3. Language Reference

This reference seeks to describe every construct in the Futhark language. It is not presented in a tutorial fashion, but rather intended for quick lookup and documentation of subtleties. For this reason, it is not written in a bottom-up manner, and some concepts may be used before they are fully defined. It is a good idea to have a basic grasp of Futhark (or some other functional programming language) before reading this reference. An ambiguous grammar is given for the full language. The text describes how ambiguities are resolved in practice (for example by applying rules of operator precedence).

This reference describes only the language itself. Documentation for the built-in prelude is available elsewhere.

3.1. Identifiers and Keywords

id          ::=  letter (letter | "_" | "'")* | "_" id
quals       ::=  (id ".")+
qualid      ::=  id | quals id
binop       ::=  opstartchar opchar*
qualbinop   ::=  binop | quals binop | "`" qualid "`"
fieldid     ::=  decimal | id
opstartchar ::=  "+" | "-" | "*" | "/" | "%" | "=" | "!" | ">" | "<" | "|" | "&" | "^"
opchar      ::=  opstartchar | "."
constructor ::=  "#" id

Many things in Futhark are named. When we are defining something, we give it an unqualified name (id). When referencing something inside a module, we use a qualified name (qualid). The constructor names of a sum type (Compound Types and Values) are identifiers prefixed with #, with no space afterwards. The fields of a record are named with fieldid. Note that a fieldid can be a decimal number. Futhark has three distinct name spaces: terms, module types, and types. Modules (including parametric modules) and values both share the term namespace.

3.2. Primitive Types and Values

literal ::=  intnumber | floatnumber | "true" | "false"

Boolean literals are written true and false. The primitive types in Futhark are the signed integer types i8, i16, i32, i64, the unsigned integer types u8, u16, u32, u64, the floating-point types f32, f64, as well as bool. An f32 is always a single-precision float and a f64 is a double-precision float.

int_type   ::=  "i8" | "i16" | "i32" | "i64" | "u8" | "u16" | "u32" | "u64"
float_type ::=  "f32" | "f64"

Numeric literals can be suffixed with their intended type. For example 42i8 is of type i8, and 1337e2f64 is of type f64. If no suffix is given, the type of the literal will be inferred based on its use. If the use is not constrained, integral literals will be assigned type i32, and decimal literals type f64. Hexadecimal literals are supported by prefixing with 0x, and binary literals by prefixing with 0b.

Floats can also be written in hexadecimal format such as 0x1.fp3, instead of the usual decimal notation. Here, 0x1.f evaluates to 1 15/16 and the p3 multiplies it by 2^3 = 8.

intnumber   ::=  (decimal | hexadecimal | binary) [int_type]
decimal     ::=  decdigit (decdigit |"_")*
hexadecimal ::=  0 ("x" | "X") hexdigit (hexdigit |"_")*
binary      ::=  0 ("b" | "B") bindigit (bindigit | "_")*
floatnumber      ::=  (pointfloat | exponentfloat) [float_type]
pointfloat       ::=  [intpart] fraction
exponentfloat    ::=  (intpart | pointfloat) exponent
hexadecimalfloat ::=  0 ("x" | "X") hexintpart hexfraction ("p"|"P") ["+" | "-"] decdigit+
intpart          ::=  decdigit (decdigit |"_")*
fraction         ::=  "." decdigit (decdigit |"_")*
hexintpart       ::=  hexdigit (hexdigit | "_")*
hexfraction      ::=  "." hexdigit (hexdigit |"_")*
exponent         ::=  ("e" | "E") ["+" | "-"] decdigit+
decdigit ::=  "0"..."9"
hexdigit ::=  decdigit | "a"..."f" | "A"..."F"
bindigit ::=  "0" | "1"

3.2.1. Compound Types and Values

type ::=    qualid
          | array_type
          | tuple_type
          | record_type
          | sum_type
          | function_type
          | type_application

Compound types can be constructed based on the primitive types. The Futhark type system is entirely structural, and type abbreviations are merely shorthands. The only exception is abstract types whose definition has been hidden via the module system (see Module System).

tuple_type ::=  "(" ")" | "(" type ("," type)+ ")"

A tuple value or type is written as a sequence of comma-separated values or types enclosed in parentheses. For example, (0, 1) is a tuple value of type (i32,i32). The elements of a tuple need not have the same type – the value (false, 1, 2.0) is of type (bool, i32, f64). A tuple element can also be another tuple, as in ((1,2),(3,4)), which is of type ((i32,i32),(i32,i32)). A tuple cannot have just one element, but empty tuples are permitted, although they are not very useful. Empty tuples are written () and are of type ().

array_type ::=  "[" [dim] "]" type
dim        ::=  qualid | decimal

An array value is written as a sequence of zero or more comma-separated values enclosed in square brackets: [1,2,3]. An array type is written as [d]t, where t is the element type of the array, and d is an integer or variable indicating the size. We can often elide d and write just [] (an anonymous size), in which case the size will be inferred. As an example, an array of three integers could be written as [1,2,3], and has type [3]i32. An empty array is written as [], and its type is inferred from its use. When writing Futhark values for such uses as futhark test (but not when writing programs), empty arrays are written empty([0]t) for an empty array of type [0]t. When using empty, all dimensions must be given a size, and at least one must be zero, e.g. empty([2][0]i32).

