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<*programming*> An algorithm for ascribing types to
expressions in some language, based on the types of the
constants of the language and a set of type inference rules
such as
f :: A -> B, x :: A
--------------------- (App)
f x :: B
This rule, called "App" for application, says that if
expression f has type A -> B and expression x has type A then
we can deduce that expression (f x) has type B. The
expressions above the line are the premises and below, the
conclusion. An alternative notation often used is:
G |- x : A
where "|-" is the turnstile symbol (LaTeX vdash) and G is a
type assignment for the free variables of expression x. The
above can be read "under assumptions G, expression x has type
A". (As in Haskell, we use a double "::" for type
declarations and a single ":" for the infix list constructor,
cons).
Given an expression
plus (head l) 1
we can label each subexpression with a type, using type
variables X, Y, etc. for unknown types:
(plus :: Int -> Int -> Int)
(((head :: [a ] -> a) (l :: Y)) :: X)
(1 :: Int)
We then use unification on type variables to match the
partial application of plus to its first argument against
the App rule, yielding a type (Int -> Int) and a substitution
X = Int. Re-using App for the application to the second
argument gives an overall type Int and no further
substitutions. Similarly, matching App against the
application (head l) we get Y = [X ]. We already know X = Int
so therefore Y = [Int ].
This process is used both to infer types for expressions and
to check that any types given by the user are consistent.
See also generic type variable, principal type.
(1995-02-03)

Type inference refers to the automatic detection of the type of an expression in a formal language. These include programming languages and mathematical type systems, but also natural languages in some branches of computer science and linguistics.

Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability.Upton, G.

Type system

In computer programming, a **type system** is a logical system comprising a set of rules that assigns a property called a type (for example, integer, floating point, string) to every "term" (a word, phrase, or other set of symbols). Usually the terms are various constructs of a computer program, such as variables, expressions, functions, or modules. A type system dictates the operations that can be performed on a term. For variables, the type system determines the allowed values of that term. Type systems formalize and enforce the otherwise implicit categories the programmer uses for algebraic data types, data structures, or other components (e.g. "string", "array of float", "function returning boolean").

Type systems are often specified as part of programming languages and built into interpreters and compilers, although the type system of a language can be extended by optional tools that perform added checks using the language's original type syntax and grammar. The main purpose of a type system in a programming language is to reduce possibilities for bugs in computer programs due to type errors. The given type system in question determines what constitutes a type error, but in general, the aim is to prevent operations expecting a certain kind of value from being used with values for which that operation does not make sense (validity errors). Type systems allow defining interfaces between different parts of a computer program, and then checking that the parts have been connected in a consistent way. This checking can happen statically (at compile time), dynamically (at run time), or as a combination of both. Type systems have other purposes as well, such as expressing business rules, enabling certain compiler optimizations, allowing for multiple dispatch, and providing a form of documentation.