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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 language constructs of a computer program, such as variables, expressions, functions, or modules.^[1] 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.^[2] 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 of 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.

Usage overview

An example of a simple type system is that of the C language. The portions of a C program are the function definitions. One function is invoked by another function. The interface of a function states the name of the function and a list of parameters that are passed to the function's code. The code of an invoking function states the name of the invoked, along with the names of variables that hold values to pass to it. During execution, the values are placed into temporary storage, then execution jumps to the code of the invoked function. The invoked function's code accesses the values and makes use of them. If the instructions inside the function are written with the assumption of receiving an integer value, but the calling code passed a floating-point value, then the wrong result will be computed by the invoked function. The C compiler checks the types of the arguments passed to a function when it is called against the types of the parameters declared in the function's definition. If the types do not match, the compiler throws a compile-time error or warning.

A compiler may also use the static type of a value to optimize the storage it needs and the choice of algorithms for operations on the value. In many C compilers the float data type, for example, is represented in 32 bits, in accord with the IEEE specification for single-precision floating point numbers. They will thus use floating-point-specific microprocessor operations on those values (floating-point addition, multiplication, etc.).

The depth of type constraints and the manner of their evaluation affect the typing of the language. A programming language may further associate an operation with various resolutions for each type, in the case of type polymorphism. Type theory is the study of type systems. The concrete types of some programming languages, such as integers and strings, depend on practical issues of computer architecture, compiler implementation, and language design.

Fundamentals

Formally, type theory studies type systems. A programming language must have the opportunity to type check using the type system whether at compile time or runtime, manually annotated or automatically inferred. As Mark Manasse concisely put it:^[3]

The fundamental problem addressed by a type theory is to ensure that programs have meaning. The fundamental problem caused by a type theory is that meaningful programs may not have meanings ascribed to them. The quest for richer type systems results from this tension.

Assigning a data type, termed typing, gives meaning to a sequence of bits such as a value in memory or some object such as a variable. The hardware of a general purpose computer is unable to discriminate between for example a memory address and an instruction code, or between a character, an integer, or a floating-point number, because it makes no intrinsic distinction between any of the possible values that a sequence of bits might mean.^{[note 1]} Associating a sequence of bits with a type conveys that meaning to the programmable hardware to form a symbolic system composed of that hardware and some program.

A program associates each value with at least one specific type, but it also can occur that one value is associated with many subtypes. Other entities, such as objects, modules, communication channels, and dependencies can become associated with a type. Even a type can become associated with a type. An implementation of a type system could in theory associate identifications called data type (a type of a value), class (a type of an object), and kind (a type of a type, or metatype). These are the abstractions that typing can go through, on a hierarchy of levels contained in a system.

When a programming language evolves a more elaborate type system, it gains a more finely grained rule set than basic type checking, but this comes at a price when the type inferences (and other properties) become undecidable, and when more attention must be paid by the programmer to annotate code or to consider computer-related operations and functioning. It is challenging to find a sufficiently expressive type system that satisfies all programming practices in a type safe manner.

A programming language compiler can also implement a dependent type or an effect system, which enables even more program specifications to be verified by a type checker. Beyond simple value-type pairs, a virtual "region" of code is associated with an "effect" component describing what is being done with what, and enabling for example to "throw" an error report. Thus the symbolic system may be a type and effect system, which endows it with more safety checking than type checking alone.

Whether automated by the compiler or specified by a programmer, a type system makes program behavior illegal if outside the type-system rules. Advantages provided by programmer-specified type systems include:

• Abstraction (or modularity) – Types enable programmers to think at a higher level than the bit or byte, not bothering with low-level implementation. For example, programmers can begin to think of a string as a set of character values instead of as a mere array of bytes. Higher still, types enable programmers to think about and express interfaces between two of any-sized subsystems. This enables more levels of localization so that the definitions required for interoperability of the subsystems remain consistent when those two subsystems communicate.
• Documentation – In more expressive type systems, types can serve as a form of documentation clarifying the intent of the programmer. For example, if a programmer declares a function as returning a timestamp type, this documents the function when the timestamp type can be explicitly declared deeper in the code to be an integer type.

