (2017)
 Matrix derivatives, from the Matrix Cookbook
 Div, Grad, Curl and All That, Petersen and Pedersen (2012)
 Matrix Differentiation (and some other stuff), Barnes (2006)
 Symbolic Matrix Derivatives, Dwyer and Macphail (1948)
Computer algebra
 Towards an API for the real numbers, Boehm (2020)
 miniKanren as a Tool for Symbolic Computation in Python, Willard (2020)
 A Design Proposal for an Object Oriented Algebraic Library, Niculescu (2003)
 On Using Generics for Implementing Algebraic Structures, Niculescu (2011)
 How to turn a scripting language into a domainspecific language for computer algebra, Jolly and Kredel (2008)
 Evaluation of a Java Computer Algebra System, Kredel (2007)
 Typesafe Abstractions for Tensor Operations, Chen (2017)
 Einstein Summation in Numpy, Bilaniuk (2016)
 Issues in Computer Algebra, NunesHarwitt
 Term Rewriting and All That, Baader and Nipkow (1998)
 Describing the syntax of programming languages using conjunctive and Boolean grammars, Okhotin (2016)
 Formal languages over GF(2), Okhotin (2019)
Symbolic mathematics
 KMath  Kotlin mathematics extensions library
 SymJa  Computer algebra language & symbolic math library for Android
 tensor  Linear algebra for tensors with symbolic and numeric scalars
 Hipparchus  An efficient, generalpurpose mathematics components library in the Java programming language
 miniKanren  A tool for symbolic computation and logic programming
 SymJava  A Java library for fast symbolicnumeric 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
 Hacker's Guide to Neural Networks, Karpathy (2014)
 Tricks from Deep Learning, Baydin et al. (2016)
 Practical Dependent Types in Haskell: TypeSafe Neural Networks, Le (2016)
 A guide to convolutional arithmetic for deep learning, Dumoulin and Visin (2018)
Type systems
 Generalized Algebraic Data Types and ObjectOriented Programming, Kennedy and Russo (2005)
 Java Generics are Turing Complete, Grigore (2016)
 Dimension Types, Kennedy (2004)
 An algebraic view of dimension types, Kennedy (1996)
 Type Inference and Unification
 Constructive mathematics and computer programming, MartinLof (1984)
 Programming in MartinLöf's Type Theory, Nordstrom et al. (1990)
Domainspecific languages
 Compiling Embedded Languages, Elliott et al. (2003)
 Implicit Staging of EDSL Expressions: A Bridge between Shallow and Deep Embedding, Scherr and Chiba (2014)
 DSL Implementation Using Staging and Monads Sheard et al. (1999)
 Deeply Reifying Running Code for Constructing a DomainSpecific Language, Chiba et al. (2016)
 Staged Abstract Interpreters, Wei et al. (2019)
 Generating Fluent Embedded DomainSpecific Languages with Subchaining, Nakamaru et al. (2019)
 Generating a Generic Fluent API in Java, Nakamarua and Chiba (2020)
 Fling – A Fluent API Generator, Gil and Roth (2019)
 Scripting an IDE for EDSL awareness, Sergey et al. (2011)
Automated testing
 DeepTest: Automated Testing of DeepNeuralNetworkdriven Autonomous Cars, Tian et al. (2018)
 QuickCheck: A Lightweight Tool for Random Testing of Haskell Programs, Claessen and Hughes (2000)
 Learning to Discover Efficient Mathematical Identities, Zaremba et al. (2014)
AD libraries
 TensorFlow.FSharp: An eDSL for writing numerical models in F# with support for interactive tensor shapechecking
 Stalin∇, a brutally optimizing compiler for the VLAD language, a pure dialect of Scheme with firstclass automatic differentiation operators
 Autograd  Efficiently computes derivatives of NumPy code
 Myia  SCT based AD, adapted from Pearlmutter & Siskind's "Reverse Mode AD in a functional framework"
 JAX  Composable transformations of Python+NumPy programs: differentiate, vectorize, JIT to GPU/TPU, and more
 Dex  Research language for array processing in the Haskell/ML family
 Nexus  Typesafe tensors, deep learning and probabilistic programming in Scala
 Tangent  "SourcetoSource 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 multistage programming
 JAutoDiff  An Automatic Differentiation Library
 DiffSharp, a functional AD library implemented in the F# language
 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.