Type system

In computer science, a type system may be defined as "a tractable syntactic framework for classifying phrases according to the kinds of values they compute". A type system associates types with each computed value. By examining the flow of these values, a type system attempts to prove that no type errors can occur. The type system in question determines what constitutes a type error, but a type system generally seeks to guarantee that operations expecting a certain kind of value are not used with values for which that operation does not make sense.

A compiler may 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. C thus uses floating-point-specific 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 varying concrete algorithms on each type in the case of type polymorphism. Type theory is the study of type systems, although the concrete type systems of programming languages originate from practical issues of computer architecture, compiler implementation, and language design.

Fundamentals

Assigning data types (typing) gives meaning to sequences of bits. Types usually have associations either with values in memory or with objects such as variables. Because any value simply consists of a sequence of bits in a computer, hardware makes no intrinsic distinction even between memory addresses, instruction code, characters, integers and floating-point numbers, being unable to discriminate between them based on bit pattern alone. Associating a sequence of bits and a type informs programs and programmers how that sequence of bits should be understood.

Major functions provided by type systems include:

• Safety – Use of types may allow a compiler to detect meaningless or probably invalid code. For example, we can identify an expression 3 / "Hello, World" as invalid because the rules of arithmetic do not specify how to divide an integer by a string. As discussed below, strong typing offers more safety, but generally does not guarantee complete safety (see type safety for more information).
• 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 4 bytes, the compiler may be able to use more efficient machine instructions.
• Documentation – In more expressive type systems, types can serve as a form of documentation, since they can illustrate the intent of the programmer. For instance, timestamps may be represented as integers—but if a programmer declares a function as returning a timestamp type rather than merely an integer type, this documents part of the meaning of the function.
• Abstraction (or modularity) – Types allow programmers to think about programs at a higher level than the bit or byte, not bothering with low-level implementation. For example, programmers can think of a string as a collection of character values instead of as a mere array of bytes. Or, types can allow programmers to express the interface between two subsystems. This helps localize the definitions required for interoperability of the subsystems and prevents inconsistencies when those subsystems communicate.

Type safety contributes to program correctness, but cannot guarantee it unless the type checking itself becomes an undecidable problem. Depending on the specific type system, a program may give the wrong result and be safely typed, producing no compiler errors. For instance, division by zero is not caught by the type checker in most programming languages; instead it is a runtime error. To prove the absence of more general defects, other kinds of formal methods, collectively known as program analyses, are in common use, as well as software testing, a widely used empirical method for finding errors that the type checker cannot detect.

A program typically associates each value with one particular type (although a type may have more than one subtype). Other entities, such as objects, modules, communication channels, dependencies, or even types themselves, can become associated with a type. Some implementations might make the following identifications (though these are technically different concepts):

• data type – a type of a value
• class – a type of an object
• kind – a type of a type

A type system, specified for each programming language, controls the ways typed programs may behave, and makes behavior outside these rules illegal. An effect system typically provides more fine-grained control than does a type system.

Formally, type theory studies type systems. More elaborate type systems (such as dependent types) allow for finer-grained program specifications to be verified by a type checker, but this comes at a price, as type inference and other properties generally become undecidable, and type checking itself is dependent on user-supplied proofs. It is challenging to find a sufficiently expressive type system that satisfies all programming practices in type safe manner. As Mark Manasse concisely put it:

The fundamental problem addressed by a type theory is to insure 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.

Type checking

The process of verifying and enforcing the constraints of types – type checking – may occur either at compile-time (a static check) or 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 which do not lose information), one can refer to the process as strongly typed, if not, as weakly typed. The terms are not used in a strict sense.

Static typing

A programming language is said to use static typing when type checking is performed during compile-time as opposed to run-time. Statically typed languages include Ada, ActionScript 3, C, C++, C#, Eiffel, F#, Fortran, Go, Haskell, JADE, Java, ML, Objective-C, Ocaml, Pascal, Perl (with respect to distinguishing scalars, arrays, hashes and subroutines) and Scala. Static typing is a limited form of program verification (see type safety): accordingly, it allows many type errors to be caught early in the development cycle. Static type checkers evaluate only the type information that can be determined at compile time, but are able to verify that the checked conditions hold for all possible executions of the program, which eliminates the need to repeat type checks every time the program is executed. Program execution may also be made more efficient (i.e. faster or taking reduced memory) by omitting runtime type checks and enabling other optimizations.

