Aith is a perfomant systems programming language with am empathises on type systems. As of now Aith is very early stages and very little is implemented. Link to typing rules.

visualization of compiler internals

visualization of type system

pure type system subset

Features

(todo: expand on all of these)

Levity Polymorphic System-F

In languages like in C++ or Rust, generics perform monomorphization. When a generic is used in these languages they will generate code for each instante of type they use.

Rather then do this, Aith uses levity polymorphism, which can be seen as a generalization of Java's type erasure generics. In Aith, a type's kind, which is the type of a type, determines how (and if) it will be represented at runtime.

First Class Inline Functions (staging)

Aith has first class inline functions, a unique (as far as I can tell) take on staging. In Aith, inline functions can take inline functions as argument and return inline functions, all of which is completely erased at runtime.

Inline functions that type check always generate valid code and inline functions are prevented from appearing at runtime though kind checking.

Linear / Unique Types

Aith supports linear types and unique types. These are types that limit how copying of variables. Linear types promise that a variable of a linear type will be used exactly once. Unique types promise that a variable of a unique type will has not been aliased.

Aith has linear types at the inline level with multiplicity in the arrow like Linear Haskell. Aith has necessarily unique types at the runtime level with multiplicity via kinds.

Effectful Regions

Aith has support for effectful regions, similar to Rust's lifetimes. Regions allow programs to reason about borrowing and scoping resources (like memory). Conceptually, an executing program has a stack of regions that it accessing at any given time (think stack frames). If a region is alive, then that region and all it's parent regions are valid.

In Aith, regions are effectful, meaning that all runtime expressions are attached to a region that they live in. These expressions can only access memory in their region or regions proven to be parents of said region.

Type Inference

Aith is built on top of Hindley Milner type inference, similiar to Haskell and OCaml. Hindley Milner is a rather fancy type inference scheme that allows the majority of a program to be without type annotations but still statically typed.

First Class Polymorpism (System-F)

Aith implements first class polymorphism at the meta level. This allows treating generic (inline) functions (or more specifically any generic term) as if they where first class. Generic inline functions can be passed to other inline functions and returned from them. As of now, first class polymorphism is only implmented at the meta level for inline functions and not for runtime functions.

As type inference for first class polymorphism is undeciable, some compromises must be made. Aith implements it's own variant of PolyML, which requires explicit boxing and unboxing and requires annotations for variables who's types are instancited.

Boolean Unification

Aith extends Hindley Milner with boolean unification. This allows certain types to contain boolean expressions (and only boolean expressions). Boolean unification is used by Aith to type check both regions and linear types.

Hindley Milner extended with boolean unification preserves it's nice properties such as principle types. This means that types that use booleans infer as nicely as types that don't. The main trade off is that now type checking now involves what is effectively a generalization of SAT solving.

Building and Running Tests

Install ghc, cabal and make. Run make to build aith, make tests to run the tests and make test.c to generate the test c source file.

Todo List

• Proper Documentation
• Algebraic data types
• Newtypes
• Add higher kinded types (System-F A)
• Runtime level higher rank polymorphism
• Simplify boolean types to DNF rather then ANF
• Refactoring and syntax changes as usual

Syntax

Files are lists of declarations, where these declarations could be a plain variable declaration or a path declaration. For example f(x) { x } is a plain declaration and example/f(x) { x } is a path declaration.

Folders concatenates all it's contents where the folder name is prepend to all the declarations. A folder named abc prepends abc/ to all it's contents.

Declarations(code)

Description	Syntax
Inline Term	inline x = e;
Inline Term Ascribe	inline x : σ = e;
Function	x(pm, pm', ...) { e }
Function Ascribe	x(pm, pm', ...) : σ in π { e }
Function Ascribe	x(pm, pm', ...) :: σ { e }
Synonym	type x = σ;
New Type Declaration	newtype x : κ;

Terms(e)

Description	Syntax
Variable	x
Variable	x @_
Variable	x @<σ, σ', ...>
Global Variable	/x/x'/...
Inline Abstraction	\pm -> e
Inline Application	e {e'}
Inline Binding	inline pm = e; e'
Extern	extern [arity] "sym"
Function Application	e (e', e'', ...)
Runtime Binding	let pm = e; e'
Tuple Construction	(e, e', ...)
Read Pointer	*e
Write Pointer	*e = (e')
Array Increment	&e[e']
Array to Pointer	&*e
Number Literal	n
Addition	e + e'
Subtraction	e - e'
Multiplication	e * e'
Divsion	e / e'
Modulus	e % e'
Equality	e == e'
Inequality	e != e'
Less	e < e'
Less or Equal	e <= e'
Greater	e > e'
Greater or Equal	e >= e'
Integer Resize	resize e
True	true
False	false
Switch	switch e { pm -> e; pm' -> e'; ... }
Poly Introduction	ς e
Poly Elimination	e @_
Poly Elimination	e @<σ, σ', ...>
Type Annotation	e : σ
Pretype Annotation	e :: σ
Continue	continue e
Break	break e
Loop	loop (let pm = e) { e' }
Unsafe Cast	cast e

Terms (Syntax Sugar) (e)

Description	Syntax	Meaning
Not	!e	if e { false } else { true }
And	e & e'	if e { e' } else { false }
Or	e | e'	if e { true } else { e' }
Do	e; e'	let () = e; e'
If	if e { e' } else { e''}	switch (e) { true -> e; false -> e'; }

Meta Patterns(pm)

Description	Syntax
Linear Variable	x
Linear Variable Abscribe	x : σ
Unrestricted Variable Abscribe	x :* σ
Polymorphic Variable Abscribe	x :^τ σ

