43:
5521:
1302:) 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
1255:
appropriately representing the designed types in code. Static typing advocates believe programs are more reliable when they have been well type-checked, whereas dynamic-typing advocates point to distributed code that has proven reliable and to small bug databases. The value of static typing increases as the strength of the type system is increased. Advocates of
6519:
105:
5399:, have a root type but also have primitive types that are not objects. 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.
4757:
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
1492:
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
4670:
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
855:
A type error occurs when an operation receives a different type of data than it expected. 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. It is an unintended condition which might manifest in multiple stages of a
1730:
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
1101:
Most type-safe languages include some form of dynamic type checking, even if they also have a static type checker. 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
4773:
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
4596:, 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
1270:
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
4843:
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.:
1254:
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
5577:
line determined a memory location's contents by its flag bits. Flag bits specify the contents of a memory location. Instruction, data type, and functions are specified by a 3 bit code in addition to its 48 bit contents. Only the MCP (Master
Control Program) could write to the flag code
4702:" 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 "
1610:
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.
4834:
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
4552:
1696:
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. The requirement that the type system does not affect the semantics of the language is difficult to fulfill.
599:
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
1606:
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.
435:. 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).
1184:
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
984:), 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.
1029:
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.
726:, and when more attention must be paid by the programmer to annotate code or to consider computer-related operations and functioning. It is challenging to find a sufficiently expressive type system that satisfies all programming practices in a
517:
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.
802:
clarifying the intent of the programmer. For example, if a programmer declares a function as returning a timestamp type, this documents the function when the timestamp type can be explicitly declared deeper in the code to be an integer
1156:
Objects in object-oriented languages are usually accessed by a reference whose static target type (or manifest type) is equal to either the object's run-time type (its latent type) or a supertype thereof. This is conformant with the
1040:
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, but
1405:
are often harder to use in statically typed languages. In some languages, such features may also be used e.g. to generate new types and behaviors on the fly, based on run-time data. Such advanced constructs are often provided by
1109:
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.
1106:, 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.
1440:. 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".
5008:
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 of
760:
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:
5017:
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 is
1282:
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. This too may reduce the edit-compile-test-debug cycle.
4633:) 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
5449:
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 are
4839:
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.
4376:
4269:
4196:
745:, which enables even more program specifications to be verified by a type checker. Beyond simple value-type pairs, a virtual "region" of code is associated with an "effect" component describing
5591:
might surface during development, which may show that more type development is needed. —"The evaluation of a well-typed program always terminates".—B. Nordström, K. Petersson, and J. M. Smith
1078:
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
5505:), 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
503:'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.
1493:
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.
4803:, 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
4007:
3853:
3543:
3111:
2987:
2878:
5621:, p. 1: "A type system is a tractable syntactic method for proving the absence of certain program behaviors by classifying phrases according to the kinds of values they compute."
4051:
3897:
3741:
3497:
3072:
2948:
2838:
4342:
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1113:
Programming languages that include dynamic type checking but not static type checking are often called "dynamically typed programming languages". For a list of such languages, see
3328:
1267:, 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.
4898:(which is dynamic typing). This allows software developers to choose either type paradigm as appropriate, from within a single language. Gradual typing uses a special type named
3231:
2745:
2624:
2226:
2130:
3194:
2711:
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818:
information. For example, if a type requires that a value must align in memory at a multiple of four bytes, the compiler may be able to use more efficient machine instructions.
2096:
2529:
2461:
2419:
907:, 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
2385:
1800:
1735:
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
1473:, 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.
1161:, 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
718:
When a programming language evolves a more elaborate type system, it gains a more finely grained rule set than basic type checking, but this comes at a price when the type
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2191:
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4118:
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2156:
1829:
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5502:
4729:
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.
1166:
1114:
5304:
5281:
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2797:
2676:
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2350:
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1003:(meaning there are correct programs, which are also rejected, even though they do not encounter runtime errors). For example, consider a program containing the code:
5364:
5324:
5261:
5221:
5158:
5118:
2040:
2012:
1977:
1057:
1900:
576:
442:
between different parts of a computer program, and then checking that the parts have been connected in a consistent way. This checking can happen statically (at
6148:
5344:
5241:
5181:
5138:
5098:
1940:
1920:
1849:
4758:
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
5823:
1021:
at run-time, most type checkers will reject the program as ill-typed, because it is difficult (if not impossible) for a static analyzer to determine that the
1454:
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
1049:
keyword, which is used to declare variables to be checked dynamically at runtime. Other languages allow writing code that is not type-safe; for example, in
1025:
branch will not be taken. Consequently, a static type checker will quickly detect type errors in rarely used code paths. Without static type checking, even
5544:
4625:, 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
1202:
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
1037:. 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.
4770:. 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.
1469:) 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
6681:
382:. 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.
973:). 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.
7052:
6686:
4641:
for interprocess communication, statically ensuring that processes cannot share objects in shared memory in order to prevent race conditions. The
1149:. Even when not interacting with type annotations or type checking, such mechanisms are materially similar to dynamic typing implementations. See
5022:, however, and rank-1 programs still have type inference; higher rank polymorphic programs are rejected unless given explicit type annotations.)
1206:
will not be subject to static type checking. Instead, the program relies on runtime type information to determine how the variable may be used.
6676:
6671:
5458:). Different languages, however, have different criteria for when two type expressions are understood to denote the same type. These different
309:
6501:
6441:
6302:
5704:
5667:
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such as files, strings, and so on, because any operation that simultaneously destroys a linear object and creates a similar object (such as
571:
is the study of type systems. The concrete types of some programming languages, such as integers and strings, depend on practical issues of
6659:
6560:
5574:
115:
4930:: the compiler draws conclusions about the types of variables based on how programmers use those variables. For example, given a function
126:
64:
6059:
5633:, p. 1: "The fundamental purpose of a type system is to prevent the occurrence of execution errors during the running of a program."
4581:, 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
1761:
Many type systems have been created that are specialized for use in certain environments with certain types of data, or for out-of-band
1133:, querying an object to discover its dynamic type and other type operations that depend on runtime type information. Another example is
904:
4984:
Numerical and string constants and expressions in code can and often do imply type in a particular context. For example, an expression
6470:
5802:
4895:
4856:
Although abstract data types and modules had been implemented in programming languages for quite some time, it wasn't until 1988 that
447:
4547:{\displaystyle \mathrm {matrix} _{\mathrm {multiply} }:\mathrm {matrix} (k,m)\times \mathrm {matrix} (m,n)\to \mathrm {matrix} (k,n)}
4762:
types being unioned. C's "union" concept is similar to union types, but is not typesafe, as it permits operations that are valid on
1125:
Some languages allow both static and dynamic typing. For example, Java and some other ostensibly statically typed languages support
936:
144:
86:
1082:(i.e., a reference to a type) containing its type information. This runtime type information (RTTI) can also be used to implement
4649:-like language) uses this type system in order to gain a lot of speed (compared to performing a deep copy) while remaining safe.
1402:
492:
that are passed to the function's code. The code of an invoking function states the name of the invoked, along with the names of
489:
4214:
1347:. Further, dynamic typing better accommodates transitional code and prototyping, such as allowing a placeholder data structure (
6810:
6109:
1721:
564:
6039:
4141:
1692:, that the choice of type system be made independent of choice of language; that a type system should be a module that can be
1053:, programmers can freely cast a value between any two types that have the same size, effectively subverting the type concept.
669:
A program associates each value with at least one specific type, but it also can occur that one value is associated with many
6927:
6732:
6664:
6626:
6003:
5388:
5384:
5376:
4646:
4638:
4589:
1742:
1323:
1315:
1303:
1264:
773:
637:
417:
4981:
elsewhere in the program that specifies a non-numeric type (such as a string or list) as an argument would signal an error.
5744:
856:
program's development. Thus a facility for detection of the error is needed in the type system. In some languages, such as
6481:
1386:
1196:
1158:
776:
implementation. For example, programmers can begin to think of a string as a set of character values instead of as a mere
686:
371:
6827:
6757:
6605:
5380:
4642:
4592:
make the value of all expressions in the language decidable so that type checking can be decidable. However, in general
1772:
The following table gives an overview over type theoretic concepts that are used in specialized type systems. The names
1407:
1393:
has stronger rules regarding type definitions (for both functions and variables). This forces a developer to write more
1351:) to be transparently used in place of a full data structure (usually for the purposes of experimentation and testing).
1319:
1307:
1137:. More generally, most programming languages include mechanisms for dispatching over different 'kinds' of data, such as
1069:
781:
439:
4292:
are based on the idea of using scalars or values to more precisely describe the type of some other value. For example,
715:, or metatype). These are the abstractions that typing can go through, on a hierarchy of levels contained in a system.
7082:
5526:
5396:
5392:
4902:
to represent statically unknown types; gradual typing replaces the notion of type equality with a new relation called
4864:
established the formal theory under the slogan: "Abstract types have existential type". The theory is a second-order
1382:
1295:
1210:
621:
493:
375:
367:
999:(meaning that it is possible to write an algorithm that determines whether a program is well-typed), then it must be
57:
51:
1731:
code re-use: in a language with polymorphism, programmers need only implement a data structure such as a list or an
595:
whether at compile time or runtime, manually annotated or automatically inferred. As Mark
Manasse concisely put it:
6837:
6705:
5819:
5534:
507:
428:
302:
4924:: the programmer must explicitly associate each variable with a specific type. Others, such as Haskell's, perform
1488:
will have the value 42. Although this may not be what the programmer anticipated, it is a well-defined result. If
7113:
7015:
6967:
6879:
6857:
6852:
6780:
6646:
5923:"Static Typing Where Possible, Dynamic Typing When Needed: The End of the Cold War Between Programming Languages"
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4570:
2804:
1134:
837:
674:
617:
511:
349:
273:
68:
6889:
6553:
5922:
5408:
4634:
3976:
3822:
3504:
3079:
2955:
2845:
1333:
Dynamic typing allows constructs that some (simple) static type checking would reject as illegal. For example,
1291:
1050:
701:
629:
530:
474:
413:
6523:
4906:
that relates the dynamic type to every other type. The consistency relation is symmetric but not transitive.