Advantages provided by compiler-specified type systems include:

• Optimization – Static type-checking may provide useful compile-time information. For example, if a type requires that a value must align in memory at a multiple of four bytes, the compiler may be able to use more efficient machine instructions.
• Safety – A type system enables the compiler to detect meaningless or invalid code. For example, we can identify an expression 3 / "Hello, World" as invalid, when the rules do not specify how to divide an integer by a string. Strong typing offers more safety, but cannot guarantee complete type safety.

Type errors

A type error occurs when an operation receives a different type of data than it expected.^[4] For example, a type error would happen if a line of code divides two integers, and is passed a string of letters instead of an integer.^[4] It is an unintended condition^{[note 2]} which might manifest in multiple stages of a program's development. Thus a facility for detection of the error is needed in the type system. In some languages, such as Haskell, for which type inference is automated, lint might be available to its compiler to aid in the detection of error.

Type safety contributes to program correctness, but might only guarantee correctness at the cost of making the type checking itself an undecidable problem (as in the Halting problem). In a type system with automated type checking, a program may prove to run incorrectly yet produce no compiler errors. Division by zero is an unsafe and incorrect operation, but a type checker which only runs at compile time does not scan for division by zero in most languages; that division would surface as a runtime error. To prove the absence of these defects, other kinds of formal methods, collectively known as program analyses, are in common use. Alternatively, a sufficiently expressive type system, such as in dependently typed languages, can prevent these kinds of errors (for example, expressing the type of non-zero numbers). In addition, software testing is an empirical method for finding errors that such a type checker would not detect.

Type checking

The process of verifying and enforcing the constraints of types—type checking—may occur at compile time (a static check) or at run-time (a dynamic check). If a language specification requires its typing rules strongly (i.e., more or less allowing only those automatic type conversions that do not lose information), one can refer to the process as strongly typed, if not, as weakly typed. The terms are not usually used in a strict sense.

Static type checking

Static type checking is the process of verifying the type safety of a program based on analysis of a program's text (source code). If a program passes a static type checker, then the program is guaranteed to satisfy some set of type safety properties for all possible inputs.

Static type checking can be considered a limited form of program verification (see type safety), and in a type-safe language, can be considered also an optimization. If a compiler can prove that a program is well-typed, then it does not need to emit dynamic safety checks, allowing the resulting compiled binary to run faster and to be smaller.

Static type checking for Turing-complete languages is inherently conservative. That is, if a type system is both sound (meaning that it rejects all incorrect programs) and decidable (meaning that it is possible to write an algorithm that determines whether a program is well-typed), then it must be incomplete (meaning there are correct programs, which are also rejected, even though they do not encounter runtime errors).^[7] For example, consider a program containing the code:

if <complex test> then <do something> else <signal that there is a type error>

Even if the expression <complex test> always evaluates to true at run-time, most type checkers will reject the program as ill-typed, because it is difficult (if not impossible) for a static analyzer to determine that the else branch will not be taken.^[8] Consequently, a static type checker will quickly detect type errors in rarely used code paths. Without static type checking, even code coverage tests with 100% coverage may be unable to find such type errors. The tests may fail to detect such type errors, because the combination of all places where values are created and all places where a certain value is used must be taken into account.

A number of useful and common programming language features cannot be checked statically, such as downcasting. Thus, many languages will have both static and dynamic type checking; the static type checker verifies what it can, and dynamic checks verify the rest.

Many languages with static type checking provide a way to bypass the type checker. Some languages allow programmers to choose between static and dynamic type safety. For example, historically C# declares variables statically,^{[9]: 77, Section 3.2} but C# 4.0 introduces the dynamic keyword, which is used to declare variables to be checked dynamically at runtime.^{[9]: 117, Section 4.1} Other languages allow writing code that is not type-safe; for example, in C, programmers can freely cast a value between any two types that have the same size, effectively subverting the type concept.