Because they evaluate type information during compilation, and therefore lack type information that is only available at run-time, static type checkers are conservative. They will reject some programs that may be well-behaved at run-time, but that cannot be statically determined to be well-typed. For example, even if an expression <complex test> always evaluates to true at run-time, a program containing the code

if <complex test> then 42 else <type error>

will be rejected as ill-typed, because a static analysis cannot determine that the else branch won't be taken. The conservative behaviour of static type checkers is advantageous when <complex test> evaluates to false infrequently: A static type checker can 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.

The most widely used statically typed languages are not formally type safe. They have "loopholes" in the programming language specification enabling programmers to write code that circumvents the verification performed by a static type checker and so address a wider range of problems. For example, most C-style languages have type punning, and Haskell has such features as unsafePerformIO: such operations may be unsafe at runtime, in that they can cause unwanted behaviour due to incorrect typing of values when the program runs.

Dynamic typing

A programming language is said to be dynamically typed when the majority of its type checking is performed at run-time as opposed to at compile-time. In dynamic typing, values have types but variables do not; that is, a variable can refer to a value of any type.

Implementations of dynamically-typed languages generally associate run-time objects with "tags" containing their type information. This run-time classification is then used to implement type checks and dispatch overloaded functions, but can also enable pervasive uses of dynamic dispatch, late binding and similar idioms which would be cumbersome at best in a statically-typed language, requiring the use of variant types or similar features.

More broadly, as explained below, dynamic typing can improve support for dynamic programming language features, such as generating types and functionality based on run-time data. (Nevertheless, dynamically typed languages need not support any or all such features, and some dynamic programming languages are statically typed.) On the other hand, dynamic typing provides fewer a priori guarantees: a dynamically typed language accepts and attempts to execute some programs which would be ruled as invalid by a static type checker, either due to errors in the program or due to static type checking being too conservative.

Dynamic typing may result in runtime type errors—that is, at runtime, a value may have an unexpected type, and an operation nonsensical for that type is applied. Such errors may occur long after the place where the programming mistake was made—that is, the place where the wrong type of data passed into a place it should not have. This may make the bug difficult to locate.

Dynamically typed language systems' run-time checks can potentially be more sophisticated than those of statically typed languages, as they can use dynamic information as well as any information from the source code. On the other hand, runtime checks only assert that conditions hold in a particular execution of the program, and the checks are repeated for every execution of the program.

Dynamically typed languages include APL, Erlang, Groovy, JavaScript, Lisp, Lua, MATLAB/GNU Octave, Perl (with respect to user-defined types but not built-in types), PHP, Prolog, Python, Ruby, Smalltalk and Tcl.

Development in dynamically typed languages is often supported by programming practices such as unit testing. Testing is a key practice in professional software development, and is particularly important in dynamically typed languages. In practice, the testing done to ensure correct program operation can detect a much wider range of errors than static type-checking, but conversely cannot search as comprehensively for the errors that static type checking is able to detect.

Combinations of dynamic and static typing

The presence of static typing in a programming language does not necessarily imply the absence of all dynamic typing mechanisms. For example, Java and some other ostensibly statically typed languages, support downcasting and other type operations which depend on runtime type checks, a form of dynamic typing. More generally, most programming languages include mechanisms for dispatching over different 'kinds' of data, such as disjoint unions, variant types and polymorphic objects: 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.

Certain languages, for example Clojure, are dynamically typed by default but allow this behaviour to be overridden through the use of explicit type hints that result in static typing. One reason to use such hints would be to achieve the performance benefits of static typing in performance-sensitive parts of code.

As of the 4.0 Release, the .NET Framework supports a variant of dynamic typing via the System.Dynamic namespace whereby a static object of type 'dynamic' is a placeholder for the .NET runtime to interrogate its dynamic facilities to resolve the object reference.

Static and dynamic type checking in practice

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

Static typing can find type errors reliably at compile time. This should increase the reliability of the delivered program. However, programmers disagree over how commonly type errors occur, and thus disagree over the proportion of those bugs which are coded that would be caught by appropriately representing the designed types in code. Static typing advocates believe programs are more reliable when they have been well type-checked, while dynamic typing advocates point to distributed code that has proven reliable and to small bug databases. The value of static typing, then, presumably increases as the strength of the type system is increased. Advocates of dependently typed 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.

Static typing usually results in compiled code that executes more quickly. When the compiler knows the exact data types that are in use, it can produce optimized machine code. Further, compilers for statically typed languages can find assembler shortcuts more easily. Some dynamically typed languages such as Common Lisp allow optional type declarations for optimization for this very reason. Static typing makes this pervasive. See optimization.