Multi-dimensional arrays are supported in Futhark, but they must be regular, meaning that all inner arrays must have the same shape. For example, [[1,2], [3,4], [5,6]] is a valid array of type [3][2]i32, but [[1,2], [3,4,5], [6,7]] is not, because there we cannot come up with integers m and n such that [m][n]i32 describes the array. The restriction to regular arrays is rooted in low-level concerns about efficient compilation. However, we can understand it in language terms by the inability to write a type with consistent dimension sizes for an irregular array value. In a Futhark program, all array values, including intermediate (unnamed) arrays, must be typeable.

sum_type ::=  constructor type* ("|" constructor type*)*

Sum types are anonymous in Futhark, and are written as the constructors separated by vertical bars. Each constructor consists of a #-prefixed name, followed by zero or more types, called its payload. Note: The current implementation of sum types is fairly inefficient, in that all possible constructors of a sum-typed value will be resident in memory. Avoid using sum types where multiple constructors have large payloads.

record_type ::=  "{" "}" | "{" fieldid ":" type ("," fieldid ":" type)* "}"

Records are mappings from field names to values, with the field names known statically. A tuple behaves in all respects like a record with numeric field names starting from zero, and vice versa. It is an error for a record type to name the same field twice.

type_application ::=  type type_arg | "*" type
type_arg         ::=  "[" [dim] "]" | type

A parametric type abbreviation can be applied by juxtaposing its name and its arguments. The application must provide as many arguments as the type abbreviation has parameters - partial application is presently not allowed. See Type Abbreviations for further details.

function_type ::=  param_type "->" type
param_type    ::=  type | "(" id ":" type ")"

Functions are classified via function types, but they are not fully first class. See Higher-order functions for the details.

stringlit  ::=  '"' stringchar '"'
stringchar ::=  <any source character except "\" or newline or quotes>

String literals are supported, but only as syntactic sugar for UTF-8 encoded arrays of u8 values. There is no character type in Futhark.

3.3. Declarations

A Futhark file or module consists of a sequence of declarations. Each declaration is processed in order, and a declaration can only refer to names bound by preceding declarations.

dec ::=    fun_bind | val_bind | type_bind | mod_bind | mod_type_bind
         | "open" mod_exp
         | "import" stringlit
         | "local" dec
         | "#[" attr "]" dec

The open declaration brings names defined in another module into scope (see also Module System). For the meaning of import, see Referring to Other Files. If a declaration is prefixed with local, whatever names it defines will not be visible outside the current module. In particular local open is used to bring names from another module into scope, without making those names available to users of the module being defined. In most cases, using module type ascription is a better idea.

3.3.1. Declaring Functions and Values

fun_bind ::=    ("let" | "entry") (id | "(" binop ")") type_param* pat+ [":" type] "=" exp
              | ("let" | "entry") pat binop pat [":" type] "=" exp
val_bind ::=  "let" id [":" type] "=" exp

Functions and values must be defined before they are used. A function declaration must specify the name, parameters, and body of the function:

let name params...: rettype = body

Hindley-Milner-style type inference is supported. A parameter may be given a type with the notation (name: type). Functions may not be recursive. You may put size annotations in the return type and parameter types; see Size Types. A function can be polymorphic by using type parameters, in the same way as for Type Abbreviations:

let reverse [n] 't (xs: [n]t): [n]t = xs[::-1]

Type parameters for a function do not need to cover the types of all parameters. The type checker will add more if necessary. For example, the following is well typed:

let pair 'a (x: a) y = (x, y)

A new type variable will be invented for the parameter y.

Shape and type parameters are not passed explicitly when calling function, but are automatically derived. If an array value v is passed for a type parameter t, all other arguments passed of type t must have the same shape as v. For example, consider the following definition:

let pair 't (x: t) (y: t) = (x, y)

The application pair [1] [2,3] will fail at run-time.

To simplify the handling of in-place updates (see In-place Updates), the value returned by a function may not alias any global variables.

3.3.2. User-Defined Operators

Infix operators are defined much like functions:

let (p1: t1) op (p2: t2): rt = ...

For example:

let (a:i32,b:i32) +^ (c:i32,d:i32) = (a+c, b+d)

We can also define operators by enclosing the operator name in parentheses and suffixing the parameters, as an ordinary function:

let (+^) (a:i32,b:i32) (c:i32,d:i32) = (a+c, b+d)

This is necessary when defining a polymorphic operator.

A valid operator name is a non-empty sequence of characters chosen from the string "+-*/%=!><&^". The fixity of an operator is determined by its first characters, which must correspond to a built-in operator. Thus, +^ binds like +, whilst *^ binds like *. The longest such prefix is used to determine fixity, so >>= binds like >>, not like >.

It is not permitted to define operators with the names && or || (although these as prefixes are accepted). This is because a user-defined version of these operators would not be short-circuiting. User-defined operators behave exactly like ordinary functions, except for being infix.

A built-in operator can be shadowed (i.e. a new + can be defined). This will result in the built-in polymorphic operator becoming inaccessible, except through the intrinsics module.

An infix operator can also be defined with prefix notation, like an ordinary function, by enclosing it in parentheses:

let (+) (x: i32) (y: i32) = x - y

This is necessary when defining operators that take type or shape parameters.

3.3.3. Entry Points

Apart from declaring a function with the keyword let, it can also be declared with entry. When the Futhark program is compiled any top-level function declared with entry will be exposed as an entry point. If the Futhark program has been compiled as a library, these are the functions that will be exposed. If compiled as an executable, you can use the --entry-point command line option of the generated executable to select the entry point you wish to run.

Any top-level function named main will always be considered an entry point, whether it is declared with entry or not.

3.3.4. Value Declarations

A named value/constant can be declared as follows:

let name: type = definition

The definition can be an arbitrary expression, including function calls and other values, although they must be in scope before the value is defined. A constant value may not have a unique type (see In-place updates). If the return type contains any anonymous sizes (see Size types), new existential sizes will be constructed for them.