4014:
3860:
3702:
3464:
3046:
2922:
2811:
1458:
to describe languages that do not allow operations or conversions that violate the rules of the type system.
652:, because it makes no intrinsic distinction between any of the possible values that a sequence of bits might
7042:
6957:
4295:
3335:
2915:
1762:
1279:
405:
335:
3662:
2233:
1626:
Some languages allow different levels of checking to apply to different regions of code. Examples include:
942:
If a language specification requires its typing rules strongly, more or less allowing only those automatic
6785:
6641:
6600:
6595:
6399:
6175:
5981:
5963:
5470:
5464:
4788:
4783:
4577:
has been created based on this type system, but because type checking for conventional dependent types is
3295:
1427:
1095:
777:
545:
462:
217:
179:
5411:
is consistent with the type expected by the context in which that expression appears. For example, in an
3201:
2718:
2597:
2205:
689:
can become associated with a type. Even a type can become associated with a type. An implementation of a
7103:
6775:
6750:
6390:
6319:
6277:
6043:
4865:
4582:
2103:
1653:
on a per-file basis allows only a variable of exact type of the type declaration will be accepted, or a
682:
649:
641:
572:
485:
454:
295:
248:
119:
that states a
Knowledge (XXG) editor's personal feelings or presents an original argument about a topic.
3170:
2690:
2569:
2075:
30:
This article is about type systems in computer programming. For the formal study of type systems, see
7118:
6577:
5412:
5019:
4998:
4796:
2492:
2424:
2392:
1746:
1299:
1150:
1073:
977:
887:
with automated type checking, a program may prove to run incorrectly yet produce no compiler errors.
678:
560:
401:
323:
6084:
4732:
The
Forsythe language includes a general implementation of intersection types. A restricted form is
7108:
7047:
7025:
6952:
6805:
6797:
6717:
6546:
6404:
6180:
5986:
5506:
5005:
2364:
1779:
1737:
1677:
1603:
1367:
1330:), so explicit type declaration is not a necessary requirement for static typing in all languages.
988:
876:
872:
723:
386:
379:
213:
6531:
5968:
1596:, equivalent to three characters after the terminating zero character of the string pointed to by
1102:
is a subtype of A. If the program tries to convert a value of type A to type B, which is known as
1033:
A number of useful and common programming language features cannot be checked statically, such as
7030:
7010:
6962:
6937:
6722:
6691:
6429:
6417:
6227:
6009:
4800:
788:-sized subsystems. This enables more levels of localization so that the definitions required for
772:) – Types enable programmers to think at a higher level than the bit or byte, not bothering with
707:
359:
354:
6170:
Kopylov, Alexei (2003). "Dependent intersection: A new way of defining records in type theory".
3405:
2161:
1397:
for a template than a Python developer would need to. More advanced run-time constructs such as
1008:
if <complex test> then <do something> else <signal that there is a type error>
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3943:
3746:
3586:
3379:
2750:
2629:
2466:
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591:
studies type systems. A programming language must have the opportunity to type check using the
563:
may further associate an operation with various resolutions for each type, in the case of type
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6822:
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matrix. We can then define typing rules such as the following rule for matrix multiplication:
3815:
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2562:
1732:
625:
458:
453:
Type systems have other purposes as well, such as expressing business rules, enabling certain
258:
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1378:
385:
Type systems formalize and enforce the otherwise implicit categories the programmer uses for
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6409:
6219:
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Proceedings of the 26th ACM SIGPLAN-SIGACT Symposium on
Principles of Programming Languages
5595:, in a functional programming language where functions are first class citizens. —From the
5349:
5309:
5246:
5206:
5143:
5103:
2025:
1982:
1947:
525:
may also use the static type of a value to optimize the storage it needs and the choice of
427:
The main purpose of a type system in a programming language is to reduce possibilities for
7067:
6932:
6884:
6817:
5699:. Handbook of Logic in Computer Science. Vol. 5. Oxford University Press. p. 2.
5596:
5468:, in which any two types that describe values with the same structure are equivalent, and
4733:
4626:
4622:
4593:
1876:
1673:
1470:
1374:
943:
880:
865:
662:
613:
549:
278:
268:
204:
6382:
4678:
Intersection types are useful for describing overloaded function types: for example, if "
875:, but might only guarantee correctness at the cost of making the type checking itself an
753:, and enabling for example to "throw" an error report. Thus the symbolic system may be a
5869:
5593:
A systematic change in variables to avoid capture of a free variable can introduce error
506:
If the instructions inside the function are written with the assumption of receiving an
7020:
6842:
6832:
6740:
6353:
5549:
5329:
5226:
5166:
5123:
5083:
5014:
5010:
4991:
4926:
4915:
4887:
4881:
4861:
4289:
4284:
3457:
3288:
3163:
2014:) describes the type which results from replacing all occurrences of the type variable
1925:
1905:
1834:
1709:
1701:
1602:. This is memory that the program is not expected to access. In C terms this is simply
1287:
1256:
1138:
895:
does not scan for division by zero in most languages; that division would surface as a
861:
735:
633:
421:
390:
345:
331:
253:
243:
208:
6493:
7097:
6942:
6451:
6374:
5045:
4690:" is the type of functions taking an integer argument and returning an integer, and "
4597:
4578:
2068:
1705:
1615:
1476:
Consider the following program of a language that is both type-safe and memory-safe:
1026:
900:
896:
799:
741:
263:
238:
6899:
6874:
6421:
6378:
6231:
6026:
5874:
5592:
4891:
4618:
4574:
2198:
1689:
1260:
1146:
1087:
932:
892:
815:
443:
283:
6455:
6013:
5741:"... any sound, decidable type system must be incomplete" —D. Remy (2017). p. 29,
4823:
and returns an integer. This could be implemented in different ways; for example:
828:
to detect meaningless or invalid code. For example, we can identify an expression
6292:
5976:
Xi, Hongwei; Pfenning, Frank (1999). "Dependent Types in
Practical Programming".
5944:
5387:
types including primitive types inherit from a single root object. Every type in
1741:. The type-theoretic foundations of polymorphism are closely related to those of
1704:. While both typing disciplines can be used to perform static analysis of code (
7077:
6922:
6869:
6696:
6264:
5751:
5554:
5520:
5474:, in which no two syntactically distinct type expressions denote the same type (
5407:
A type checker for a statically typed language must verify that the type of any
5037:
5031:
4614:
4609:
1766:
1449:
1363:
1359:
1355:
1348:
1272:
1177:
1126:
1103:
1091:
1042:
1034:
981:
970:
966:
842:
588:
568:
222:
174:
31:
17:
6131:
5897:
6982:
6977:
6894:
6862:
6767:
6710:
6335:
6189:
5516:
5049:
4975:
must be numbers—since addition is only defined for numbers. Thus, any call to
4750:
4745:
2683:
1635:
478:
432:
5677:
7057:
7035:
6992:
6987:
6654:
6610:
6569:
5929:
5482:
1750:
1398:
1248:
1162:
916:
891:
is an unsafe and incorrect operation, but a type checker which only runs at
780:
of bytes. Higher still, types enable programmers to think about and express
727:
719:
695:
670:
555:
The depth of type constraints and the manner of their evaluation affect the
537:
526:
394:
6149:"Sure. It's called "gradual typing", and I would qualify it as trendy. ..."
4600:
when fed programs that do not type-check, causing the compilation to fail.
1153:
for more discussion of the interactions between static and dynamic typing.
6518:
5995:
792:
of the subsystems remain consistent when those two subsystems communicate.
657:
6972:
4869:
2357:
1195:, a pedagogic environment based on Lisp, and a precursor of the language
1191:
1130:
825:
522:
409:
358:(a word, phrase, or other set of symbols). Usually the terms are various
6223:
5962:(PhD). Department of Mathematical Sciences, Carnegie Mellon University.
5845:
1480:
var x := 5; var y := "37"; var z := x + y;
991:
languages is inherently conservative. That is, if a type system is both
1343:
function is possible with static typing, but requires advanced uses of
1173:
857:
833:
645:
341:
199:
6413:
5855:
5428:
must be consistent with the declared or inferred type of the variable
757:, which endows it with more safety checking than type checking alone.
5036:
A type system that assigns types to terms in type environments using
1665:
1339:
functions, which execute arbitrary data as code, become possible. An
1181:
919:
method for finding errors that such a type checker would not detect.
868:
might be available to its compiler to aid in the detection of error.
4675:
signed or unsigned chars, because it is compatible with both types.
397:, such as "string", "array of float", "function returning boolean".
6029:
is an example of a language that is both type-safe and memory-safe.
1390:
1327:
1275:
allow optional type declarations for optimization for this reason.
514:, then the wrong result will be computed by the invoked function.
1311:
1278:
By contrast, dynamic typing may allow compilers to run faster and
840:. Strong typing offers more safety, but cannot guarantee complete
6291:
Barendregt, Henk; Dekkers, Wil; Statman, Richard (20 June 2013).
5509:
in a language may also have implications for type compatibility.
552:
on those values (floating-point addition, multiplication, etc.).
6590:
1708:), optional type systems do not enforce type safety at runtime (
1639:
1344:
1335:
927:
The process of verifying and enforcing the constraints of types—
807:
Advantages provided by compiler-specified type systems include:
798:– In more expressive type systems, types can serve as a form of
6542:
6152:
Is there a language that allows both static and dynamic typing?
4920:
Many static type systems, such as those of C and Java, require
6585:
1769:
and are only available as part of prototype research systems.
1650:
1247:
The choice between static and dynamic typing requires certain
946:
that do not lose information, one can refer to the process as
609:
546:
IEEE specification for single-precision floating point numbers
541:
98:
36:
6538:
416:
that perform added checks using the language's original type
6383:"On Understanding Types, Data Abstraction, and Polymorphism"
4872:, but with existential instead of universal quantification.