By contrast, dynamic typing may allow compilers to run more quickly and allow interpreters to dynamically load new code, since changes to source code in dynamically typed languages may result in less checking to perform and less code to revisit. This too may reduce the edit-compile-test-debug cycle.

Statically typed languages which lack type inference (such as Java and C) require that programmers declare the types they intend a method or function to use. This can serve as additional documentation for the program, which the compiler will not permit the programmer to ignore or permit to drift out of synchronization. However, a language can be statically typed without requiring type declarations (examples include Haskell, Scala and to a lesser extent C#), so explicit type declaration is not a necessary requirement for static typing in all languages.

Dynamic typing allows constructs that some static type checking would reject as illegal. For example, eval functions, which execute arbitrary data as code, become possible. Furthermore, 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-fledged data structure (usually for the purposes of experimentation and testing).

Dynamic typing is used in duck typing which can support easier code reuse.

Dynamic typing typically makes metaprogramming more effective and easier to use. For example, C++ templates are typically more cumbersome to write than the equivalent Ruby or Python code. More advanced run-time constructs such as metaclasses and introspection are often more difficult 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 typing

Main article: Strongly typed programming language

A type system is said to feature strong typing when it specifies one or more restrictions on how operations involving values of different data types can be intermixed. A computer language that implements strong typing will prevent the successful execution of an operation on arguments which have the wrong type.

Weak typing means that a language implicitly converts (or casts) types when used. Consider the following example:

var x := 5;    // (1)  (x is an integer)
var y := "37"; // (2)  (y is a string)
x + y;         // (3)  (?)

In a weakly typed language, the result of this operation is unclear. Some languages, such as Visual Basic, would produce runnable code producing the result 42: the system would convert the string "37" into the number 37 to forcibly make sense of the operation. Other languages like JavaScript would produce the result "537": the system would convert the number 5 to the string "5" and then concatenate the two. In both Visual Basic and JavaScript, the resulting type is determined by rules that take both operands into consideration. In some languages, such as AppleScript, the type of the resulting value is determined by the type of the left-most operand only.

A C cast gone wrong exemplifies the problem that can occur if strong typing is absent; if a programmer casts a value from one type to another in C, not only must the compiler allow the code at compile time, but the runtime must allow it as well. This may permit more compact and faster C code, but it can make debugging more difficult.

Safely and unsafely typed systems

Main article: Type safety

A third way of categorizing the type system of a programming language uses the safety of typed operations and conversions. Computer scientists consider a language "type-safe" if it does not allow operations or conversions which lead to erroneous conditions.

Some observers use the term memory-safe language (or just safe language) to describe languages that do not allow undefined operations to occur. 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.

var x := 5;     // (1)
var y := "37";  // (2)
var z := x + y; // (3)

In languages like Visual Basic, variable z in the example acquires the value 42. While the programmer may or may not have intended this, the language defines the result specifically, and the program does not crash or assign an ill-defined value to z. In this respect, such languages are type-safe; however, if the value of y was a string that could not be converted to a number (e.g. "hello world"), the results would be undefined. Such languages are type-safe (in that they will not crash) but can easily produce undesirable results.

Now let us look at the same example in C:

int x = 5;
char y[] = "37";
char* z = x + y;

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. The content of that location is undefined, and might lie outside addressable memory. The mere computation of such a pointer may result in undefined behavior (including the program crashing) according to C standards, and in typical systems dereferencing z at this point could cause the program to crash. We have a well-typed, but not memory-safe program—a condition that cannot occur in a type-safe language.

Polymorphism and types

Main article: Type polymorphism

The term "polymorphism" refers to the ability of code (in particular, methods 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.

Duck typing

Main article: Duck typing

In "duck typing", a statement calling a method m on an object does not rely on the declared type of the object; only that the object, of whatever type, must implement the method called. One way of looking at this is that in duck typing systems the type of an object is intrinsic to the object and is determined by what methods it implements, and hence that a duck typing system is by definition type-safe since one can only invoke operations an object actually implements. Another way of looking at this is that the object is a member of several types, including a type that describes the fact that it "has a method m." Type checking however occurs only on demand at runtime, every time the method m needs to be executed, not at compile-time or load-time.

Duck typing differs from structural typing in that, if the part (of the whole module structure) needed for a given local computation is present at runtime, the duck type system is satisfied in its type identity analysis. On the other hand, a structural type system would require the analysis of the whole module structure at compile-time to determine type identity or type dependence.

Duck typing differs from a nominative type system in a number of aspects. The most prominent ones are that, for duck typing, type information is determined at runtime (as contrasted to compile-time) and the name of the type is irrelevant to determine type identity or type dependence; only partial structure information is required for that, for a given point in the program execution.