3.3.5. Type Abbreviations

type_bind  ::=  "type" ["^" | "~"] id type_param* "=" type
type_param ::=  "[" id "]" | "'" id | "'~" id | "'^" id

Type abbreviations function as shorthands for the purpose of documentation or brevity. After a type binding type t1 = t2, the name t1 can be used as a shorthand for the type t2. Type abbreviations do not create distinct types: the types t1 and t2 are entirely interchangeable.

If the right-hand side of a type contains anonymous sizes, it must be declared “size-lifted” with type~. If it (potentially) contains a function, it must be declared “fully lifted” with type^. A lifted type can also contain anonymous sizes. Lifted types cannot be put in arrays. Fully lifted types cannot be returned from conditional or loop expressions.

A type abbreviation can have zero or more parameters. A type parameter enclosed with square brackets is a size parameter, and can be used in the definition as an array dimension size, or as a dimension argument to other type abbreviations. When passing an argument for a shape parameter, it must be enclosed in square brackets. Example:

type two_intvecs [n] = ([n]i32, [n]i32)

let x: two_intvecs [2] = (iota 2, replicate 2 0)

Size parameters work much like shape declarations for arrays. Like shape declarations, they can be elided via square brackets containing nothing. All size parameters must be used in the definition of the type abbreviation.

A type parameter prefixed with a single quote is a type parameter. It is in scope as a type in the definition of the type abbreviation. Whenever the type abbreviation is used in a type expression, a type argument must be passed for the parameter. Type arguments need not be prefixed with single quotes:

type two_vecs [n] 't = ([n]t, [n]t)
type two_intvecs [n] = two_vecs [n] i32
let x: two_vecs [2] i32 = (iota 2, replicate 2 0)

A size-lifted type parameter is prefixed with '~, and a fully lifted type parameter with '^. These have the same rules and restrictions as lifted type abbreviations.

3.4. Expressions

Expressions are the basic construct of any Futhark program. An expression has a statically determined type, and produces a value at runtime. Futhark is an eager/strict language (“call by value”).

The basic elements of expressions are called atoms, for example literals and variables, but also more complicated forms.

atom     ::=    literal
              | qualid ("." fieldid)*
              | stringlit
              | "(" ")"
              | "(" exp ")" ("." fieldid)*
              | "(" exp ("," exp)* ")"
              | "{" "}"
              | "{" field ("," field)* "}"
              | qualid "[" index ("," index)* "]"
              | "(" exp ")" "[" index ("," index)* "]"
              | quals "." "(" exp ")"
              | "[" exp ("," exp)* "]"
              | "[" exp [".." exp] "..." exp "]"
              | "(" qualbinop ")"
              | "(" exp qualbinop ")"
              | "(" qualbinop exp ")"
              | "(" ( "." field )+ ")"
              | "(" "." "[" index ("," index)* "]" ")"
exp      ::=    atom
              | exp qualbinop exp
              | exp exp
              | constructor exp*
              | exp ":" type
              | exp ":>" type
              | exp [ ".." exp ] "..." exp
              | exp [ ".." exp ] "..<" exp
              | exp [ ".." exp ] "..>" exp
              | "if" exp "then" exp "else" exp
              | "let" pat "=" exp "in" exp
              | "let" id "[" index ("," index)* "]" "=" exp "in" exp
              | "let" id type_param* pat+ [":" type] "=" exp "in" exp
              | "(" "\" pat+ [":" type] "->" exp ")"
              | "loop" pat [("=" exp)] loopform "do" exp
              | "#[" attr "]" exp
              | "unsafe" exp
              | "assert" atom atom
              | exp "with" "[" index ("," index)* "]" "=" exp
              | exp "with" fieldid ("." fieldid)* "=" exp
              | "match" exp ("case" pat "->" exp)+
field    ::=    fieldid "=" exp
              | id
pat      ::=    id
              | literal
              |  "_"
              | "(" ")"
              | "(" pat ")"
              | "(" pat ("," pat)+ ")"
              | "{" "}"
              | "{" fieldid ["=" pat] ["," fieldid ["=" pat]] "}"
              | constructor pat*
              | pat ":" type
loopform ::=    "for" id "<" exp
              | "for" pat "in" exp
              | "while" exp
index    ::=    exp [":" [exp]] [":" [exp]]
              | [exp] ":" exp [":" [exp]]
              | [exp] [":" exp] ":" [exp]

Some of the built-in expression forms have parallel semantics, but it is not guaranteed that the the parallel constructs in Futhark are evaluated in parallel, especially if they are nested in complicated ways. Their purpose is to give the compiler as much freedom and information is possible, in order to enable it to maximise the efficiency of the generated code.

3.4.1. Resolving Ambiguities

The above grammar contains some ambiguities, which in the concrete implementation is resolved via a combination of lexer and grammar transformations. For ease of understanding, they are presented here in natural text.

  • An expression x.y may either be a reference to the name y in the module x, or the field y in the record x. Modules and values occupy the same name space, so this is disambiguated by the type of x.

  • A type ascription (exp : type) cannot appear as an array index, as it conflicts with the syntax for slicing.

  • In f [x], there is am ambiguity between indexing the array f at position x, or calling the function f with the singleton array x. We resolve this the following way:

    • If there is a space between f and the opening bracket, it is treated as a function application.

    • Otherwise, it is an array index operation.

  • An expression (-x) is parsed as the variable x negated and enclosed in parentheses, rather than an operator section partially applying the infix operator -.

  • The following table describes the precedence and associativity of infix operators. All operators in the same row have the same precedence. The rows are listed in increasing order of precedence. Note that not all operators listed here are used in expressions; nevertheless, they are still used for resolving ambiguities.






