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Incomputer programming, atype system is alogical system comprising a set of rules that assigns a property called atype (for example,integer,floating point,string) to everyterm (a word, phrase, or other set of symbols). Usually the terms are variouslanguage constructs of acomputer program, such asvariables,expressions,functions, ormodules.[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 foralgebraic data types,data structures, or otherdata types, such as "string", "array of float", "function returning boolean".
Type systems are often specified as part ofprogramming languages and built intointerpreters andcompilers, although the type system of a language can be extended byoptional tools that perform added checks using the language's original typesyntax andgrammar.
The main purpose of a type system in a programming language is to reduce possibilities forbugs in computer programs due totype 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 defininginterfaces 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 (atcompile time), dynamically (atrun time), or as a combination of both.
Type systems have other purposes as well, such as expressing business rules, enabling certaincompiler optimizations, allowing formultiple dispatch, and providing a form ofdocumentation.
An example of a simple type system is that of theC language. The portions of a C program are thefunction definitions. One function is invoked by another function.
Theinterface of a function states the name of the function and a list ofparameters that are passed to the function's code. The code of an invoking function states the name of the invoked, along with the names ofvariables that hold values to pass to it.
During acomputer program's 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 aninteger value, but the calling code passed afloating-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.
Acompiler may also use the static type of a value to optimize the storage it needs and the choice ofalgorithms for operations on the value. In manyC compilers thefloatdata type, for example, is represented in 32bits, in accord with theIEEE specification for single-precision floating point numbers. They will thus use floating-point-specificmicroprocessor operations on those values (floating-point addition, multiplication, etc.).
The depth of type constraints and the manner of their evaluation affect thetyping of the language. Aprogramming language may further associate an operation with various resolutions for each type, in the case of typepolymorphism.Type theory is the study of type systems. The concrete types of some programming languages, such as integers and strings, depend on practical issues ofcomputer architecture, compiler implementation, andlanguage design.
Formally,type theory studies type systems. A programming language must have the opportunity to type check using thetype 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, termedtyping, gives meaning to a sequence ofbits such as a value inmemory or someobject such as avariable. Thehardware of ageneral purpose computer is unable to discriminate between for example amemory address and aninstruction code, or between acharacter, aninteger, or afloating-point number, because it makes no intrinsic distinction between any of the possible values that a sequence of bits mightmean.[note 1] Associating a sequence of bits with a type conveys thatmeaning to the programmable hardware to form asymbolic 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 manysubtypes. Other entities, such asobjects,modules,communication channels, anddependencies can become associated with a type. Even a type can become associated with a type. An implementation of atype system could in theory associate identifications calleddata type (a type of a value),class (a type of an object), andkind (atype 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 typeinferences (and other properties) becomeundecidable, 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 atype safe manner.
A programming language compiler can also implement adependent type or aneffect 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 describingwhat is being donewith what, and enabling for example to "throw" an error report. Thus the symbolic system may be atype 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 renders program behavior illegal if it falls outside the type-system rules. Advantages provided by programmer-specified type systems include:
Advantages provided by compiler-specified type systems include:
3 / "Hello, World" as invalid, when the rules do not specify how to divide aninteger by astring. Strong typing offers more safety, but cannot guarantee completetype safety.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 asHaskell, for whichtype inference is automated,lint might be available to its compiler to aid in the detection of error.
Type safety contributes toprogram correctness, but might only guarantee correctness at the cost of making the type checking itself anundecidable problem (as in theHalting problem). In atype 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 atcompile time does not scan for division by zero in most languages; that division would surface as aruntime error. To prove the absence of these defects, other kinds offormal methods, collectively known asprogram 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, expressingthe type of non-zero numbers). In addition,software testing is anempirical method for finding errors that such a type checker would not detect.
The process of verifying and enforcing the constraints of types—type checking—may occur atcompile time (a static check) or atrun-time (a dynamic check).
If a language specification requires its typing rules strongly, more or less allowing only those automatictype conversions that do not lose information, one can refer to the process asstrongly typed; if not, asweakly typed.
The terms are not usually used in a strict sense.
Static type checking is the process of verifying thetype 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 ofprogram verification (seetype 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 forTuring-complete languages is inherently conservative. That is, if a type system is bothsound (meaning that it rejects all incorrect programs) anddecidable (meaning that it is possible to write an algorithm that determines whether a program is well-typed), then it must beincomplete (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 totrue 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 theelse 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, evencode 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 asdowncasting. 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 butC# 4.0 introduces thedynamic 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, inC, programmers can freely cast a value between any two types that have the same size, effectively subverting the type concept.