Duck typing uses the premise that (referring to a value) "if it walks like a duck, and quacks like a duck, then it is a duck" (this is a reference to the duck test which is attributed to James Whitcomb Riley). The term may have been coined by Alex Martelli in a 2000 message to the comp.lang.python newsgroup (see Python).

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.

Dependent types

Dependent types are based on the idea of using scalars or values to more precisely describe the type of some other value. For example, "matrix(3,3)" might be the type of a 3×3 matrix. We can then define typing rules such as the following rule for matrix multiplication:

matrix_multiply : matrix(k,m) × matrix(m,n) → matrix(k,n)

where k, m, n are arbitrary positive integer values. A variant of ML called Dependent ML has been created based on this type system, but because type-checking conventional dependent types is undecidable, not all programs using them can be type-checked without some kind of limits. Dependent ML limits the sort of equality it can decide to Presburger arithmetic. Other languages such as Epigram make the value of all expressions in the language decidable so that type checking can be decidable, it is also possible to make the language Turing complete at the price of undecidable type checking like in Cayenne.

Linear types

Linear types, based on the theory of linear logic, and closely related to uniqueness types, are types assigned to values having the property that they have one and only one reference to them at all times. These are valuable for describing large immutable values such as strings, files, and so on, because any operation that simultaneously destroys a linear object and creates a similar object (such as 'str = str + "a"') can be optimized "under the hood" into an in-place mutation. Normally this is not possible because such mutations could cause side effects on parts of the program holding other references to the object, violating referential transparency. They are also used in the prototype operating system Singularity for interprocess communication, statically ensuring that processes cannot share objects in shared memory in order to prevent race conditions. The Clean language (a Haskell-like language) uses this type system in order to gain a lot of speed while remaining safe.

Intersection types

Intersection types are types describing values that belong to both of two other given types with overlapping value sets. For example, in most implementations of C the signed char has range -128 to 127 and the unsigned char has range 0 to 255, so the intersection type of these two types would have range 0 to 127. Such an intersection type could be safely passed into functions expecting either signed or unsigned chars, because it is compatible with both types.

Intersection types are useful for describing overloaded function types: For example, if "int → int" is the type of functions taking an integer argument and returning an integer, and "float → float" is the type of functions taking a float argument and returning a float, then the intersection of these two types can be used to describe functions that do one or the other, based on what type of input they are given. Such a function could be passed into another function expecting an "int → int" function safely; it simply would not use the "float → float" functionality.

In a subclassing hierarchy, the intersection of a type and an ancestor type (such as its parent) is the most derived type. The intersection of sibling types is empty.

The Forsythe language includes a general implementation of intersection types. A restricted form is refinement types.

Union types

Union types are types describing values that belong to either of two types. For example, in C, the signed char has range -128 to 127, and the unsigned char has range 0 to 255, so the union of these two types would have range -128 to 255. Any function handling this union type would have to deal with integers in this complete range. More generally, the only valid operations on a union type are operations that are valid on both types being unioned. C's "union" concept is similar to union types, but is not typesafe because it permits operations that are valid on either type, rather than both. Union types are important in program analysis, where they are used to represent symbolic values whose exact nature (e.g., value or type) is not known.

In a subclassing hierarchy, the union of a type and an ancestor type (such as its parent) is the ancestor type. The union of sibling types is a subtype of their common ancestor (that is, all operations permitted on their common ancestor are permitted on the union type, but they may also have other valid operations in common).

Existential types

Existential types are frequently used in connection with record types to represent modules and abstract data types, due to their ability to separate implementation from interface. For example, the type "T = ∃X { a: X; f: (X → int); }" describes a module interface that has a data member of type X and a function that takes a parameter of the same type X and returns an integer. This could be implemented in different ways; for example:

• intT = { a: int; f: (int → int); }
• floatT = { a: float; f: (float → int); }

These types are both subtypes of the more general existential type T and correspond to concrete implementation types, so any value of one of these types is a value of type T. Given a value "t" of type "T", we know that "t.f(t.a)" is well-typed, regardless of what the abstract type X is. This gives flexibility for choosing types suited to a particular implementation while clients that use only values of the interface type—the existential type—are isolated from these choices.