1668:
allows the compiler to require a conversion between objects.
488:
of a function states the name of the function and a list of
412:, although the type system of a language can be extended by
4264:{\textstyle (x:\sigma )\times \tau =\sum _{x:\sigma }\tau }
1381:
are typically more cumbersome to write than the equivalent
832:
as invalid, when the rules do not specify how to divide an
481:
definitions. One function is invoked by another function.
334:
comprising a set of rules that assigns a property called a
116:
personal reflection, personal essay, or argumentative essay
1680:
can also be used to achieve a higher level of strictness.
656:. Associating a sequence of bits with a type conveys that
5820:"Dynamic Method Dispatch or Runtime Polymorphism in Java"
5478:, types must have the same "name" in order to be equal).
4191:{\textstyle (x:\sigma )\to \tau =\prod _{x:\sigma }\tau }
976:
Static type checking can be considered a limited form of
1115:
the category for dynamically typed programming languages
899:. To prove the absence of these defects, other kinds of
4637:. They are also used in the prototype operating system
1802:
range over types. The following notation will be used:
1056:
For a list of languages with static type checking, see
122:
4217:
4144:
4017:
3863:
1590:
will point to a memory address five characters beyond
5391:
inherits from the Object class. Some languages, like
5352:
5332:
5312:
5292:
5269:
5249:
5229:
5209:
5189:
5169:
5146:
5126:
5106:
5086:
5066:
4379:
4350:
4298:
4100:
4059:
3979:
3946:
3905:
3825:
3775:
3749:
3705:
3665:
3615:
3589:
3551:
3507:
3467:
3408:
3382:
3338:
3298:
3239:
3204:
3173:
3119:
3082:
3049:
2995:
2958:
2925:
2890:
2848:
2814:
2779:
2753:
2721:
2693:
2658:
2632:
2600:
2572:
2537:
2495:
2469:
2427:
2395:
2367:
2332:
2306:
2268:
2236:
2208:
2164:
2138:
2106:
2078:
2028:
1985:
1950:
1928:
1908:
1879:
1857:
1837:
1811:
1782:
995:(meaning that it rejects all incorrect programs) and
965:
Static type checking is the process of verifying the
733:
A programming language compiler can also implement a
6174:. LICS 2003. IEEE Computer Society. pp. 86–95.
4569:
are arbitrary positive integer values. A variant of
969:
of a program based on analysis of a program's text (
7001:
6910:
6796:
6766:
6731:
6619:
6576:
6203:Mitchell, John C.; Plotkin, Gordon D. (July 1988).
4995:
4985:
4976:
4970:
4964:
4958:
4952:
4931:
4721:
4715:
4709:
4703:
4697:
4691:
4685:
4679:
1765:. Frequently, these are based on ideas from formal
1597:
1591:
1585:
1489:
1485:
1231:
1214:
5729:invited lecture at TFP12, at St Andrews University
5722:"Some History of Functional Programming Languages"
5691:Nordström, B.; Petersson, K.; Smith, J.M. (2001).
5358:
5338:
5318:
5298:
5275:
5255:
5235:
5215:
5195:
5175:
5152:
5132:
5112:
5092:
5072:
4546:
4362:
4336:
4263:
4190:
4112:
4086:
4045:
4001:
3958:
3932:
3891:
3847:
3802:
3761:
3735:
3689:
3642:
3601:
3575:
3537:
3491:
3444:
3394:
3368:
3322:
3275:
3225:
3188:
3146:
3105:
3066:
3022:
2981:
2942:
2902:
2872:
2832:
2791:
2765:
2739:
2705:
2670:
2644:
2618:
2584:
2549:
2523:
2481:
2455:
2413:
2379:
2344:
2318:
2292:
2254:
2220:
2185:
2150:
2124:
2090:
2034:
2006:
1971:
1934:
1914:
1894:
1863:
1843:
1823:
1794:
1700:Optional typing is related to, but distinct from,
1362:). Many languages with static typing also feature
1064:Dynamic type checking and runtime type information
957:The terms are not usually used in a strict sense.
473:An example of a simple type system is that of the
5485:, the compatibility relation is more complex: If
5183:, decide whether there exists a type environment
693:could in theory associate identifications called
6463:CRC Handbook of Computer Science and Engineering
6172:18th IEEE Symposium on Logic in Computer Science
5540:Covariance and contravariance (computer science)
5383:have a unified type system. This means that all
5004:Type inference is in general possible, if it is
4714:" function safely; it simply would not use the "
1432:Languages are often colloquially referred to as
1410:; many of these are dynamically typed, although
632:is unable to discriminate between for example a
597:
6324:(5th ed.). ECMA. December 2017. ECMA-334.
5660:Programming languages: Concepts and constructs
5462:of types vary widely, two extreme cases being
4910:Explicit or implicit declaration and inference
1169:return types and argument types respectively.
1165:. In some languages subtypes may also possess
6554:
5662:(2nd ed.). Addison-Wesley. p. 142.
4890:, variables may be assigned a type either at
4791:types are frequently used in connection with
548:. They will thus use floating-point-specific
303:
8:
6488:. Vol. 77. Elsevier. pp. 149–184.
6263:Siek, Jeremy; Taha, Walid (September 2006).
5436:, is specific to each programming language.
4997:might imply a list of integers—typically an
2867:
2855:
2827:
2815:
1243:Static and dynamic type checking in practice
666:composed of that hardware and some program.
400:Type systems are often specified as part of
6532:"What to Know Before Debating Type Systems"
6048:(11th ed.). ECMA. June 2020. ECMA-262.
5653:
5651:
5545:Polymorphism in object-oriented programming
5497:can be used in a context where one of type
4753:are types describing values that belong to
4666:are types describing values that belong to
1058:the category for statically typed languages
6561:
6547:
6539:
6297:. Cambridge University Press. p. 66.
1121:Combining static and dynamic type checking
814:– Static type-checking may provide useful
310:
296:
156:
6403:
6179:
6125:
6123:
6114:PHP Manual: Language Reference: Functions
5985:
5967:
5351:
5331:
5311:
5291:
5268:
5248:
5228:
5208:
5188:
5168:
5145:
5125:
5105:
5085:
5065:
4509:
4471:
4433:
4402:
4401:
4381:
4378:
4349:
4299:
4297:
4246:
4216:
4173:
4143:
4099:
4058:
4028:
4016:
4002:{\displaystyle \bigcup _{x:\sigma }\tau }
3984:
3978:
3945:
3904:
3874:
3862:
3848:{\displaystyle \bigcap _{x:\sigma }\tau }
3830:
3824:
3774:
3748:
3704:
3664:
3614:
3588:
3550:
3538:{\displaystyle M:(x:\sigma )\times \tau }
3506:
3466:
3407:
3381:
3337:
3297:
3238:
3203:
3172:
3118:
3106:{\displaystyle M:\exists {}\alpha .\tau }
3092:
3081:
3053:
3048:
2994:
2982:{\displaystyle M:\forall {}\alpha .\tau }
2968:
2957:
2929:
2924:
2889:
2873:{\displaystyle M:\langle x:\tau \rangle }
2847:
2813:
2778:
2752:
2720:
2692:
2657:
2631:
2599:
2571:
2536:
2506:
2494:
2468:
2438:
2426:
2394:
2366:
2331:
2305:
2267:
2235:
2207:
2163:
2137:
2105:
2077:
2027:
1984:
1949:
1927:
1907:
1878:
1856:
1836:
1810:
1781:
195:
145:Learn how and when to remove this message
87:Learn how and when to remove this message
6040:"4.2.2 The Strict Variant of ECMAScript"
5960:Dependent Types in Practical Programming
5788:
5786:
5784:
5745:"Type systems for programming languages"
5630:
5403:Compatibility: equivalence and subtyping
4830:floatT = { a: float; f: (float → int); }
4046:{\textstyle M:\bigcup _{x:\sigma }\tau }
3892:{\textstyle M:\bigcap _{x:\sigma }\tau }
2053:
50:This article includes a list of general
6266:Gradual Typing for Functional Languages
6243:
6241:
6045:ECMAScript® 2020 Language Specification
5943:Laucher, Amanda; Snively, Paul (2012).
5611:
5566:
4127:
3736:{\displaystyle M:(x:\sigma )\cap \tau }
3492:{\displaystyle (x:\sigma )\times \tau }
3067:{\displaystyle \exists {}\alpha .\tau }
2943:{\displaystyle \forall {}\alpha .\tau }
2833:{\displaystyle \langle x:\tau \rangle }
660:to the programmable hardware to form a
230:
187:
166:
159:
6274:Scheme and Functional Programming 2006
6205:"Abstract Types Have Existential Type"
6165:
6163:
6161:
5775:
5642:
5618:
5422:, the inferred type of the expression
4963:together, the compiler can infer that
4337:{\displaystyle \mathrm {matrix} (3,3)}
3369:{\displaystyle M:(x:\sigma )\to \tau }
477:. The portions of a C program are the
5432:. This notion of consistency, called
5326:, decide whether there exists a term
3690:{\displaystyle (x:\sigma )\cap \tau }
2255:{\displaystyle M:\sigma \times \tau }
1496:Now consider a similar example in C:
1286:Statically typed languages that lack
529:for operations on the value. In many
510:value, but the calling code passed a
7:
5032:Type theory § Decision problems
4594:proof of decidability is undecidable
3323:{\displaystyle (x:\sigma )\to \tau }
1370:that also enable easier code reuse.
1098:(reflection), and similar features.