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 atype tag (i.e., a reference to a type) containing its type information. This runtime type information (RTTI) can also be used to implementdynamic 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 asdowncasting, 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".
Some languages allow both static and dynamic typing. For example, Java and some other ostensibly statically typed languages supportdowncasting types to theirsubtypes, querying an object to discover its dynamic type and other type operations that depend on runtime type information. Another example isC++ RTTI. More generally, most programming languages include mechanisms for dispatching over different 'kinds' of data, such asdisjoint unions,runtime polymorphism, andvariant types. Even when not interacting with type annotations or type checking, such mechanisms are materially similar to dynamic typing implementations.Seeprogramming 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 theLiskov 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 orsubtype polymorphism. In some languages subtypes may also possesscovariant or contravariant return types and argument types respectively.
Certain languages, for exampleClojure,Common Lisp, orCython 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 bygradual typing. The programming environmentDrRacket, a pedagogic environment based on Lisp, and a precursor of the languageRacket 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 isdynamic 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
InRust, thedynstd::any::Any type provides dynamic typing of'static types.[13]
The choice between static and dynamic typing requires certaintrade-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 ofdependent typing,[who?] implemented in languages such asDependent ML andEpigram, 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 asCommon Lisp allow optional type declarations for optimization for this reason.
By contrast, dynamic typing may allow compilers to run faster andinterpreters 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 lacktype inference (such asC andJava prior toversion 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 includeHaskell,Scala,OCaml,F#,Swift, and to a lesser extentC# andC++), 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. Aneval function is possible with static typing, but requires advanced uses ofalgebraic 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 allowsduck typing (which enableseasier code reuse). Many[specify] languages with static typing also featureduck typing or other mechanisms likegeneric programming that also enable easier code reuse.
Dynamic typing typically makesmetaprogramming easier to use. For example,C++ templates are typically more cumbersome to write than the equivalentRuby orPython code sinceC++ has stronger rules regarding type definitions (for both functions and variables). This forces a developer to write moreboilerplate code for a template than a Python developer would need to. More advanced run-time constructs such asmetaclasses andintrospection 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 bydynamic programming languages; many of these are dynamically typed, althoughdynamic typing need not be related todynamic programming languages.
Languages are often colloquially referred to asstrongly typed orweakly 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".
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 termtype-safe language to describe languages that do not allow operations or conversions that violate the rules of the type system.
Computer scientists use the termmemory-safe language (or justsafe 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 willcheck 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 variablez will have the value 42. Although this may not be what the programmer anticipated, it is a well-defined result. Ify 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:
intx=5;chary[]="37";char*z=x+y;printf("%c\n",*z);
In this examplez will point to a memory address five characters beyondy, equivalent to three characters after the terminating zero character of the string pointed to byy. This is memory that the program is not expected to access. In C terms this is simplyundefined 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.
Some languages allow different levels of checking to apply to different regions of code. Examples include:
use strict directive inJavaScript[18][19][20] andPerl applies stronger checking.declare(strict_types=1) inPHP[21] on a per-file basis allows only a variable of exact type of the type declaration will be accepted, or aTypeError will be thrown.Option Strict On inVB.NET allows the compiler to require a conversion between objects.Additional tools such aslint andIBM Rational Purify can also be used to achieve a higher level of strictness.
It has been proposed, chiefly byGilad Bracha, that the choice of type system be made independent of choice of language; that a type system should be a module that can beplugged 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]
The termpolymorphism 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 anassociative 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 polymorphismgeneric programming. The type-theoretic foundations of polymorphism are closely related to those ofabstraction,modularity and (in some cases)subtyping.
Many type systems have been created that are specialized for use in certain environments with certain types of data, or for out-of-bandstatic program analysis. Frequently, these are based on ideas from formaltype 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 namesM, N, O range over terms and the names range over types.The following notation will be used:
| Type notion | Notation | Meaning |
|---|---|---|
| Function | If and, then. | |
| Product | If, then is a pair s.t. and. | |
| Sum | If, then is the first injection s.t., or is the second injection s.t.. | |
| Intersection | If, then and. | |
| Union | If, then or. | |
| Record | If, thenM has a member. | |
| Polymorphic | If, then for any typeσ. | |
| Existential | If, then for some typeσ. | |
| Recursive | If, then. | |
| Dependent function[a] | If and, then. | |
| Dependent pair[b] | If, then is a pair s.t. and. | |
| Dependent intersection[24] | If, then and. | |
| Familial intersection[24] | If, then for any term. | |
| Familial union[24] | If, then for some term. |
Dependent types are based on the idea of using scalars or values to more precisely describe the type of some other value. For example, might be the type of a matrix. We can then define typing rules such as the following rule for matrix multiplication:
wherek,m,n are arbitrary positive integer values. A variant ofML calledDependent ML has been created based on this type system, but because type checking for conventional dependent types isundecidable, not all programs using them can be type-checked without some kind of limits. Dependent ML limits the sort of equality it can decide toPresburger arithmetic.