In general it's impossible for the typechecker to infer which existential type a given module belongs to. In the above example intT { a: int; f: (int → int); } could also have the type ∃X { a: X; f: (int → int); }. The simplest solution is to annotate every module with its intended type, e.g.:

• intT = { a: int; f: (int → int); } as ∃X { a: X; f: (X → int); }

Although abstract data types and modules had been implemented in programming languages for quite some time, it wasn't until 1988 that John C. Mitchell and Gordon Plotkin established the formal theory under the slogan: "Abstract [data] types have existential type". The theory is a second-order typed lambda calculus similar to System F, but with existential instead of universal quantification.

Explicit or implicit declaration and inference

Further information: Type inference

Many static type systems, such as those of C and Java, require type declarations: The programmer must explicitly associate each variable with a particular type. Others, such as Haskell's, perform type inference: The compiler draws conclusions about the types of variables based on how programmers use those variables. For example, given a function f(x,y) which adds x and y together, the compiler can infer that x and y must be numbers – since addition is only defined for numbers. Therefore, any call to f elsewhere in the program that specifies a non-numeric type (such as a string or list) as an argument would signal an error.

Numerical and string constants and expressions in code can and often do imply type in a particular context. For example, an expression 3.14 might imply a type of floating-point, while [1, 2, 3] might imply a list of integers – typically an array.

Type inference is in general possible if it is decidable in the type theory in question. Moreover, even if inference is undecidable in general for a given type theory, inference is often possible for a large subset of real-world programs. Haskell's type system, a version of Hindley-Milner, is a restriction of System Fω to so-called rank-1 polymorphic types, in which type inference is decidable. Most Haskell compilers allow arbitrary-rank polymorphism as an extension, but this makes type inference undecidable. (Type checking is decidable, however, and rank-1 programs still have type inference; higher rank polymorphic programs are rejected unless given explicit type annotations.)

Types of types

Main article: Data type

A type of types is a kind. Kinds appear explicitly in typeful programming, such as a type constructor in the Haskell language.

Types fall into several broad categories:

• Primitive types – the simplest kind of type; e.g., integer and floating-point number
  • Boolean
  • Integral types – types of whole numbers; e.g., integers and natural numbers
  • Floating point types – types of numbers in floating-point representation
• Reference types
• Option types
  • Nullable types
• Composite types – types composed of basic types; e.g., arrays or records.

  Abstract data types

• Algebraic types
• Subtype
• Derived type
• Object types; e.g., type variable
• Partial type
• Recursive type
• Function types; e.g., binary functions
• universally quantified types, such as parameterized types
• existentially quantified types, such as modules
• Refinement types – types which identify subsets of other types
• Dependent types – types which depend on terms (values)
• Ownership types – types which describe or constrain the structure of object-oriented systems

Compatibility: equivalence and subtyping

A type-checker for a statically typed language must verify that the type of any expression is consistent with the type expected by the context in which that expression appears. For instance, in an assignment statement of the form x := e, the inferred type of the expression e must be consistent with the declared or inferred type of the variable x. This notion of consistency, called compatibility, is specific to each programming language.

If the type of e and the type of x are the same and assignment is allowed for that type, then this is a valid expression. In the simplest type systems, therefore, the question of whether two types are compatible reduces to that of whether they are equal (or equivalent). Different languages, however, have different criteria for when two type expressions are understood to denote the same type. These different equational theories of types vary widely, two extreme cases being structural type systems, in which any two types are equivalent that describe values with the same structure, and nominative type systems, in which no two syntactically distinct type expressions denote the same type (i.e., types must have the same "name" in order to be equal).

In languages with subtyping, the compatibility relation is more complex. In particular, if A is a subtype of B, then a value of type A can be used in a context where one of type B is expected, even if the reverse is not true. Like equivalence, the subtype relation is defined differently for each programming language, with many variations possible. The presence of parametric or ad hoc polymorphism in a language may also have implications for type compatibility.

Programming style

Some programmers prefer statically typed languages; others prefer dynamically typed languages. Statically typed languages alert programmers to type errors during compilation, and they may perform better at runtime. Advocates of dynamically typed languages claim they better support rapid prototyping and that type errors are only a small subset of errors in a program. Likewise, there is often no need to manually declare all types in statically typed languages with type inference; thus, the need for the programmer to explicitly specify types of variables is automatically lowered for such languages; and some dynamic languages have run-time optimisers that can generate fast code approaching the speed of static language compilers, often by using partial type inference.

See also

• Covariance and contravariance (computer science)
• Type theory
• Operator overloading
• Polymorphism in object-oriented programming
• Programming language
• Type signature
• Signedness
• Type system cross reference list

Further reading

• What To Know Before Debating Type Systems, by Chris Smith
• Laurence Tratt, Dynamically Typed Languages, Advances in Computers, vol. 77, pages 149-184, July 2009