In most programming languages, all values have a type which describes the kind of data a particular value describes. In some languages, the type is known only at runtime; these languages are dynamically typed. In other languages, the type is known at compile time; these languages are statically typed. In statically typed languages, the input and output types of functions and local variables ordinarily must be explicitly provided by type annotations. For example, in C:
The beginning of this function definition,
int addone(int x) declares that
addone is a function which takes one argument, an integer, and returns an integer.
int result; declares that the local variable
result is an integer. In a proposed language where type inference is available, the code might be written like this instead:
val result; /*inferred-type result (in proposed language)*/
val result2; /*inferred-type result #2 */
result = x+1;
result2 = x+1.0; /* this line won't work */
}This looks very similar to how code is written in a dynamically typed language, yet all types are known at compile time. In the imaginary language in which the last example is written,
+always takes two integers and returns one integer. (This is how it works in, for example, OCaml.) From this, the type inferencer can infer that the value of
x+1is an integer, therefore
resultis an integer, therefore the return value of
addoneis an integer. Similarly, since
+requires that both of its arguments be integers,
xmust be an integer, and therefore
addoneaccepts one integer as an argument.
However, in the subsequent line, result2 is calculated by adding a decimal "
1.0" with floating-point arithmetic, causing a conflict in the use of
x for both integer and floating-point expressions. Such a situation would generate a compile-time error message. In older languages, result2 might have been implicitly declared as a floating-point variable, from implicitly converting integer
x in the expression, simply because a decimal point was accidentally placed after the integer 1. Such a situation shows the difference between type inference, which does not involve type conversion, and implicit type conversion, which forces data to the higher-precision data type, often without restrictions.
Type inference refers to the ability to deduce automatically, either partially or fully, the type of a value derived from the eventual evaluation of an expression. As this process is systematically performed at compile time, the compiler is often able to infer the type of a variable or the type signature of a function, without explicit type annotations having been given. In many cases, it is possible to omit type annotations from a program completely if the type inference system is robust enough, or the program or language simple enough.
To obtain the information required to infer correctly the type of an expression lacking an explicit type annotation, the compiler either gathers this information as an aggregate and subsequent reduction of the type annotations given for its subexpressions (which may themselves be variables or functions), or through an implicit understanding of the type of various atomic values (e.g., : Unit; true : Bool; 42 : Integer; 3.14159 : Real; etc.). It is through recognition of the eventual reduction of expressions to implicitly typed atomic values that the compiler for a type inferring language is able to compile a program completely without type annotations. In the case of highly complex forms of higher order programming and polymorphism, it is not always possible for the compiler to infer as much, however, and type annotations are occasionally necessary for disambiguation.
map, which applies a procedure to each element of a list, and may be defined as:
map f  = 
map f (first:rest) = f first : map f rest
From this, it is evident that the function
map takes a list as its second argument, that its first argument
f is a function that can be applied to the type of elements of that list, and that the result of
map is constructed as a list with elements that are results of
So assuming that a list contains elements of the same type, we can reliably construct a type signature
map :: (a -> b) -> [a] -> [b]where the syntax "
a -> b" denotes a function that takes an
aas its parameter and produces a
a -> b -> c" is equivalent to "
a -> (b -> c)".
Note that the inferred type of
map is parametrically polymorphic: The type of the arguments and results of
f are not inferred, but left as type variables, and so
map can be applied to functions and lists of various types, as long as the actual types match in each invocation.
The origin of this algorithm is the type inference algorithm for the simply typed lambda calculus, which was devised by Haskell B. Curry and Robert Feys in 1958. In 1969 Roger Hindley extended this work and proved that their algorithm always inferred the most general type. In 1978 Robin Milner , independently of Hindley's work, provided an equivalent algorithm, Algorithm W. In 1982 Luis Damas finally proved that Milner's algorithm is complete and extended it to support systems with polymorphic references.
where is a primitive expression (such as "3") and is a primitive type (such as "Integer").
We want to construct a function which maps a pair , where is a type environment and is a term, to some type . We assume that this function is already defined on primitives. The other cases are:
Note that so far we do not specify what to do when we fail to meet the various conditions. This is because, in the simple type checking algorithm, the check simply fails whenever anything goes wrong.
Now, we develop a more sophisticated algorithm that can deal with type variables and constraints on them. Therefore, we extend the set T of primitive types to include an infinite supply of variables, denoted by lowercase Greek letters
See for more details.
Unifying the empty set of equations is easy enough: , where is the identity substitution.
Unifying a variable with a type goes this way: , where is the substitution composition operator, and is the set of remaining constraints with the new substitution applied to it.
Of course, .
The interesting case remains as .
A simple example would be a[i] = b[i] (assume this to be c-like syntax for this example). First Hindley-Milner would find that i must be of type int, further more that 'a' must be an "array of " and 'b' must an "array of ". Now, since there is an assignment of to , must be of the same type (assuming no implicit type conversions) as . In the very least must be a supertype of .
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