Other languages such asEpigram make the value of all expressions in the language decidable so that type checking can be decidable. However, in generalproof 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 aninfinite loop when fed programs that do not type-check, causing the compilation to fail.
Linear types, based on the theory oflinear logic, and closely related touniqueness 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 largeimmutable values such as files, strings, and so on, because any operation that simultaneously destroys a linear object and creates a similar object (such asstr = 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, violatingreferential transparency. They are also used in the prototype operating systemSingularity for interprocess communication, statically ensuring that processes cannot share objects in shared memory in order to prevent race conditions. TheClean language (aHaskell-like language) uses this type system in order to gain a lot of speed (compared to performing a deep copy) while remaining safe.
Intersection types are types describing values that belong toboth 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 expectingeither 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 isrefinement types.
Union types are types describing values that belong toeither 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 onboth types being unioned. C's "union" concept is similar to union types, but is not typesafe, as it permits operations that are valid oneither type, rather thanboth. 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 are frequently used in connection withrecord types to representmodules andabstract 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 nameda of typeX and a function namedf that takes a parameter of thesame typeX and returns an integer. This could be implemented in different ways; for example:
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 typeX 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.:
Although abstract data types and modules had been implemented in programming languages for quite some time, it wasn't until 1988 thatJohn C. Mitchell andGordon Plotkin established the formal theory under the slogan: "Abstract [data] types have existential type".[25] The theory is a second-ordertyped lambda calculus similar toSystem F, but with existential instead of universal quantification.
In a type system withGradual typing, variables may be assigned a type either atcompile-time (which is static typing), or atrun-time (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 nameddynamic to represent statically unknown types; gradual typing replaces the notion of type equality with a new relation calledconsistency that relates the dynamic type to every other type. The consistency relation is symmetric but not transitive.[27]
Many static type systems, such as those of C and Java, requiretype declarations: the programmer must explicitly associate each variable with a specific type. Others, such as Haskell's, performtype inference: the compiler draws conclusions about the types of variables based on how programmers use those variables. For example, given a functionf(x,y) that addsx andy together, the compiler can infer thatx andy must be numbers—since addition is only defined for numbers. Thus, any call tof 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 expression3.14 might imply a type offloating-point, while[1,2,3] might imply a list of integers—typically anarray.
Type inference is in general possible, if it iscomputable in the type system in question. Moreover, even if inference is not computable in general for a given type system, inference is often possible for a large subset of real-world programs. Haskell's type system, a version ofHindley–Milner, is a restriction ofSystem Fω to so-called rank-1 polymorphic types, in which type inference is computable. Most Haskell compilers allow arbitrary-rank polymorphism as an extension, but this makes type inference not computable. (Type checking isdecidable, however, and rank-1 programs still have type inference; higher rank polymorphic programs are rejected unless given explicit type annotations.)
A type system that assigns types to terms in type environments usingtyping rules is naturally associated with thedecision problems oftype checking,typability, andtype inhabitation.[28]
Some languages likeC# orScala have a unified type system.[29] This means that allC# types including primitive types inherit from a single root object. Every type inC# inherits from the Object class. Some languages, likeJava andRaku, have a root type but also have primitive types that are not objects.[30] Java provides wrapper object types that exist together with the primitive types so developers can use either the wrapper object types or the simpler non-object primitive types. Raku automatically converts primitive types to objects when their methods are accessed.[31]
A type checker for a statically typed language must verify that the type of anyexpression is consistent with the type expected by the context in which that expression appears. For example, in anassignment statement of the formx :=e,the inferred type of the expressione must be consistent with the declared or inferred type of the variablex. This notion of consistency, calledcompatibility, is specific to each programming language.
If the type ofe and the type ofx are the same, and assignment is allowed for that type, then this is a valid expression. Thus, in the simplest type systems, the question of whether two types are compatible reduces to that of whether they areequal (orequivalent). Different languages, however, have different criteria for when two type expressions are understood to denote the same type. These differentequational theories of types vary widely, two extreme cases beingstructural type systems, in which any two types that describe values with the same structure are equivalent, andnominative 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 withsubtyping, the compatibility relation is more complex: IfB is a subtype ofA, then a value of typeB can be used in a context where one of typeA is expected (covariant), 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 orad hoc polymorphism in a language may also have implications for type compatibility.
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