540:, for example, is represented in 32
6066:. Developer.mozilla.org. 2013-07-03
4848:intT = { a: int; f: (int → int); }
3226:{\displaystyle M:\mu \alpha .\tau }
2740:{\displaystyle M:\sigma \cup \tau }
2619:{\displaystyle M:\sigma \cap \tau }
2221:{\displaystyle \sigma \times \tau }
1259:, implemented in languages such as
1186:
5293:
5270:
5190:
5067:
4827:intT = { a: int; f: (int → int); }
4525:
4522:
4519:
4516:
4513:
4510:
4487:
4484:
4481:
4478:
4475:
4472:
4449:
4446:
4443:
4440:
4437:
4434:
4424:
4421:
4418:
4415:
4412:
4409:
4406:
4403:
4397:
4394:
4391:
4388:
4385:
4382:
4315:
4312:
4309:
4306:
4303:
4300:
3089:
3050:
2965:
2926:
2125:{\displaystyle M:\sigma \to \tau }
56:it lacks sufficient corresponding
25:
6318:"8.2.4 Type system unification".
5040:is naturally associated with the
3189:{\displaystyle \mu \alpha .\tau }
2706:{\displaystyle \sigma \cup \tau }
2585:{\displaystyle \sigma \cap \tau }
1688:It has been proposed, chiefly by
1461:Computer scientists use the term
608:, gives meaning to a sequence of
496:that hold values to pass to it.
450:), or as a combination of both.
6517:
6482:"5. Dynamically Typed Languages"
5697:Algebraic and Logical Structures
5519:
4894:(which is static typing), or at
2091:{\displaystyle \sigma \to \tau }
1622:Variable levels of type checking
1354:Dynamic typing typically allows
1230:type provides dynamic typing of
103:
41:
6434:Types and Programming Languages
5826:from the original on 2020-12-07
2524:{\displaystyle M=\iota _{2}(N)}
2456:{\displaystyle M=\iota _{1}(N)}
2414:{\displaystyle M:\sigma +\tau }
1776:range over terms and the names
1722:Polymorphism (computer science)
1373:Dynamic typing typically makes
1172:Certain languages, for example
575:, compiler implementation, and
6248:Siek, Jeremy (24 March 2014).
6212:ACM Trans. Program. Lang. Syst
5921:Meijer, Erik; Drayton, Peter.
5731:. See the section on Algol 60.
5346:that can be assigned the type
4815:that takes a parameter of the
4541:
4529:
4506:
4503:
4491:
4465:
4453:
4331:
4319:
4230:
4218:
4160:
4157:
4145:
4081:
4069:
3927:
3915:
3797:
3785:
3724:
3712:
3678:
3666:
3637:
3625:
3570:
3558:
3526:
3514:
3480:
3468:
3439:
3427:
3418:
3412:
3360:
3357:
3345:
3314:
3311:
3299:
3270:
3249:
3141:
3129:
3017:
3005:
2518:
2512:
2450:
2444:
2287:
2275:
2174:
2168:
2116:
2082:
2001:
1989:
1966:
1954:
1889:
1883:
1484:In this example, the variable
1189:. The programming environment
722:(and other properties) become
604:Assigning a data type, termed
1:
6627:Arbitrary-precision or bignum
6494:10.1016/S0065-2458(09)01205-4
6480:Tratt, Laurence (July 2009).
5720:Turner, D.A. (12 June 2012).
4774:valid operations in common).
2531:is the second injection s.t.
2380:{\displaystyle \sigma +\tau }
1795:{\displaystyle \sigma ,\tau }
1444:Type safety and memory safety
1416:dynamic programming languages
1408:dynamic programming languages
1159:Liskov substitution principle
6461:. In Allen B. Tucker (ed.).
2463:is the first injection s.t.
1422:Strong and weak type systems
1377:easier to use. For example,
1070:Dynamic programming language
909:the type of non-zero numbers
824:– A type system enables the
438:Type systems allow defining
431:in computer programs due to
6465:(2nd ed.). CRC Press.
5527:Computer programming portal
871:Type safety contributes to
705:(a type of an object), and
7135:
6294:Lambda Calculus with Types
6085:"Strict Mode (JavaScript)"
6060:"Strict mode – JavaScript"
5844:Wright, Andrew K. (1995).
5693:"Martin-Löf's Type Theory"
5535:Comparison of type systems
5120:, decide whether the term
5029:
4913:
4879:
4852:∃X { a: X; f: (X → int); }
4781:
4743:
4656:
4607:
4282:
3445:{\displaystyle M(N):\tau }
2186:{\displaystyle M(N):\tau }
1719:
1642:applies stronger checking.
1614:For more information, see
1447:
1425:
1167:covariant or contravariant
1067:
673:. Other entities, such as
461:, and providing a form of
29:
6968:Strongly typed identifier
6354:"Numerics, § Auto-boxing"
6321:C# Language Specification
6250:"What is gradual typing?"
6190:10.1109/LICS.2003.1210048
5980:. ACM. pp. 214–227.
5286:Given a type environment
5243:can be assigned the type
5140:can be assigned the type
5060:Given a type environment
4621:, and closely related to
4617:, based on the theory of
4363:{\displaystyle 3\times 3}
4113:{\displaystyle N:\sigma }
3959:{\displaystyle N:\sigma }
3762:{\displaystyle M:\sigma }
3602:{\displaystyle N:\sigma }
3395:{\displaystyle N:\sigma }
2766:{\displaystyle M:\sigma }
2645:{\displaystyle M:\sigma }
2482:{\displaystyle N:\sigma }
2319:{\displaystyle N:\sigma }
2151:{\displaystyle N:\sigma }
1824:{\displaystyle M:\sigma }
1672:Additional tools such as
1661:
1366:or other mechanisms like
1322:, and to a lesser extent
987:Static type checking for
550:microprocessor operations
5870:"dynamic (C# Reference)"
5850:(PhD). Rice University.
5818:Miglani, Gaurav (2018).
5575:Burroughs ALGOL computer
5366:in the type environment.
5263:in the type environment
5160:in the type environment.
4635:referential transparency
4588:Other languages such as
1757:Specialized type systems
1498:
1478:
903:, collectively known as
630:general purpose computer
27:Computer science concept
7043:Parametric polymorphism
5797:(4 ed.). Manning.
5493:, then a value of type
5471:nominative type systems
5465:structural type systems
5299:{\displaystyle \Gamma }
5276:{\displaystyle \Gamma }
5196:{\displaystyle \Gamma }
5073:{\displaystyle \Gamma }
4344:might be the type of a
4087:{\displaystyle M:\tau }
3933:{\displaystyle M:\tau }
3803:{\displaystyle M:\tau }
3643:{\displaystyle O:\tau }
3576:{\displaystyle M=(N,O)}
3276:{\displaystyle M:\tau }
3147:{\displaystyle M:\tau }
3023:{\displaystyle M:\tau }
2903:{\displaystyle x:\tau }
2792:{\displaystyle M:\tau }
2671:{\displaystyle M:\tau }
2550:{\displaystyle N:\tau }
2345:{\displaystyle O:\tau }
2293:{\displaystyle M=(N,O)}
1902:is that application of
1864:{\displaystyle \sigma }
1763:static program analysis
1647:declare(strict_types=1)
1414:need not be related to
1013:Even if the expression
935:(a static check) or at
71:more precise citations.
6154:. stackoverflow. 2012.
5360:
5340:
5320:
5300:
5277:
5257:
5237:
5217:
5197:
5177:
5154:
5134:
5114:
5094:
5074:
5013:, is a restriction of
4990:might imply a type of
4886:In a type system with
4784:Existential quantifier
4548:
4364:
4338:
4265:
4192:
4136:dependent product type
4114:
4088:
4047:
4003:
3960:
3934:
3893:
3849:
3804:
3763:
3737:
3691:
3656:Dependent intersection
3644:
3603:
3577:
3539:
3493:
3446:
3396:
3370:
3324:
3277:
3227:
3190:
3148:
3107:
3068:
3024:
2983:
2944:
2904:
2874:
2834:
2793:
2767:
2741:
2707:
2672:
2646:
2620:
2586:
2551:
2525:
2483:
2457:
2415:
2381:
2346:
2320:
2294:
2256:
2222:
2187:
2152:
2126:
2092:
2036:
2008:
1973:
1936:
1916:
1896:
1865:
1845:
1825:
1796:
1716:Polymorphism and types
1428:Strong and weak typing
1096:reflective programming
755:type and effect system
683:communication channels
602:
455:compiler optimizations
180:Strong vs. weak typing
125:by rewriting it in an
6530:Smith, Chris (2011).
6486:Advances in Computers
6391:ACM Computing Surveys
6278:University of Chicago
5996:10.1145/292540.292560
5847:Practical Soft Typing
5361:
5359:{\displaystyle \tau }
5341:
5321:
5319:{\displaystyle \tau }
5301:
5278:
5258:
5256:{\displaystyle \tau }
5238:
5218:
5216:{\displaystyle \tau }
5198:
5178:
5155:
5153:{\displaystyle \tau }
5135:
5115:
5113:{\displaystyle \tau }
5095:
5075:
4914:Further information:
4866:typed lambda calculus
4811:and a function named
4583:Presburger arithmetic
4549:
4365:
4339:
4266:
4193:
4115:
4089:
4048:
4004:
3961:
3935:
3894:
3850:
3816:Familial intersection
3805:
3764:
3738:
3692:
3645:
3604:
3578:
3540:
3494:
3447:
3397:
3371:
3325:
3278:
3228:
3191:
3149:
3108:
3069:
3025:
2984:
2945:
2905:
2875:
2835:
2794:
2768:
2742:
2708:
2673:
2647:
2621:
2587:
2552:
2526:
2484:
2458:
2416:
2382:
2347:
2321:
2295:
2257:
2223:
2188:
2153:
2127:
2093:
2037:
2035:{\displaystyle \tau }
2018:(resp. term variable
2009:
2007:{\displaystyle \tau }
1974:
1972:{\displaystyle \tau }
1937:
1917:
1897:
1866:
1846:
1826:
1797:
1684:Optional type systems
699:(a type of a value),
650:floating-point number
573:computer architecture
544:, in accord with the
402:programming languages
6526:at Wikimedia Commons
6358:Perl 6 Documentation
6340:Perl 6 Documentation
5958:Xi, Hongwei (1998).