    :, :>






    <= >= > < == !=


    & ^ |


    << >>


    + -


    * / % // %%









3.4.2. Patterns

We say that a pattern is irrefutable if it can never fail to match a value of the appropriate type. Concretely, this means that it does not require any specific sum type constructor (unless the type in question has only a single constructor), or any specific numeric or boolean literal. Patterns used in function parameters and let bindings must be irrefutable. Patterns used in case need not be irrefutable.

A pattern _ matches any value. A pattern consisting of a literal value (e.g. a numeric constant) matches exactly that value.

3.4.3. Semantics of Simple Expressions literal

Evaluates to itself. qualid

A variable name; evaluates to its value in the current environment. stringlit

Evaluates to an array of type []i32 that contains the code points of the characters as integers. ()

Evaluates to an empty tuple. ( e )

Evaluates to the result of e. (e1, e2, ..., eN)

Evaluates to a tuple containing N values. Equivalent to the record literal {0=e1, 1=e2, ..., N-1=eN}. {f1, f2, ..., fN}

A record expression consists of a comma-separated sequence of field expressions. Each field expression defines the value of a field in the record. A field expression can take one of two forms:

f = e: defines a field with the name f and the value resulting from evaluating e.

f: defines a field with the name f and the value of the variable f in scope.

Each field may only be defined once. a[i]

Return the element at the given position in the array. The index may be a comma-separated list of indexes instead of just a single index. If the number of indices given is less than the rank of the array, an array is returned.

The array a must be a variable name or a parenthesised expression. Furthermore, there may not be a space between a and the opening bracket. This disambiguates the array indexing a[i], from a [i], which is a function call with a literal array. a[i:j:s]

Return a slice of the array a from index i to j, the former inclusive and the latter exclusive, taking every s-th element. The s parameter may not be zero. If s is negative, it means to start at i and descend by steps of size s to j (not inclusive).

It is generally a bad idea for s to be non-constant. Slicing of multiple dimensions can be done by separating with commas, and may be intermixed freely with indexing.

If s is elided it defaults to 1. If i or j is elided, their value depends on the sign of s. If s is positive, i become 0 and j become the length of the array. If s is negative, i becomes the length of the array minus one, and j becomes minus one. This means that a[::-1] is the reverse of the array a.

In the general case, the size of the array produced by a slice is unknown (see Size types). In a few cases, the size is known statically:

  • a[0:n] has size n

  • a[:n] has size n

  • a[0:n:1] has size n

  • a[:n:1] has size n

This holds only if n is a variable or constant. [x, y, z]

Create an array containing the indicated elements. Each element must have the same type and shape. x..y...z

Construct an integer array whose first element is x and which proceeds stride of y-x until reaching z (inclusive). The ..y part can be elided in which case a stride of 1 is used. A run-time error occurs if z is lesser than x or y, or if x and y are the same value.

In the general case, the size of the array produced by a range is unknown (see Size types). In a few cases, the size is known statically:

  • 1..2...n has size n

This holds only if n is a variable or constant. x..y..<z

Construct an integer array whose first elements is x, and which proceeds upwards with a stride of y until reaching z (exclusive). The ..y part can be elided in which case a stride of 1 is used. A run-time error occurs if z is lesser than x or y, or if x and y are the same value.

  • 0..1..<n has size n

  • 0..<n has size n

This holds only if n is a variable or constant. x..y..>z

Construct an integer array whose first elements is x, and which proceeds downwards with a stride of y until reaching z (exclusive). The ..y part can be elided in which case a stride of -1 is used. A run-time error occurs if z is greater than x or y, or if x and y are the same value. e.f

Access field f of the expression e, which must be a record or tuple. m.(e)

Evaluate the expression e with the module m locally opened, as if by open. This can make some expressions easier to read and write, without polluting the global scope with a declaration-level open. x binop y

Apply an operator to x and y. Operators are functions like any other, and can be user-defined. Futhark pre-defines certain “magical” overloaded operators that work on many different types. Overloaded functions cannot be defined by the user. Both operands must have the same type. The predefined operators and their semantics are:


Power operator, defined for all numeric types.

//, %%

Division and remainder on integers, with rounding towards zero.

*, /, %, +, -

The usual arithmetic operators, defined for all numeric types. Note that / and % rounds towards negative infinity when used on integers - this is different from in C.

^, &, |, >>, <<

Bitwise operators, respectively bitwise xor, and, or, arithmetic shift right and left, and logical shift right. Shift amounts must be non-negative and the operands must be integers. Note that, unlike in C, bitwise operators have higher priority than arithmetic operators. This means that x & y == z is understood as (x & y) == z, rather than x & (y == z) as it would in C. Note that the latter is a type error in Futhark anyhow.

==, !=

Compare any two values of builtin or compound type for equality.

<, <=. >, >=

Company any two values of numeric type for equality. x && y

Short-circuiting logical conjunction; both operands must be of type bool. x || y

Short-circuiting logical disjunction; both operands must be of type bool. f x

Apply the function f to the argument x. #c x y z

Apply the sum type constructor #x to the payload x, y, and z. A constructor application is always assumed to be saturated, i.e. its entire payload provided. This means that constructors may not be partially applied. e : t

Annotate that e is expected to be of type t, failing with a type error if it is not. If t is an array with shape declarations, the correctness of the shape declarations is checked at run-time.