For a list of languages with static type checking, see the category for statically typed languages.

Dynamic type checking and runtime type information

Dynamic type checking is the process of verifying the type safety of a program at runtime. Implementations of dynamically type-checked languages generally associate each runtime object with a type tag (i.e., a reference to a type) containing its type information. This runtime type information (RTTI) can also be used to implement dynamic dispatch, late binding, downcasting, reflective programming (reflection), and similar features.

Most type-safe languages include some form of dynamic type checking, even if they also have a static type checker.^[10] The reason for this is that many useful features or properties are difficult or impossible to verify statically. For example, suppose that a program defines two types, A and B, where B is a subtype of A. If the program tries to convert a value of type A to type B, which is known as downcasting, then the operation is legal only if the value being converted is actually a value of type B. Thus, a dynamic check is needed to verify that the operation is safe. This requirement is one of the criticisms of downcasting.

By definition, dynamic type checking may cause a program to fail at runtime. In some programming languages, it is possible to anticipate and recover from these failures. In others, type-checking errors are considered fatal.

Programming languages that include dynamic type checking but not static type checking are often called "dynamically typed programming languages". For a list of such languages, see the category for dynamically typed programming languages.

Combining static and dynamic type checking

Some languages allow both static and dynamic typing. For example, Java and some other ostensibly statically typed languages support downcasting types to their subtypes, querying an object to discover its dynamic type and other type operations that depend on runtime type information. Another example is C++ RTTI. More generally, most programming languages include mechanisms for dispatching over different 'kinds' of data, such as disjoint unions, runtime polymorphism, and variant types. Even when not interacting with type annotations or type checking, such mechanisms are materially similar to dynamic typing implementations. See programming language for more discussion of the interactions between static and dynamic typing.

Objects in object-oriented languages are usually accessed by a reference whose static target type (or manifest type) is equal to either the object's run-time type (its latent type) or a supertype thereof. This is conformant with the Liskov substitution principle, which states that all operations performed on an instance of a given type can also be performed on an instance of a subtype. This concept is also known as subsumption or subtype polymorphism. In some languages subtypes may also possess covariant or contravariant return types and argument types respectively.

Certain languages, for example Clojure, Common Lisp, or Cython are dynamically type checked by default, but allow programs to opt into static type checking by providing optional annotations. One reason to use such hints would be to optimize the performance of critical sections of a program. This is formalized by gradual typing. The programming environment DrRacket, a pedagogic environment based on Lisp, and a precursor of the language Racket is also soft-typed.^[11]

Conversely, as of version 4.0, the C# language provides a way to indicate that a variable should not be statically type checked. A variable whose type is dynamic will not be subject to static type checking. Instead, the program relies on runtime type information to determine how the variable may be used.^{[12][9]: 113–119}

In Rust, the dyn std::any::Any type provides dynamic typing of 'static types.^[13]

Static and dynamic type checking in practice

The choice between static and dynamic typing requires certain trade-offs.

Static typing can find type errors reliably at compile time, which increases the reliability of the delivered program. However, programmers disagree over how commonly type errors occur, resulting in further disagreements over the proportion of those bugs that are coded that would be caught by appropriately representing the designed types in code.^[14][15] Static typing advocates^[who?] believe programs are more reliable when they have been well type-checked, whereas dynamic-typing advocates^[who?] point to distributed code that has proven reliable and to small bug databases.^{[citation needed]} The value of static typing increases as the strength of the type system is increased. Advocates of dependent typing,^[who?] implemented in languages such as Dependent ML and Epigram, have suggested that almost all bugs can be considered type errors, if the types used in a program are properly declared by the programmer or correctly inferred by the compiler.^[16]

Static typing usually results in compiled code that executes faster. When the compiler knows the exact data types that are in use (which is necessary for static verification, either through declaration or inference) it can produce optimized machine code. Some dynamically typed languages such as Common Lisp allow optional type declarations for optimization for this reason.