5413:assignment statement
5375:Some languages like
5350:
5330:
5310:
5290:
5267:
5247:
5227:
5207:
5187:
5167:
5144:
5124:
5104:
5084:
5064:
4671:functions expecting
4377:
4348:
4296:
4215:
4207:Also referred to as
4142:
4134:Also referred to as
4098:
4057:
4015:
3977:
3944:
3903:
3861:
3823:
3773:
3747:
3703:
3663:
3613:
3587:
3549:
3505:
3465:
3406:
3380:
3336:
3296:
3237:
3202:
3171:
3117:
3080:
3047:
2993:
2956:
2923:
2888:
2846:
2812:
2777:
2751:
2719:
2691:
2656:
2630:
2598:
2570:
2535:
2493:
2467:
2425:
2393:
2365:
2330:
2304:
2266:
2234:
2206:
2162:
2136:
2104:
2076:
2026:
1983:
1948:
1926:
1906:
1895:{\displaystyle M(N)}
1877:
1855:
1835:
1809:
1780:
1749:and (in some cases)
1463:memory-safe language
1345:algebraic data types
1199:is also soft-typed.
1163:subtype polymorphism
1151:programming language
1143:runtime polymorphism
1074:Interpreted language
1017:always evaluates to
1015:<complex test>
978:program verification
961:Static type checking
561:programming language
512:floating-point value
387:algebraic data types
324:computer programming
7048:Primitive data type
6953:Recursive data type
6806:Algebraic data type
6682:Quadruple precision
6430:Pierce, Benjamin C.
6224:10.1145/44501.45065
5793:Skeet, Jon (2019).
5757:on 14 November 2017
5507:ad hoc polymorphism
5460:equational theories
5371:Unified type system
5223:such that the term
4801:abstract data types
1738:generic programming
1678:IBM Rational Purify
1604:undefined behaviour
1368:generic programming
939:(a dynamic check).
877:undecidable problem
873:program correctness
612:such as a value in
559:of the language. A
446:), dynamically (at
360:language constructs
7011:Abstract data type
6692:Extended precision
6651:Reduced precision
5658:Sethi, R. (1996).
5481:In languages with
5356:
5336:
5316:
5296:
5273:
5253:
5233:
5213:
5193:
5173:
5150:
5130:
5110:
5090:
5070:
4766:type, rather than
4664:Intersection types
4653:Intersection types
4544:
4360:
4334:
4261:
4257:
4209:dependent sum type
4188:
4184:
4110:
4084:
4043:
4039:
3999:
3995:
3956:
3930:
3889:
3885:
3845:
3841:
3800:
3759:
3733:
3687:
3640:
3599:
3573:
3535:
3489:
3442:
3392:
3366:
3320:
3289:Dependent function
3273:
3223:
3186:
3144:
3103:
3064:
3020:
2979:
2940:
2900:
2870:
2830:
2789:
2763:
2737:
2703:
2668:
2642:
2616:
2582:
2547:
2521:
2479:
2453:
2411:
2377:
2342:
2316:
2290:
2252:
2218:
2183:
2148:
2122:
2088:
2032:
2004:
1969:
1932:
1912:
1892:
1861:
1841:
1821:
1792:
1471:check array bounds
1456:type-safe language
830:3 / "Hello, World"
127:encyclopedic style
114:is written like a
7091:
7090:
6823:Associative array
6687:Octuple precision
6522:Media related to
6503:978-0-12-374812-6
6443:978-0-262-16209-8
6414:10.1145/6041.6042
6381:(December 1985).
6304:978-0-521-76614-2
6280:. pp. 81–92.
6132:"Pluggable Types"
5902:doc.rust-lang.org
5898:"std::any — Rust"
5706:978-0-19-154627-3
5669:978-0-201-59065-4
5589:leaky abstraction
5339:{\displaystyle e}
5236:{\displaystyle e}
5176:{\displaystyle e}
5133:{\displaystyle e}
5093:{\displaystyle e}
5054:type inhabitation
5042:decision problems
5026:Decision problems
4922:type declarations
4778:Existential types
4726:" functionality.
4659:Intersection type
4242:
4169:
4124:
4123:
4024:
3980:
3870:
3826:
1935:{\displaystyle N}
1915:{\displaystyle M}
1844:{\displaystyle M}
1733:associative array
1360:easier code reuse
948:strongly typed; i
459:multiple dispatch
320:
319:
155:
154:
147:
97:
96:
89:
16:(Redirected from
7126:
7114:Program analysis
7063:Type constructor
6948:Opaque data type
6880:Record or Struct
6677:Double precision
6672:Single precision
6563:
6556:
6549:
6540:
6535:
6521:
6507:
6476:
6460:
6447:
6425:
6407:
6387:
6362:
6361:
6350:
6344:
6343:
6332:
6326:
6325:
6315:
6309:
6308:
6288:
6282:
6281:
6271:
6260:
6254:
6253:
6245:
6236:
6235:
6209:
6200:
6194:
6193:
6183:
6167:
6156:
6155:
6145:
6139:
6138:
6136:
6127:
6118:
6117:
6106:
6100:
6099:
6097:
6096:
6081:
6075:
6074:
6072:
6071:
6056:
6050:
6049:
6036:
6030:
6024:
6018:
6017:
5989:
5973:
5971:
5955:
5949:
5948:
5945:"Types vs Tests"
5940:
5934:
5933:
5927:
5918:
5912:
5911:
5909:
5908:
5894:
5888:
5887:
5885:
5883:
5866:
5860:
5859:
5841:
5835:
5834:
5832:
5831:
5815:
5809:
5808:
5790:
5779:
5773:
5767:
5766:
5764:
5762:
5756:
5750:. Archived from
5749:
5739:
5733:
5732:
5726:
5717:
5711:
5710:
5688:
5682:
5681:
5655:
5646:
5640:
5634:
5628:
5622:
5616:
5600:
5585:
5579:
5571:
5529:
5524:
5523:
5500:
5496:
5492:
5489:is a subtype of
5488:
5448:
5445:and the type of
5443:
5431:
5426:
5421:
5365:
5363:
5362:
5357:
5345:
5343:
5342:
5337:
5325:
5323:
5322:
5317:
5305:
5303:
5302:
5297:
5282:
5280:
5279:
5274:
5262:
5260:
5259:
5254:
5242:
5240:
5239:
5234:
5222:
5220:
5219:
5214:
5202:
5200:
5199:
5194:
5182:
5180:
5179:
5174:
5159:
5157:
5156:
5151:
5139:
5137:
5136:
5131:
5119:
5117:
5116:
5111:
5099:
5097:
5096:
5091:
5079:
5077:
5076:
5071:
4996:
4989:
4988:
4980:
4979:
4974:
4973:
4968:
4967:
4962:
4961:
4956:
4955:
4950:
4949:
4946:
4943:
4940:
4937:
4934:
4858:John C. Mitchell
4734:refinement types
4725:
4724:
4719:
4718:
4713:
4712:
4707:
4706:
4701:
4700:
4695:
4694:
4689:
4688:
4683:
4682:
4632:
4627:immutable values
4623:uniqueness types
4568:
4564:
4560:
4553:
4551:
4550:
4545:
4528:
4490:
4452:
4429:
4428:
4427:
4400:
4369:
4367:
4366:
4361:
4343:
4341:
4340:
4335:
4318:
4272:
4270:
4268:
4267:
4262:
4256:
4205:
4199:
4197:
4195:
4194:
4189:
4183:
4132:
4119:
4117:
4116:
4111:
4093:
4091:
4090:
4085:
4052:
4050:
4049:
4044:
4038:
4008:
4006:
4005:
4000:
3994:
3965:
3963:
3962:
3957:
3939:
3937:
3936:
3931:
3898:
3896:
3895:
3890:
3884:
3854:
3852:
3851:
3846:
3840:
3809:
3807:
3806:
3801:
3768:
3766:
3765:
3760:
3742:
3740:
3739:
3734:
3696:
3694:
3693:
3688:
3649:
3647:
3646:
3641:
3608:
3606:
3605:
3600:
3582:
3580:
3579:
3574:
3544:
3542:
3541:
3536:
3498:
3496:
3495:
3490:
3451:
3449:
3448:
3443:
3401:
3399:
3398:
3393:
3375:
3373:
3372:
3367:
3329:
3327:
3326:
3321:
3282:
3280:
3279:
3274:
3232:
3230:
3229:
3224:
3195:
3193:
3192:
3187:
3157:
3153:
3151:
3150:
3145:
3112:
3110:
3109:
3104:
3093:
3073:
3071:
3070:
3065:
3054:
3033:
3029:
3027:
3026:
3021:
2988:
2986:
2985:
2980:
2969:
2949:
2947:
2946:
2941:
2930:
2909:
2907:
2906:
2901:
2883:
2879:
2877:
2876:
2871:
2839:
2837:
2836:
2831:
2798:
2796:
2795:
2790:
2772:
2770:
2769:
2764:
2746:
2744:
2743:
2738:
2712:
2710:
2709:
2704:
2677:
2675:
2674:
2669:
2651:
2649:
2648:
2643:
2625:
2623:
2622:
2617:
2591:
2589:
2588:
2583:
2556:
2554:
2553:
2548:
2530:
2528:
2527:
2522:
2511:
2510:
2488:
2486:
2485:
2480:
2462:
2460:
2459:
2454:
2443:
2442:
2420:
2418:
2417:
2412:
2386:
2384:
2383:
2378:
2351:
2349:
2348:
2343:
2325:
2323:
2322:
2317:
2299:
2297:
2296:
2291:
2261:
2259:
2258:
2253:
2227:
2225:
2224:
2219:
2192:
2190:
2189:
2184:
2157:
2155:
2154:
2149:
2131:
2129:
2128:
2123:
2097:
2095:
2094:
2089:
2054:
2049:
2045:
2041:
2039:
2038:
2033:
2021:
2017:
2013:
2011:
2010:
2005:
1978:
1976:
1975:
1970:
1941:
1939:
1938:
1933:
1921:
1919:
1918:
1913:
1901:
1899:
1898:
1893:
1870:
1868:
1867:
1862:
1850:
1848:
1847:
1842:
1830:
1828:
1827:
1822:
1801:
1799:
1798:
1793:
1775:
1663:
1662:Option Strict On
1656:
1648:
1633:
1601:
1600:
1595:
1594:
1589:
1588:
1584:In this example
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1491:
1487:
1395:boilerplate code
1257:dependent typing
1238:
1237:
1234:
1229:
1228:
1224:
1220:
1217:
1205:
1084:dynamic dispatch
1048:
1024:
1020:
1016:
1009:
944:type conversions
913:software testing
911:). In addition,
905:program analyses
889:Division by zero
831:
790:interoperability
638:instruction code
501:computer program
364:computer program
312:
305:
298:
231:Minor categories
188:Major categories
167:General concepts
157:
150:
143:
139:
136:
130:
107:
106:
99:
92:
85:
81:
78:
72:
67:this article by
58:inline citations
45:
44:
37:
21:
18:Existential type
7134:
7133:
7129:
7128:
7127:
7125:
7124:
7123:
7094:
7093:
7092:
7087:
7068:Type conversion
7003:
6997:
6933:Enumerated type
6906:
6792:
6786:null-terminated
6762:
6727:
6615:
6572:
6567:
6529:
6514:
6504:
6479:
6473:
6458:
6450:
6444:
6428:
6385:
6373:
6370:
6368:Further reading
6365:
6352:
6351:
6347:
6334:
6333:
6329:
6317:
6316:
6312:
6305:
6290:
6289:
6285:
6269:
6262:
6261:
6257:
6247:
6246:
6239:
6207:
6202:
6201:
6197:
6169:
6168:
6159:
6147:
6146:
6142:
6134:
6129:
6128:
6121:
6110:"Strict typing"
6108:
6107:
6103:
6094:
6092:
6083:
6082:
6078:
6069:
6067:
6058:
6057:
6053:
6038:
6037:
6033:
6025:
6021:
6006:
5975:
5974:
5957:
5956:
5952:
5942:
5941:
5937:
5925:
5920:
5919:
5915:
5906:
5904:
5896:
5895:
5891:
5881:
5879:
5868:
5867:
5863:
5843:
5842:
5838:
5829:
5827:
5817:
5816:
5812:
5805:
5792:
5791:
5782:
5774:
5770:
5760:
5758:
5754:
5747:
5742:
5740:
5736:
5724:
5719:
5718:
5714:
5707:
5690:
5689:
5685:
5670:
5657:
5656:
5649:
5641:
5637:
5629:
5625:
5617:
5613:
5609:
5604:
5603:
5597:lambda calculus
5587:For example, a
5586:
5582:
5572:
5568:
5563:
5525:
5518:
5515:
5498:
5494:
5490:
5486:
5446:
5441:
5439:If the type of
5429:
5424:
5416:
5405:
5373:
5348:
5347:
5328:
5327:
5308:
5307:
5288:
5287:
5265:
5264:
5245:
5244:
5225:
5224:
5205:
5204:
5185:
5184:
5165:
5164:
5142:
5141:
5122:
5121:
5102:
5101:
5082:
5081:
5062:
5061:
5034:
5028:
4986:
4977:
4971:
4965:
4959:
4953:
4947:
4944:
4941:
4938:
4935:
4932:
4918:
4912:
4884:
4878:
4786:
4780:
4748:
4742:
4722:
4716:
4710:
4704:
4698:
4692:
4686:
4680:
4661:
4655:
4631:str = str + "a"
4630:
4612:
4606:
4566:
4562:
4558:
4380:
4375:
4374:
4346:
4345:
4294:
4293:
4290:Dependent types
4287:
4281:
4279:Dependent types
4276:
4275:
4213:
4212:
4206:
4202:
4140:
4139:
4133:
4129:
4096:
4095:
4055:
4054:
4013:
4012:
3975:
3974:
3971:Familial union
3942:
3941:
3901:
3900:
3859:
3858:
3821:
3820:
3771:
3770:
3745:
3744:
3701:
3700:
3661:
3660:
3611:
3610:
3585:
3584:
3583:is a pair s.t.
3547:
3546:
3503:
3502:
3463:
3462:
3404:
3403:
3378:
3377:
3334:
3333:
3294:
3293:
3235:
3234:
3200:
3199:
3169:
3168:
3155:
3115:
3114:
3078:
3077:
3045:
3044:
3031:
2991:
2990:
2954:
2953:
2921:
2920:
2886:
2885:
2881:
2844:
2843:
2810:
2809:
2775:
2774:
2749:
2748:
2717:
2716:
2689:
2688:
2654:
2653:
2628:
2627:
2596:
2595:
2568:
2567:
2533:
2532:
2502:
2491:
2490:
2465:
2464:
2434:
2423:
2422:
2391:
2390:
2363:
2362:
2328:
2327:
2302:
2301:
2300:is a pair s.t.
2264:
2263:
2232:
2231:
2204:
2203:
2160:
2159:
2134:
2133:
2102:
2101:
2074:
2073:
2047:
2043:
2024:
2023:
2019:
2015:
1981:
1980:
1946:
1945:
1924:
1923:
1904:
1903:
1875:
1874:
1853:
1852:
1833:
1832:
1807:
1806:
1778:
1777:
1773:
1759:
1724:
1718:
1686:
1657:will be thrown.
1654:
1646:
1631:
1624:
1598:
1592:
1586:
1582:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1542:
1539:
1536:
1533:
1530:
1527:
1524:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1482:
1481:
1452:
1446:
1430:
1424:
1375:metaprogramming
1358:(which enables
1245:
1235:
1232:
1226:
1222:
1218:
1215:
1203:
1139:disjoint unions
1129:types to their
1123:
1076:
1066:
1046:
1045:introduces the
1022:
1018:
1014:
1007:
989:Turing-complete
963:
925:
881:Halting problem
853:
829:
784:between two of
663:symbolic system
640:, or between a
585:
577:language design
471:
457:, allowing for
404:and built into
391:data structures
316:
151:
140:
134:
131:
123:help improve it
120:
108:
104:
93:
82:
76:
73:
63:Please help to
62:
46:
42:
35:
28:
23:
22:
15:
12:
11:
5:
7132:
7130:
7122:
7121:
7116:
7111:
7106:
7096:
7095:
7089:
7088:
7086:
7085:
7080:
7075:
7070:
7065:
7060:
7055:
7050:
7045:
7040:
7039:
7038:
7028:
7023:
7021:Data structure
7018:
7013:
7007:
7005:
6999:
6998:
6996:
6995:
6990:
6985:
6980:
6975:
6970:
6965:
6960:
6955:
6950:
6945:
6940:
6935:
6930:
6925:
6920:
6914:
6912:
6908:
6907:
6905:
6904:
6903:
6902:
6892:
6887:
6882:
6877:
6872:
6867:
6866:
6865:
6855:
6850:
6845:
6840:
6835:
6830:
6825:
6820:
6815:
6814:
6813:
6802:
6800:
6794:
6793:
6791:
6790:
6789:
6788:
6778:
6772:
6770:
6764:
6763:
6761:
6760:
6755:
6754:
6753:
6748:
6737:
6735:
6729:
6728:
6726:
6725:
6720:
6715:
6714:
6713:
6703:
6702:
6701:
6700:
6699:
6689:
6684:
6679:
6674:
6669:
6668:
6667:
6662:
6660:Half precision
6657:
6647:Floating point
6644:
6639:
6634:
6629:
6623:
6621:
6617:
6616:
6614:
6613:
6608:
6603:
6598:
6593:
6588:
6582:
6580:
6574:
6573:
6568:
6566:
6565:
6558:
6551:
6543:
6537:
6536:
6527:
6513:
6512:External links
6510:
6509:
6508:
6502:
6477:
6472:978-1584883609
6471:
6456:"Type systems"
6452:Cardelli, Luca
6448:
6442:
6426:
6405:10.1.1.117.695
6398:(4): 471–523.
6375:Cardelli, Luca
6369:
6366:
6364:
6363:
6345:
6336:"Native Types"
6327:
6310:
6303:
6283:
6255:
6237:
6218:(3): 470–502.
6195:
6181:10.1.1.89.4223
6157:
6140:
6119:
6101:
6076:
6051:
6031:
6019:
6004:
5987:10.1.1.69.2042
5950:
5935:
5913:
5889:
5861:
5836:
5810:
5804:978-1617294532
5803:
5780:
5768:
5743:Remy, Didier.
5734:
5712:
5705:
5683:
5668:
5647:
5645:, p. 208.