Due to ambiguities, this syntactic form cannot appear as an array index expression unless it is first enclosed in parentheses. However, as an array index must always be of type i32, there is never a reason to put an explicit type ascription there. e :> t

Coerce the size of e to t. The type of t must match the type of e, except that the sizes may be statically different. At run-time, it will be verified that the sizes are the same. ! x

Logical negation if x is of type bool. Bitwise negation if x is of integral type. - x

Numerical negation of x, which must be of numeric type. #[attr] e

Apply the given attribute to the expression. Attributes are an ad-hoc and optional mechanism for providing extra information, directives, or hints to the compiler. See Attributes for more information. unsafe e

Elide safety checks and assertions (such as bounds checking) that occur during execution of e. This is useful if the compiler is otherwise unable to avoid bounds checks (e.g. when using indirect indexes), but you really do not want them there. Make very sure that the code is correct; eliding such checks can lead to memory corruption.

This construct is deprecated. Use the #[unsafe] attribute instead. assert cond e

Terminate execution with an error if cond evaluates to false, otherwise produce the result of evaluating e. Unless e produces a value that is used subsequently (it can just be a variable), dead code elimination may remove the assertion. a with [i] = e

Return a, but with the element at position i changed to contain the result of evaluating e. Consumes a. r with f = e

Return the record r, but with field f changed to have value e. The type of the field must remain unchanged. Type inference is limited: r must have a completely known type up to f. This sometimes requires extra type annotations to make the type of r known. if c then a else b

If c evaluates to true, evaluate a, else evaluate b.

3.4.4. Binding Expressions let pat = e in body

Evaluate e and bind the result to the irrefutable pattern pat (see Patterns) while evaluating body. The in keyword is optional if body is a let expression. let a[i] = v in body

Write v to a[i] and evaluate body. The given index need not be complete and can also be a slice, but in these cases, the value of v must be an array of the proper size. This notation is Syntactic sugar for let a = a with [i] = v in a. let f params... = e in body

Bind f to a function with the given parameters and definition (e) and evaluate body. The function will be treated as aliasing any free variables in e. The function is not in scope of itself, and hence cannot be recursive. loop pat = initial for x in a do loopbody

  1. Bind pat to the initial values given in initial.

  2. For each element x in a, evaluate loopbody and rebind pat to the result of the evaluation.

  3. Return the final value of pat.

The = initial can be left out, in which case initial values for the pattern are taken from equivalently named variables in the environment. I.e., loop (x) = ... is equivalent to loop (x = x) = .... loop pat = initial for x < n do loopbody

Equivalent to loop (pat = initial) for x in [0..1..<n] do loopbody. loop pat = initial while cond do loopbody

  1. Bind pat to the initial values given in initial.

  2. If cond evaluates to true, bind pat to the result of evaluating loopbody, and repeat the step.

  3. Return the final value of pat. match x case p1 -> e1 case p2 -> e2

Match the value produced by x to each of the patterns in turn, picking the first one that succeeds. The result of the corresponding expression is the value of the entire match expression. All the expressions associated with a case must have the same type (but not necessarily match the type of x). It is a type error if there is not a case for every possible value of x - inexhaustive pattern matching is not allowed.

3.4.5. Function Expressions \x y z: t -> e

Produces an anonymous function taking parameters x, y, and z, returns type t, and whose body is e. Lambdas do not permit type parameters; use a named function if you want a polymorphic function. (binop)

An operator section that is equivalent to \x y -> x *binop* y. (x binop)

An operator section that is equivalent to \y -> x *binop* y. (binop y)

An operator section that is equivalent to \x -> x *binop* y. (.a.b.c)

An operator section that is equivalent to \x -> x.a.b.c. (.[i,j])

An operator section that is equivalent to \x -> x[i,j].

3.5. Higher-order functions

At a high level, Futhark functions are values, and can be used as any other value. However, to ensure that the compiler is able to compile the higher-order functions efficiently via defunctionalisation, certain type-driven restrictions exist on how functions can be used. These also apply to any record or tuple containing a function (a functional type):

  • Arrays of functions are not permitted.

  • A function cannot be returned from an if expression.

  • A loop parameter cannot be a function.

Further, type parameters are divided into non-lifted (bound with an apostrophe, e.g. 't), size-lifted ('~t), and fully lifted ('^t). Only fully lifted type parameters may be instantiated with a functional type. Within a function, a lifted type parameter is treated as a functional type.

See also In-place updates for details on how uniqueness types interact with higher-order functions.

3.6. Type Inference

Futhark supports Hindley-Milner-style type inference, so in many cases explicit type annotations can be left off. Record field projection cannot in isolation be fully inferred, and may need type annotations where their inputs are bound. The same goes when constructing sum types, as Futhark cannot assume that a given constructor only belongs to a single type. Further, unique types (see In-place updates) must be explicitly annotated.

3.7. Size Types

Futhark supports a simple system of size-dependent types that statically verifies that the sizes of arrays passed to a function are compatible. The focus is on simplicity, not completeness.

Whenever a pattern occurs (in let, loop, and function parameters), as well as in return types, size annotations may be used to express invariants about the shapes of arrays that are accepted or produced by the function. For example:

let f [n] (a: [n]i32) (b: [n]i32): [n]i32 =
  map2 (+) a b

We use a size parameter, [n], to explicitly quantify sizes. The [n] parameter is not explicitly passed when calling f. Rather, its value is implicitly deduced from the arguments passed for the value parameters. An array can contain anonymous dimensions, e.g. []i32, for which the type checker will invent fresh size parameters, which ensures that all arrays have a (symbolic) size.

A size annotation can also be an integer constant (with no suffix). Size parameters can be used as ordinary variables within the scope of the parameters. The type checker verifies that the program obeys any constraints imposed by size annotations.

Size-dependent types are supported, as the names of parameters can be used in the return type of a function:

let replicate 't (n: i32) (x: t): [n]t = ...

An application replicate 10 0 will have type [10]i32.