Runtime Patterns(pm)

Description	Syntax
Variable	x
Variable Abscribe	x : σ
Tuple Destruction	(pm, pm', ...)
True	true
False	false

Scheme(ς)

Description	Syntax
TypeScheme	<pmσ, pmσ', ...>

Types(σ, τ, π, κ, ρ)

Description	Syntax
Hole	_
Variable	x
Linear Inline Function	σ -> τ
Unrestricted Inline Function	σ -* τ
Polymorphic Inline Function	σ -π τ
Poly	ς σ
Function Pointer	function(σ, σ', ...) -> τ uses π
Tuple	(σ, σ', ...)
Effect	σ in π
Unique	unique σ
Shared	σ @ π
Pointer	σ*
Array	σ[]
Number	ρ integer(ρ')
Boolean	boolean
IO Region	io
Step	step<σ, τ>
Meta Type	metatype
Type	type<σ, τ>
Boxed	boxed
Capacity	capacity
Region	region
Pointer Representation	pointer
Struct Representation	struct (ρ, ρ', ...)
Union Representation	union (ρ, ρ', ...)
Word Representation	ρ word
Signed	signed
Unsigned	unsigned
Byte Size	8bit
Short Size	16bit
Int Size	32bit
Long Size	64bit
Native Size	native
Representation	representation
Signedness	signedness
Size	size
Type True	true
Type False	false
Type And	σ & τ
Type Or	σ | τ
Type Not	!σ
Type Xor	σ (+) τ

Types (Internal) (σ, τ, π, κ, ρ)

Description	Syntax
Unification	unification
Kind	kind<σ>
Syntactic	syntactic
Propositional	propositional
Top	/|\
Function Literal Type	function literal(σ) -> τ uses π
Label	label
Ambiguous Label	ambiguous

Types (Syntax Sugar) (σ, τ, π, κ, ρ)

Description	Syntax	Meaning
Byte	byte	signed integer(8bit)
Short	short	signed integer(16bit)
Int	int	signed integer(32bit)
Long	long	signed integer(64bit)
PtrDiff	ptrdiff	signed integer(native)
UByte	ubyte	unsigned integer(8bit)
UShort	ushort	unsigned integer(16bit)
UInt	uint	unsigned integer(32bit)
ULong	ulong	unsigned integer(64bit)
Integer	integer(σ)	signed integer(σ)
Natural	natural(σ)	unsigned integer(σ)
Native Integer	integer	signed integer(native)
Native Natural	natural	unsigned integer(native)

Type Pattern(pmσ)

Description	Syntax
Variable	x : κ
Concrete Variable	x :* κ

C Compiler Requirements

This list may be incomplete.

• All pointers must have the same representation (including function pointers).
• Signed integers must have 2's complement wrapping. (-fwrapv on gcc)

Papers

Useful / Inspirational papers:

Implemented

Linear / Unique Types

• Linear Haskell (video)
  • Implemented for inline functions

Levity polymorphism

• Levity Polymorphism (video)
  • Implemented with more restrictive representation lambda.

Compiler Design

• Invertible Syntax Descriptions: Unifying Parsing and Pretty Printing
  • Implemented with prisms instead of partial isomorphisms.

Reduction

• Demonstrating Lambda Calculus Reduction
  • Applicative order reduction used for inline function reduction.

Unification

• Unification Theory
  • Reference for unification and E-unification
• Embedding Boolean Expressions into Logic Programming

Type Inference (First Class Polymorphism)

PolyML is implemented, but with scope type variables are used rather then schematic ones.

• Semi-Explicit First-Class Polymorphism for ML

Compiler Design

• How OCaml type checker works -- or what polymorphism and garbage collection have in common
  • Only implemented for checking escaping skolem variables. Still need to implement let generalization.

Regions

• • Monadic and Substructural Type Systems for Region-Based Memory Management
    • General guide for regions.
  • Monadic Regions
  • A Step-Indexed Model of Substructural State
  • Linear Regions Are All You Need

Inspirational, and / or Formerly Used

General Introductions

• Practical Foundations for Programming Languages
• Type Systems

Pure Type Systems

• An Introduction to Generalized Type Systems
• The Structural Theory of Pure Type Systems (video)

Linear / Unique Types

• • Making Uniqueness Typing Less Unique
  • Uniqueness Typing Redefined
  • Equality-Based Uniqueness Typing
  • Uniqueness Typing Simplified
  • Modelling Unique and Affine Typing using Polymorphism
• A taste of linear logic
• Linearity and Uniqueness: An Entente Cordiale
• The Best of Both Worlds: Linear Functional Programming without Compromise (video)

Effect Tracking

• Polymorphic Types and Effects with Boolean Unification

Type Inference (First Class Polymorphism)

• • HMF: Simple type inference for first-class polymorphism
  • Flexible types: robust type inference for first-class polymorphism
• FreezeML : Complete and Easy Type Inference for First-Class Polymorphism (video)
• A Quick Look at Impredicativity (video)
• QML : Explicit First-Class Polymorphism for ML

Type Inference (Subtyping)

• • Algebraic Subtyping
  • Polymorphism, Subtyping, and Type Inference in MLsub (video)
• The Simple Essence of Algebraic Subtyping (video)

Unification

• Unification Under a Mixed Prefix

Internals

• System F with Type Equality Coercions
• Generalizing Hindley-Milner Type Inference Algorithms

Copyright

Copyright © Freddy A Cubas, "Superstar64"

Licensed under GPL3 only. See LICENSE for more info.