5635:
5623:
5610:
5608:
5605:
5602:
5601:
5580:
5565:
5564:
5562:
5559:
5558:
5557:
5552:
5550:Type signature
5547:
5542:
5537:
5531:
5530:
5514:
5511:
5404:
5401:
5372:
5369:
5368:
5367:
5355:
5335:
5315:
5295:
5284:
5272:
5252:
5232:
5212:
5192:
5172:
5161:
5149:
5129:
5109:
5089:
5069:
5030:Main article:
5027:
5024:
5011:Hindley–Milner
4992:floating-point
4927:type inference
4916:Type inference
4911:
4908:
4888:Gradual typing
4882:Gradual typing
4880:Main article:
4877:
4876:Gradual typing
4874:
4862:Gordon Plotkin
4854:
4853:
4832:
4831:
4828:
4782:Main article:
4779:
4776:
4744:Main article:
4741:
4738:
4657:Main article:
4654:
4651:
4608:Main article:
4605:
4602:
4555:
4554:
4543:
4540:
4537:
4534:
4531:
4527:
4524:
4521:
4518:
4515:
4512:
4508:
4505:
4502:
4499:
4496:
4493:
4489:
4486:
4483:
4480:
4477:
4474:
4470:
4467:
4464:
4461:
4458:
4455:
4451:
4448:
4445:
4442:
4439:
4436:
4432:
4426:
4423:
4420:
4417:
4414:
4411:
4408:
4405:
4399:
4396:
4393:
4390:
4387:
4384:
4359:
4356:
4353:
4333:
4330:
4327:
4324:
4321:
4317:
4314:
4311:
4308:
4305:
4302:
4285:Dependent type
4283:Main article:
4280:
4277:
4274:
4273:
4260:
4255:
4252:
4249:
4245:
4241:
4238:
4235:
4232:
4229:
4226:
4223:
4220:
4200:
4187:
4182:
4179:
4176:
4172:
4168:
4165:
4162:
4159:
4156:
4153:
4150:
4147:
4126:
4125:
4122:
4121:
4109:
4106:
4103:
4094:for some term
4083:
4080:
4077:
4074:
4071:
4068:
4065:
4062:
4042:
4037:
4034:
4031:
4027:
4023:
4020:
4009:
3998:
3993:
3990:
3987:
3983:
3972:
3968:
3967:
3955:
3952:
3949:
3929:
3926:
3923:
3920:
3917:
3914:
3911:
3908:
3888:
3883:
3880:
3877:
3873:
3869:
3866:
3855:
3844:
3839:
3836:
3833:
3829:
3818:
3812:
3811:
3799:
3796:
3793:
3790:
3787:
3784:
3781:
3778:
3758:
3755:
3752:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3697:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3658:
3652:
3651:
3639:
3636:
3633:
3630:
3627:
3624:
3621:
3618:
3598:
3595:
3592:
3572:
3569:
3566:
3563:
3560:
3557:
3554:
3534:
3531:
3528:
3525:
3522:
3519:
3516:
3513:
3510:
3499:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3460:
3458:Dependent pair
3454:
3453:
3441:
3438:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3391:
3388:
3385:
3365:
3362:
3359:
3356:
3353:
3350:
3347:
3344:
3341:
3330:
3319:
3316:
3313:
3310:
3307:
3304:
3301:
3291:
3285:
3284:
3272:
3269:
3266:
3263:
3260:
3257:
3254:
3251:
3248:
3245:
3242:
3222:
3219:
3216:
3213:
3210:
3207:
3196:
3185:
3182:
3179:
3176:
3166:
3160:
3159:
3154:for some type
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3122:
3102:
3099:
3096:
3091:
3088:
3085:
3074:
3063:
3060:
3057:
3052:
3042:
3036:
3035:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2998:
2978:
2975:
2972:
2967:
2964:
2961:
2950:
2939:
2936:
2933:
2928:
2918:
2912:
2911:
2899:
2896:
2893:
2869:
2866:
2863:
2860:
2857:
2854:
2851:
2840:
2829:
2826:
2823:
2820:
2817:
2807:
2801:
2800:
2788:
2785:
2782:
2762:
2759:
2756:
2736:
2733:
2730:
2727:
2724:
2713:
2702:
2699:
2696:
2686:
2680:
2679:
2667:
2664:
2661:
2641:
2638:
2635:
2615:
2612:
2609:
2606:
2603:
2592:
2581:
2578:
2575:
2565:
2559:
2558:
2546:
2543:
2540:
2520:
2517:
2514:
2509:
2505:
2501:
2498:
2478:
2475:
2472:
2452:
2449:
2446:
2441:
2437:
2433:
2430:
2410:
2407:
2404:
2401:
2398:
2387:
2376:
2373:
2370:
2360:
2354:
2353:
2341:
2338:
2335:
2315:
2312:
2309:
2289:
2286:
2283:
2280:
2277:
2274:
2271:
2251:
2248:
2245:
2242:
2239:
2228:
2217:
2214:
2211:
2201:
2195:
2194:
2182:
2179:
2176:
2173:
2170:
2167:
2147:
2144:
2141:
2121:
2118:
2115:
2112:
2109:
2098:
2087:
2084:
2081:
2071:
2065:
2064:
2061:
2058:
2052:
2051:
2031:
2003:
2000:
1997:
1994:
1991:
1988:
1968:
1965:
1962:
1959:
1956:
1953:
1943:
1931:
1911:
1891:
1888:
1885:
1882:
1872:
1860:
1840:
1820:
1817:
1814:
1791:
1788:
1785:
1758:
1755:
1720:Main article:
1717:
1714:
1710:dynamic typing
1702:gradual typing
1685:
1682:
1670:
1669:
1658:
1643:
1623:
1620:
1525:"37"
1499:
1479:
1448:Main article:
1445:
1442:
1434:strongly typed
1426:Main article:
1423:
1420:
1412:dynamic typing
1288:type inference
1244:
1241:
1187:gradual typing
1122:
1119:
1065:
1062:
1011:
1010:
962:
959:
931:—may occur at
924:
921:
901:formal methods
864:is automated,
862:type inference
852:
849:
848:
847:
819:
805:
804:
793:
749:is being done
736:dependent type
713:type of a type
634:memory address
584:
581:
533:compilers the
470:
469:Usage overview
467:
414:optional tools
346:floating point
340:(for example,
332:logical system
318:
317:
315:
314:
307:
300:
292:
289:
288:
287:
286:
281:
276:
271:
266:
261:
256:
251:
249:Flow-sensitive
246:
241:
233:
232:
228:
227:
226:
225:
220:
211:
202:
190:
189:
185:
184:
183:
182:
177:
169:
168:
164:
163:
153:
152:
111:
109:
102:
95:
94:
49:
47:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7131:
7120:
7117:
7115:
7112:
7110:
7107:
7105:
7102:
7101:
7099:
7084:
7081:
7079:
7076:
7074:
7071:
7069:
7066:
7064:
7061:
7059:
7056:
7054:
7051:
7049:
7046:
7044:
7041:
7037:
7034:
7033:
7032:
7029:
7027:
7024:
7022:
7019:
7017:
7014:
7012:
7009:
7008:
7006:
7000:
6994:
6991:
6989:
6986:
6984:
6981:
6979:
6976:
6974:
6971:
6969:
6966:
6964:
6961:
6959:
6956:
6954:
6951:
6949:
6946:
6944:
6943:Function type
6941:
6939:
6936:
6934:
6931:
6929:
6926:
6924:
6921:
6919:
6916:
6915:
6913:
6909:
6901:
6898:
6897:
6896:
6893:
6891:
6888:
6886:
6883:
6881:
6878:
6876:
6873:
6871:
6868:
6864:
6861:
6860:
6859:
6856:
6854:
6851:
6849:
6846:
6844:
6841:
6839:
6836:
6834:
6831:
6829:
6826:
6824:
6821:
6819:
6816:
6812:
6809:
6808:
6807:
6804:
6803:
6801:
6799:
6795:
6787:
6784:
6783:
6782:
6779:
6777:
6774:
6773:
6771:
6769:
6765:
6759:
6756:
6752:
6749:
6747:
6744:
6743:
6742:
6739:
6738:
6736:
6734:
6730:
6724:
6721:
6719:
6716:
6712:
6709:
6708:
6707:
6704:
6698:
6695:
6694:
6693:
6690:
6688:
6685:
6683:
6680:
6678:
6675:
6673:
6670:
6666:
6663:
6661:
6658:
6656:
6653:
6652:
6650:
6649:
6648:
6645:
6643:
6640:
6638:
6635:
6633:
6630:
6628:
6625:
6624:
6622:
6618:
6612:
6609:
6607:
6604:
6602:
6599:
6597:
6594:
6592:
6589:
6587:
6584:
6583:
6581:
6579:
6578:Uninterpreted
6575:
6571:
6564:
6559:
6557:
6552:
6550:
6545:
6544:
6541:
6533:
6528:
6525:
6520:
6516:
6515:
6511:
6505:
6499:
6495:
6491:
6487:
6483:
6478:
6474:
6468:
6464:
6457:
6453:
6449:
6445:
6439:
6436:. MIT Press.