3.7.1. Unknown sizes

Since sizes must be constants or variables, there are many cases where the type checker cannot assign a precise size to the result of some operation. For example, the type of concat should conceptually be:

val concat [n] [m] 't : [n]t -> [m]t -> [n+m]t

But this is not presently allowed. Instead, the return type contains an anonymous size:

val concat [n] [m] 't : [n]t -> [m]t -> []t

When an application concat xs ys is found, the result will be of type [k]t, where k is a fresh unknown size variable that is considered distinct from every other size in the program. It is often necessary to perform a size coercion (see Size coercion) to convert an unknown size to a known size.

Generally, unknown sizes are constructed whenever the true size cannot be expressed. The following lists all possible sources of unknown sizes. Size going out of scope

An unknown size is created when the proper size of an array refers to a name that has gone out of scope:

let c = a + b
in replicate c 0

The type of replicate c 0 is [c]i32, but since c is locally bound, the type of the entire expression is [k]i32 for some fresh k. Compound expression passed as function argument

Intuitively, the type of replicate (x+y) 0 should be [x+y]i32, but since sizes must be names or constants, this is not expressible. Therefore an unknown size variable is created and the size of the expression becomes [k]i32. Compound expression used as range bound

While a simple range expression such as 0..<n can be assigned type [n]i32, a range expression 0..<(n+1) will give produce an unknown size. Complex slicing

Most complex array slicing, such as xs[a:b], will have an unknown size. Exceptions are listed in the reference for slice expressions. Complex ranges

Most complex ranges, such as a..<b, will have an known size. Exceptions exist for general ranges and “upto” ranges. Anonymous size in function return type

Whenever the result of a function application would have an anonymous size, that size is replaced with a fresh unknown size variable.

For example, filter has the following type:

val filter [n] 'a : (p: a -> bool) -> (as: [n]a) -> []a

Naively, an application filter f xs seems like it would have type []a, but a fresh unknown size k will be created and the actual type will be [k]a. Branches of if return arrays of different sizes

When an if (or match) expression has branches that returns array of different sizes, the differing sizes will be replaced with fresh unknown sizes. For example:

if b then [[1,2], [3,4]]
     else [[5,6]]

This expression will have type [k][2]i32, for some fresh k.

Important: The check whether the sizes differ is done when first encountering the if or match during type checking. At this point, the type checker may not realise that the two sizes are actually equal, even though constraints later in the function force them to be. This can always be resolved by adding type annotations. An array produced by a loop does not have a known size

If the size of some loop parameter is not maintained across a loop iteration, the final result of the loop will contain unknown sizes. For example:

loop xs = [1] for i < n do xs ++ xs

Similar to conditionals, the type checker may sometimes be too cautious in assuming that some size may change during the loop. Adding type annotations to the loop parameter can be used to resolve this.

3.7.2. Size coercion

Size coercion, written with :>, can be used to perform a runtime-checked coercion of one size to another. Since size annotations can refer only to variables and constants, this is necessary when writing more complicated size functions:

let concat_to 'a (m: i32) (a: []a) (b: []a) : [m]a =
  a ++ b :> [m]a

Only expression-level type annotations give rise to run-time checks. Despite their similar syntax, parameter and return type annotations must be valid at compile-time, or type checking will fail.

3.7.3. Causality restriction

Conceptually, size parameters are assigned their value by reading the sizes of concrete values passed along as parameters. This means that any size parameter must be used as the size of some parameter. This is an error:

let f [n] (x: i32) = n

The following is not an error:

let f [n] (g: [n]i32 -> [n]i32) = ...

However, using this function comes with a constraint: whenever an application f x occurs, the value of the size parameter must be inferable. Specifically, this value must have been used as the size of an array before the f x application is encountered. The notion of “before” is subtle, as there is no evaluation ordering of a Futhark expression, except that a let-binding is always evaluated before its body, the argument to a function is always evaluated before the function itself, and the left operand to an operator is evaluated before the right.

The causality restriction only occurs when a function has size parameters whose first use is not as a concrete array size. For example, it does not apply to uses of the following function:

let f [n] (arr: [n]i32) (g: [n]i32 -> [n]i32) = ...

This is because the proper value of n can be read directly from the actual size of the array.

3.7.4. Empty array literals

Just as with size-polymorphic functions, when constructing an empty array, we must know the exact size of the (missing) elements. For example, in the following program we are forcing the elements of a to be the same as the elements of b, but the size of the elements of b are not known at the time a is constructed:

let main (b: bool) (xs: []i32) =
  let a = [] : [][]i32
  let b = [filter (>0) xs]
  in a[0] == b[0]

The result is a type error.

3.7.5. Sum types

When constructing a value of a sum type, the compiler must still be able to determine the size of the constructors that are not used. This is illegal:

type sum = #foo ([]i32) | #bar ([]i32)

let main (xs: *[]i32) =
  let v : sum = #foo xs
  in xs

3.7.6. Modules

When matching a module with a module type (see Module System), a non-lifted abstract type (i.e. one that is declared with type rather than type^) may not be implemented by a type abbreviation that contains any anonymous sizes. This is to ensure that if we have the following:

module m : { type t } = ...

Then we can construct an array of values of type m.t without worrying about constructing an irregular array.

