Type system

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.

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 also be considered 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 runtime, 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.

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".

Combining static and dynamic type checking

Certain 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 runtime 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 trait object provides dynamic typing of types containing no or only 'static lifetimes.^[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 required 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 runtime 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 runtime 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 also. 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.

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]

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, ${\displaystyle \mathrm {matrix} (3,3)}$ might be the type of a ${\displaystyle 3\times 3}$ matrix. We can then define typing rules such as the following rule for matrix multiplication:

${\displaystyle \mathrm {matrix} _{\mathrm {multiply} }:\mathrm {matrix} (k,m)\times \mathrm {matrix} (m,n)\to \mathrm {matrix} (k,n)}$

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 for 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. However, in general proof of decidability is undecidable, so many programs require hand-written annotations that may be very non-trivial. As this impedes the development process, many language implementations provide an easy way out in the form of an option to disable this condition. This, however, comes at the cost of making the type-checker run in an infinite loop when fed programs that do not type-check, causing the compilation to fail.

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 files, strings, 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, as 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 to prevent race conditions. The Clean language (a Haskell-like language) uses this type system to gain a lot of speed (compared to performing a deep copy) 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 a -128 to 127 range, and the unsigned char has a 0 to 255 range, so the union of these two types would have an overall "virtual" range of -128 to 255 that may be used partially depending on which union member is accessed. 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, as 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 named a of type X and a function named f 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: (X → 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".^[25] The theory is a second-order typed lambda calculus similar to System F, but with existential instead of universal quantification.

Gradual typing

In a type system with Gradual typing, variables may be assigned a type either at compile-time (which is static typing), or at runtime (which is dynamic typing).^[26] This allows software developers to choose either type paradigm as appropriate, from within a single language.^[26] Gradual typing uses a special type named dynamic to represent statically unknown types; gradual typing replaces the notion of type equality with a new relation called consistency that relates the dynamic type to every other type. The consistency relation is symmetric but not transitive.^[27]