By contrast, dynamic typing may allow compilers to run faster and interpreters to dynamically load new code, because changes to source code in dynamically typed languages may result in less checking to perform and less code to revisit.^{[clarification needed]} This too may reduce the edit-compile-test-debug cycle.

Statically typed languages that lack type inference (such as C and Java prior to version 10) require that programmers declare the types that a method or function must use. This can serve as added program documentation, that is active and dynamic, instead of static. This allows a compiler to prevent it from drifting out of synchrony, and from being ignored by programmers. However, a language can be statically typed without requiring type declarations (examples include Haskell, Scala, OCaml, F#, Swift, and to a lesser extent C# and C++), so explicit type declaration is not a necessary requirement for static typing in all languages.

Dynamic typing allows constructs that some (simple) static type checking would reject as illegal. For example, eval functions, which execute arbitrary data as code, become possible. An eval function is possible with static typing, but requires advanced uses of algebraic data types. Further, dynamic typing better accommodates transitional code and prototyping, such as allowing a placeholder data structure (mock object) to be transparently used in place of a full data structure (usually for the purposes of experimentation and testing).

Dynamic typing typically allows duck typing (which enables easier code reuse). Many^[specify] languages with static typing also feature duck typing or other mechanisms like generic programming that also enable easier code reuse.

Dynamic typing typically makes metaprogramming easier to use. For example, C++ templates are typically more cumbersome to write than the equivalent Ruby or Python code since C++ has stronger rules regarding type definitions (for both functions and variables). This forces a developer to write more boilerplate code for a template than a Python developer would need to. More advanced run-time constructs such as metaclasses and introspection are often harder to use in statically typed languages. In some languages, such features may also be used e.g. to generate new types and behaviors on the fly, based on run-time data. Such advanced constructs are often provided by dynamic programming languages; many of these are dynamically typed, although dynamic typing need not be related to dynamic programming languages.

Strong and weak type systems

Languages are often colloquially referred to as strongly typed or weakly typed. In fact, there is no universally accepted definition of what these terms mean. In general, there are more precise terms to represent the differences between type systems that lead people to call them "strong" or "weak".

Type safety and memory safety

A third way of categorizing the type system of a programming language is by the safety of typed operations and conversions. Computer scientists use the term type-safe language to describe languages that do not allow operations or conversions that violate the rules of the type system.

Computer scientists use the term memory-safe language (or just safe language) to describe languages that do not allow programs to access memory that has not been assigned for their use. For example, a memory-safe language will check array bounds, or else statically guarantee (i.e., at compile time before execution) that array accesses out of the array boundaries will cause compile-time and perhaps runtime errors.

Consider the following program of a language that is both type-safe and memory-safe:^[17]

var x := 5;   
var y := "37"; 
var z := x + y;

In this example, the variable z will have the value 42. Although this may not be what the programmer anticipated, it is a well-defined result. If y were a different string, one that could not be converted to a number (e.g. "Hello World"), the result would be well-defined as well. Note that a program can be type-safe or memory-safe and still crash on an invalid operation. This is for languages where the type system is not sufficiently advanced to precisely specify the validity of operations on all possible operands. But if a program encounters an operation that is not type-safe, terminating the program is often the only option.

Now consider a similar example in C:

int x = 5;
char y = "37";
char* z = x + y;
printf("%c\n", *z);

In this example z will point to a memory address five characters beyond y, equivalent to three characters after the terminating zero character of the string pointed to by y. This is memory that the program is not expected to access. In C terms this is simply undefined behaviour and the program may do anything; with a simple compiler it might actually print whatever byte is stored after the string "37". As this example shows, C is not memory-safe. As arbitrary data was assumed to be a character, it is also not a type-safe language.