3.7.7. Higher-order functions

When a higher-order function takes a functional argument whose return type is a non-lifted type parameter, any instantiation of that type parameter must have a non-anonymous size. If the return type is a lifted type parameter, then the instantiation may contain anonymous sizes. This is why the type of map guarantees regular arrays:

val map [n] 'a 'b : (a -> b) -> [n]a -> [n]b

The type parameter b can only be replaced with a type that has non-anonymous sizes, which means they must be the same for every application of the function. In contrast, this is the type of the pipeline operator:

val (|>) '^a -> '^b : a -> (a -> b) -> b

The provided function can return something with an anonymous size (such as filter). A function whose return type has an unknown size

If a function (named or anonymous) is inferred to have a return type that contains an unknown size variable created within the function body, that size variable will be replaced with an anonymous size. In most cases this is not important, but it means that an expression like the following is ill-typed:

map (\xs -> iota (length xs)) (xss : [n][m]i32)

This is because the (length xs) expression gives rise to some fresh size k. The lambda is then assigned the type [n]t -> [k]i32, which is immediately turned into [n]t -> []i32 because k was generated inside its body. A function of this type cannot be passed to map, as explained before. The solution is to bind length to a name before the lambda.

3.8. In-place Updates

In-place updates do not provide observable side effects, but they do provide a way to efficiently update an array in-place, with the guarantee that the cost is proportional to the size of the value(s) being written, not the size of the full array.

The a with [i] = v language construct, and derived forms, performs an in-place update. The compiler verifies that the original array (a) is not used on any execution path following the in-place update. This involves also checking that no alias of a is used. Generally, most language constructs produce new arrays, but some (slicing) create arrays that alias their input arrays.

When defining a function parameter or return type, we can mark it as unique by prefixing it with an asterisk. For example:

let modify (a: *[]i32) (i: i32) (x: i32): *[]i32 =
  a with [i] = a[i] + x

For bulk in-place updates with multiple values, use the scatter function in the basis library. In the parameter declaration a: *[i32], the asterisk means that the function modify has been given “ownership” of the array a, meaning that any caller of modify will never reference array a after the call again. This allows the with expression to perform an in-place update.

After a call modify a i x, neither a or any variable that aliases a may be used on any following execution path.

3.8.1. Alias Analysis

The rules used by the Futhark compiler to determine aliasing are intuitive in the intra-procedural case. Aliases are associated with entire arrays. Aliases of a record are tuple are tracked for each element, not for the record or tuple itself. Most constructs produce fresh arrays, with no aliases. The main exceptions are if, loop, function calls, and variable literals.

  • After a binding let a = b, that simply assigns a new name to an existing variable, the variable a aliases b. Similarly for record projections and patterns.

  • The result of an if aliases the union of the aliases of the components.

  • The result of a loop aliases the initial values, as well as any aliases that the merge parameters may assume at the end of an iteration, computed to a fixed point.

  • The aliases of a value returned from a function is the most interesting case, and depends on whether the return value is declared unique (with an asterisk *) or not. If it is declared unique, then it has no aliases. Otherwise, it aliases all arguments passed for non-unique parameters.

3.8.2. In-place Updates and Higher-Order Functions

Uniqueness typing generally interacts poorly with higher-order functions. The issue is that we cannot control how many times a function argument is applied, or to what, so it is not safe to pass a function that consumes its argument. The following two conservative rules govern the interaction between uniqueness types and higher-order functions:

  1. In the expression let p = e1 in ..., if any in-place update takes place in the expression e1, the value bound by p must not be or contain a function.

  2. A function that consumes one of its arguments may not be passed as a higher-order argument to another function.

3.9. Module System

mod_bind      ::=  "module" id mod_param* "=" [":" mod_type_exp] "=" mod_exp
mod_param     ::=  "(" id ":" mod_type_exp ")"
mod_type_bind ::=  "module" "type" id type_param* "=" mod_type_exp

Futhark supports an ML-style higher-order module system. Modules can contain types, functions, and other modules and module types. Module types are used to classify the contents of modules, and parametric modules are used to abstract over modules (essentially module-level functions). In Standard ML, modules, module types and parametric modules are called structs, signatures, and functors, respectively. Module names exist in the same name space as values, but module types are their own name space.

Named modules are declared as:

module name = ...

A named module type is defined as:

module type name = ...

Where a module expression can be the name of another module, an application of a parametric module, or a sequence of declarations enclosed in curly braces:

module Vec3 = {
  type t = ( f32 , f32 , f32 )
  let add(a: t) (b: t): t =
    let (a1, a2, a3) = a in
    let (b1, b2, b3) = b in
    (a1 + b1, a2 + b2 , a3 + b3)

module AlsoVec3 = Vec3

Functions and types within modules can be accessed using dot notation:

type vector = Vec3.t
let double(v: vector): vector = Vec3.add v v

We can also use open Vec3 to bring the names defined by Vec3 into the current scope. Multiple modules can be opened simultaneously by separating their names with spaces. In case several modules define the same names, the ones mentioned last take precedence. The first argument to open may be a full module expression.

Named module types are defined as:

module type ModuleTypeName = ...

A module type expression can be the name of another module type, or a sequence of specifications, or specs, enclosed in curly braces. A spec can be a value spec, indicating the presence of a function or value, an abstract type spec, or a type abbreviation spec. For example:

module type Addable = {
  type t                 -- abstract type spec
  type two_ts = (t,t)    -- type abbreviation spec
  val add: t -> t -> t   -- value spec

This module type specifies the presence of an abstract type t, as well as a function operating on values of type t. We can use module type ascription to restrict a module to what is exposed by some module type:

module AbstractVec = Vec3 : Addable

The definition of AbstractVec.t is now hidden. In fact, with this module type, we can neither construct values of type AbstractVec.T or convert them to anything else, making this a rather useless use of abstraction. As a derived form, we can write module M: S = e to mean module M = e : S.

In a value spec, sizes in types on the left-hand side of a function arrow must not be anonymous. For example, this is forbidden:

val sum: []t -> t

Instead write:

val sum [n]: [n]t -> t

But this is allowed, because the empty size is not to the left of a function arrow:

val evens [n]: [n]i32 -> []i32

Parametric modules allow us to write definitions that abstract over modules. For example:

module Times = \(M: Addable) -> {
  let times (x: M.t) (k: i32): M.t =
    loop x' = x for i < k do
      M.add x' x

We can instantiate Times with any module that fulfils the module type Addable and get back a module that defines a function times:

module Vec3Times = Times Vec3

Now Vec3Times.times is a function of type Vec3.t -> int -> Vec3.t. As a derived form, we can write module M p = e to mean module M = \p -> e.

3.9.1. Module Expressions

mod_exp ::=    qualid
             | mod_exp ":" mod_type_exp
             | "\" "(" id ":" mod_type_exp ")" [":" mod_type_exp] "->" mod_exp
             | mod_exp mod_exp
             | "(" mod_exp ")"
             | "{" dec* "}"
             | "import" stringlit

A module expression produces a module. Modules are collections of bindings produced by declarations (dec). In particular, a module may contain other modules or module types. qualid

Evaluates to the module of the given name. (mod_exp)

Evaluates to mod_exp. mod_exp : mod_type_exp

Module ascription evaluates the module expression and the module type expression, verifies that the module implements the module type, then returns a module that exposes only the functionality described by the module type. This is how internal details of a module can be hidden. \(p: mt1): mt2 -> e

Constructs a parametric module (a function at the module level) that accepts a parameter of module type mt1 and returns a module of type mt2. The latter is optional, but the parameter type is not. e1 e2

Apply the parametric module m1 to the module m2. { decs }

Returns a module that contains the given definitions. The resulting module defines any name defined by any declaration that is not local, in particular including names made available via open. import "foo"

Returns a module that contains the definitions of the file "foo" relative to the current file. See Referring to Other Files.

3.9.2. Module Type Expressions

mod_type_exp ::=    qualid
                  | "{" spec* "}"
                  | mod_type_exp "with" qualid type_param* "=" type
                  | "(" mod_type_exp ")"
                  | "(" id ":" mod_type_exp ")" "->" mod_type_exp
                  | mod_type_exp "->" mod_type_exp
spec      ::=    "val" id type_param* ":" spec_type
               | "val" binop type_param* ":" spec_type
               | "type" ["^"] id type_param* "=" type
               | "type" ["^"] id type_param*
               | "module" id ":" mod_type_exp
               | "include" mod_type_exp
               | "#[" attr "]" spec
spec_type ::=  type | type "->" spec_type

Module types classify modules, with the only (unimportant) difference in expressivity being that modules can contain module types, but module types cannot specify that a module must contain a specific module type. They can specify of course that a module contains a submodule of a specific module type.

3.10. Referring to Other Files

You can refer to external files in a Futhark file like this:

import "file"

The above will include all non-local top-level definitions from file.fut is and make them available in the current file (but will not export them). The .fut extension is implied.

You can also include files from subdirectories:

import "path/to/a/file"

The above will include the file path/to/a/file.fut relative to the including file.

Qualified imports are also possible, where a module is created for the file:

module M = import "file"

In fact, a plain import "file" is equivalent to:

local open import "file"

3.11. Attributes

attr ::=    id
          | id "(" [attr ("," attr)*] ")"

An expression, declaration, or module type spec can be prefixed with an attribute, written as #[attr]. This may affect how it is treated by the compiler or other tools. In no case will attributes affect or change the semantics of a program, but it may affect how well it compiles and runs (or in some cases, whether it compiles or runs at all). Unknown attributes are silently ignored. Most have no effect in the interpreter. An attribute can be either an atom, written as just an identifier, or compound, consisting of an identifier and a comma-separated sequence of attributes. The latter is used for grouping and encoding of more complex information.

3.11.1. Expression attributes

Many expression attributes affect second-order array combinators (SOACS). These must be applied to a fully saturated function application or they will have no effect. If two SOACs with contradictory attributes are combined through fusion, it is unspecified which attributes take precedence.

The following expression attributes are supported. incremental_flattening(no_outer)

When using incremental flattening, do not generate the “only outer parallelism” version for the attributed SOACs. incremental_flattening(no_intra)

When using incremental flattening, do not generate the “intra-group parallelism” version for the attributed SOACs. incremental_flattening(only_intra)

When using incremental flattening, only generate the “intra-group parallelism” version of the attributed SOACs. Beware: the resulting program will fail to run if the inner parallelism does not fit on the device. incremental_flattening(only_inner)

When using incremental flattening, do not generate multiple versions for this SOAC, but do exploit inner parallelism (which may give rise to multiple versions at deeper levels). noinline

Do not inline the attributed function application. If used within a parallel construct (e.g. map), this will likely prevent the GPU backends from generating working code. sequential

Fully sequentialise the attributed SOAC. sequential_outer

Turn the outer parallelism in the attributed SOAC sequential, but preserve any inner parallelism. sequential_inner

Exploit only outer parallelism in the attributed SOAC. unsafe

Do not perform any dynamic safety checks (such as bound checks) during execution of the attributed expression. warn(safety_checks)

Make the compiler issue a warning if the attributed expression (or its subexpressions) requires safety checks (such as bounds checking) at run-time. This is used for performance-critical code where you want to be told when the compiler is unable to statically verify the safety of all operations.

3.11.2. Declaration attributes

The following declaration attributes are supported. noinline

Do not inline any calls to this function. If the function is then used within a parallel construct (e.g. map), this will likely prevent the GPU backends from generating working code.

3.11.3. Spec attributes

No spec attributes are currently supported by the compiler itself, although they are syntactically permitted and may be used by other tools.