In general, type-safety and memory-safety go hand in hand. For example, a language that supports pointer arithmetic and number-to-pointer conversions (like C) is neither memory-safe nor type-safe, because it allows arbitrary memory to be accessed as if it were valid memory of any type.

For more information, see memory safety.

Variable levels of type checking

Some languages allow different levels of checking to apply to different regions of code. Examples include:

• The use strict directive in JavaScript^[18][19][20] and Perl applies stronger checking.
• The declare(strict_types=1) in PHP^[21] on a per-file basis allows only a variable of exact type of the type declaration will be accepted, or a TypeError will be thrown.
• The Option Strict On in VB.NET allows the compiler to require a conversion between objects.

Additional tools such as lint and IBM Rational Purify can also be used to achieve a higher level of strictness.

Optional type systems

It has been proposed, chiefly by Gilad Bracha, that the choice of type system be made independent of choice of language; that a type system should be a module that can be plugged into a language as needed. He believes this is advantageous, because what he calls mandatory type systems make languages less expressive and code more fragile.^[22] The requirement that the type system does not affect the semantics of the language is difficult to fulfill.

Optional typing is related to, but distinct from, gradual typing. While both typing disciplines can be used to perform static analysis of code (static typing), optional type systems do not enforce type safety at runtime (dynamic typing).^[22][23]

Polymorphism and types

The term polymorphism refers to the ability of code (especially, functions or classes) to act on values of multiple types, or to the ability of different instances of the same data structure to contain elements of different types. Type systems that allow polymorphism generally do so in order to improve the potential for code re-use: in a language with polymorphism, programmers need only implement a data structure such as a list or an associative array once, rather than once for each type of element with which they plan to use it. For this reason computer scientists sometimes call the use of certain forms of polymorphism generic programming. The type-theoretic foundations of polymorphism are closely related to those of abstraction, modularity and (in some cases) subtyping.

Specialized type systems

Many type systems have been created that are specialized for use in certain environments with certain types of data, or for out-of-band static program analysis. Frequently, these are based on ideas from formal type theory and are only available as part of prototype research systems.

The following table gives an overview over type theoretic concepts that are used in specialized type systems. The names M, N, O range over terms and the names ${\displaystyle \sigma ,\tau }$ range over types. The following notation will be used:

• ${\displaystyle M:\sigma }$ means that ${\displaystyle M}$ has type ${\displaystyle \sigma }$;
• ${\displaystyle M(N)}$ is that application of ${\displaystyle M}$ on ${\displaystyle N}$;
• ${\displaystyle \tau }$ (resp. ${\displaystyle \tau }$) describes the type which results from replacing all occurrences of the type variable α (resp. term variable x) in ${\displaystyle \tau }$ by the type σ (resp. term N).

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Type notion	Notation	Meaning
Function	${\displaystyle \sigma \to \tau }$	If ${\displaystyle M:\sigma \to \tau }$ and ${\displaystyle N:\sigma }$, then ${\displaystyle M(N):\tau }$.
Product	${\displaystyle \sigma \times \tau }$	If ${\displaystyle M:\sigma \times \tau }$, then ${\displaystyle M=(N,O)}$ is a pair s.t. ${\displaystyle N:\sigma }$ and ${\displaystyle O:\tau }$.
Sum	${\displaystyle \sigma +\tau }$	If ${\displaystyle M:\sigma +\tau }$, then ${\displaystyle M=\iota _{1}(N)}$ is the first injection s.t. ${\displaystyle N:\sigma }$, or ${\displaystyle M=\iota _{2}(N)}$ is the second injection s.t. ${\displaystyle N:\tau }$.
Intersection	${\displaystyle \sigma \cap \tau }$	If ${\displaystyle M:\sigma \cap \tau }$, then ${\displaystyle M:\sigma }$ and ${\displaystyle M:\tau }$.
Union