6435:
6431:
6427:
6423:
6419:
6415:
6411:
6406:
6401:
6397:
6393:
6392:
6384:
6380:
6379:Wegner, Peter
6376:
6372:
6371:
6367:
6359:
6355:
6349:
6346:
6341:
6337:
6331:
6328:
6323:
6322:
6314:
6311:
6306:
6300:
6296:
6295:
6287:
6284:
6279:
6275:
6268:
6267:
6259:
6256:
6251:
6244:
6242:
6238:
6233:
6229:
6225:
6221:
6217:
6213:
6206:
6199:
6196:
6191:
6187:
6182:
6177:
6173:
6166:
6164:
6162:
6158:
6153:
6150:
6144:
6141:
6133:
6126:
6124:
6120:
6115:
6111:
6105:
6102:
6090:
6086:
6080:
6077:
6065:
6061:
6055:
6052:
6047:
6046:
6041:
6035:
6032:
6028:
6023:
6020:
6015:
6011:
6007:
6001:
5997:
5993:
5988:
5983:
5979:
5970:
5969:10.1.1.41.548
5965:
5961:
5954:
5951:
5946:
5939:
5936:
5931:
5924:
5917:
5914:
5903:
5899:
5893:
5890:
5877:
5876:
5871:
5865:
5862:
5857:
5853:
5849:
5848:
5840:
5837:
5825:
5821:
5814:
5811:
5806:
5800:
5796:
5789:
5787:
5785:
5781:
5777:
5772:
5769:
5753:
5746:
5738:
5735:
5730:
5723:
5716:
5713:
5708:
5702:
5698:
5694:
5687:
5684:
5679:
5675:
5671:
5665:
5661:
5654:
5652:
5648:
5644:
5639:
5636:
5632:
5631:Cardelli 2004
5627:
5624:
5620:
5615:
5612:
5606:
5598:
5594:
5590:
5584:
5581:
5576:
5570:
5567:
5560:
5556:
5553:
5551:
5548:
5546:
5543:
5541:
5538:
5536:
5533:
5532:
5528:
5522:
5517:
5512:
5510:
5508:
5504:
5501:is expected (
5484:
5479:
5477:
5473:
5472:
5467:
5466:
5461:
5457:
5453:
5444:
5437:
5435:
5434:compatibility
5427:
5420:
5414:
5410:
5402:
5400:
5398:
5394:
5390:
5386:
5382:
5378:
5370:
5353:
5333:
5313:
5285:
5250:
5230:
5210:
5170:
5163:Given a term
5162:
5147:
5127:
5107:
5100:, and a type
5087:
5059:
5058:
5057:
5055:
5051:
5047:
5046:type checking
5043:
5039:
5033:
5025:
5023:
5021:
5016:
5012:
5007:
5002:
5000:
4993:
4982:
4929:
4928:
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4917:
4909:
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4905:
4901:
4897:
4893:
4889:
4883:
4875:
4873:
4871:
4867:
4863:
4859:
4851:
4847:
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4841:
4838:
4829:
4826:
4825:
4824:
4822:
4818:
4814:
4810:
4806:
4802:
4798:
4795:to represent
4794:
4790:
4785:
4777:
4775:
4771:
4769:
4765:
4761:
4756:
4752:
4747:
4739:
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4727:
4676:
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4660:
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4648:
4644:
4640:
4636:
4628:
4624:
4620:
4616:
4611:
4603:
4601:
4599:
4598:infinite loop
4595:
4591:
4586:
4584:
4580:
4576:
4572:
4538:
4535:
4532:
4500:
4497:
4494:
4468:
4462:
4459:
4456:
4430:
4373:
4372:
4371:
4357:
4354:
4351:
4328:
4325:
4322:
4291:
4286:
4278:
4258:
4253:
4250:
4247:
4243:
4239:
4236:
4233:
4227:
4224:
4221:
4210:
4204:
4201:
4185:
4180:
4177:
4174:
4170:
4166:
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4154:
4151:
4148:
4137:
4131:
4128:
4107:
4104:
4101:
4078:
4075:
4072:
4066:
4063:
4060:
4040:
4035:
4032:
4029:
4025:
4021:
4018:
4010:
3996:
3991:
3988:
3985:
3981:
3973:
3970:
3969:
3953:
3950:
3947:
3940:for any term
3924:
3921:
3918:
3912:
3909:
3906:
3886:
3881:
3878:
3875:
3871:
3867:
3864:
3856:
3842:
3837:
3834:
3831:
3827:
3819:
3817:
3814:
3813:
3794:
3791:
3788:
3782:
3779:
3776:
3756:
3753:
3750:
3730:
3727:
3721:
3718:
3715:
3709:
3706:
3698:
3684:
3681:
3675:
3672:
3669:
3659:
3657:
3654:
3653:
3634:
3631:
3628:
3622:
3619:
3616:
3596:
3593:
3590:
3567:
3564:
3561:
3555:
3552:
3532:
3529:
3523:
3520:
3517:
3511:
3508:
3500:
3486:
3483:
3477:
3474:
3471:
3461:
3459:
3456:
3455:
3436:
3433:
3430:
3424:
3421:
3415:
3409:
3389:
3386:
3383:
3363:
3354:
3351:
3348:
3342:
3339:
3331:
3317:
3308:
3305:
3302:
3292:
3290:
3287:
3286:
3267:
3264:
3261:
3258:
3255:
3252:
3246:
3243:
3240:
3220:
3217:
3214:
3211:
3208:
3205:
3197:
3183:
3180:
3177:
3174:
3167:
3165:
3162:
3161:
3138:
3135:
3132:
3126:
3123:
3120:
3100:
3097:
3094:
3086:
3083:
3075:
3061:
3058:
3055:
3043:
3041:
3038:
3037:
3030:for any type
3014:
3011:
3008:
3002:
2999:
2996:
2976:
2973:
2970:
2962:
2959:
2951:
2937:
2934:
2931:
2919:
2917:
2914:
2913:
2897:
2894:
2891:
2884:has a member
2864:
2861:
2858:
2852:
2849:
2841:
2824:
2821:
2818:
2808:
2806:
2803:
2802:
2786:
2783:
2780:
2760:
2757:
2754:
2734:
2731:
2728:
2725:
2722:
2714:
2700:
2697:
2694:
2687:
2685:
2682:
2681:
2665:
2662:
2659:
2639:
2636:
2633:
2613:
2610:
2607:
2604:
2601:
2593:
2579:
2576:
2573:
2566:
2564:
2561:
2560:
2544:
2541:
2538:
2515:
2507:
2503:
2499:
2496:
2476:
2473:
2470:
2447:
2439:
2435:
2431:
2428:
2408:
2405:
2402:
2399:
2396:
2388:
2374:
2371:
2368:
2361:
2359:
2356:
2355:
2339:
2336:
2333:
2313:
2310:
2307:
2284:
2281:
2278:
2272:
2269:
2249:
2246:
2243:
2240:
2237:
2229:
2215:
2212:
2209:
2202:
2200:
2197:
2196:
2180:
2177:
2171:
2165:
2145:
2142:
2139:
2119:
2113:
2110:
2107:
2099:
2085:
2079:
2072:
2070:
2067:
2066:
2062:
2059:
2056:
2055:
2029:
1998:
1995:
1992:
1986:
1963:
1960:
1957:
1951:
1944:
1929:
1909:
1886:
1880:
1873:
1858:
1838:
1818:
1815:
1812:
1805:
1804:
1803:
1789:
1786:
1783:
1770:
1768:
1764:
1756:
1754:
1752:
1748:
1744:
1740:
1739:
1734:
1729:
1723:
1715:
1713:
1711:
1707:
1706:static typing
1703:
1698:
1695:
1691:
1683:
1681:
1679:
1675:
1667:
1659:
1652:
1644:
1641:
1637:
1634:directive in
1629:
1628:
1627:
1621:
1619:
1617:
1616:memory safety
1612:
1608:
1605:
1497:
1494:
1477:
1474:
1472:
1468:
1467:safe language
1464:
1459:
1457:
1451:
1443:
1441:
1439:
1435:
1429:
1421:
1419:
1417:
1413:
1409:
1404:
1403:introspection
1400:
1396:
1392:
1388:
1384:
1380:
1379:C++ templates
1376:
1371:
1369:
1365:
1361:
1357:
1352:
1350:
1346:
1342:
1338:
1337:
1331:
1329:
1325:
1321:
1317:
1313:
1309:
1305:
1301:
1297:
1293:
1289:
1284:
1281:
1276:
1274:
1268:
1266:
1262:
1258:
1252:
1250:
1242:
1240:
1212:
1207:
1200:
1198:
1194:
1193:
1188:
1183:
1179:
1175:
1170:
1168:
1164:
1160:
1154:
1152:
1148:
1147:variant types
1144:
1140:
1136:
1132:
1128:
1120:
1118:
1116:
1111:
1107:
1105:
1099:
1097:
1093:
1089:
1085:
1081:
1075:
1071:
1063:
1061:
1059:
1054:
1052:
1044:
1038:
1036:
1031:
1028:
1027:code coverage
1006:
1005:
1004:
1002:
998:
994:
990:
985:
983:
979:
974:
972:
968:
960:
958:
955:
953:
949:
945:
940:
938:
934:
930:
929:type checking
923:Type checking
922:
920:
918:
914:
910:
906:
902:
898:
897:runtime error
894:
890:
886:
882:
878:
874:
869:
867:
863:
859:
850:
845:
844:
839:
835:
827:
823:
820:
817:
813:
810:
809:
808:
801:
800:documentation
797:
796:Documentation
794:
791:
787:
783:
779:
775:
771:
767:
764:
763:
762:
758:
756:
752:
748:
744:
743:
742:effect system
738:
737:
731:
729:
725:
721:
716:
714:
710:
709:
704:
703:
698:
697:
692:
688:
684:
680:
676:
672:
667:
665:
664:
659:
655:
651:
647:
643:
639:
635:
631:
627:
623:
619:
615:
611:
607:
601:
596:
594:
590:
582:
580:
578:
574:
570:
566:
562:
558:
553:
551:
547:
543:
539:
536:
532:
528:
524:
519:
515:
513:
509:
504:
502:
497:
495:
491:
487:
482:
480:
476:
468:
466:
464:
463:documentation
460:
456:
451:
449:
445:
441:
436:
434:
430:
425:
423:
419:
415:
411:
407:
403:
398:
396:
392:
388:
383:
381:
377:
373:
369:
365:
361:
357:
356:
351:
347:
343:
339:
338:
333:
329:
325:
313:
308:
306:
301:
299:
294:
293:
291:
290:
285:
282:
280:
277:
275:
274:Substructural
272:
270:
267:
265:
262:
260:
257:
255:
252:
250:
247:
245:
242:
240:
237:
236:
235:
234:
229:
224:
221:
219:
215:
212:
210:
206:
203:
201:
197:
194:
193:
192:
191:
186:
181:
178:
176:
173:
172:
171:
170:
165:
162:
158:
149:
146:
138:
128:
124:
118:
117:
112:This article
110:
101:
100:
91:
88:
80:
70:
66:
60:
59:
53:
48:
39:
38:
33:
19:
7104:Type systems
7072:
6848:Intersection
6524:Type systems
6485:
6462:
6433:
6395:
6389:
6357:
6348:
6339:
6330:
6320:
6313:
6293:
6286:
6273:
6265:
6258:
6215:
6211:
6198:
6171:
6151:
6143:
6113:
6104:
6093:. Retrieved
6088:
6079:
6068:. Retrieved
6063:
6054:
6044:
6034:
6027:Visual Basic
6022:
5977:
5959:
5953:
5938:
5932:Corporation.
5916:
5905:. Retrieved
5901:
5892:
5880:. Retrieved
5875:MSDN Library
5873:
5864:
5846:
5839:
5828:. Retrieved
5813:
5794:
5771:
5759:. Retrieved
5752:the original
5737:
5728:
5715:
5696:
5686:
5659:
5638:
5626:
5614:
5583:
5569:
5480:
5475:
5469:
5463:
5459:
5455:
5451:
5440:
5438:
5433:
5423:
5418:
5415:of the form
5406:
5374:
5038:typing rules
5035:
5003:
4983:
4925:
4921:
4919:
4903:
4899:
4892:compile-time
4885:
4855:
4849:
4842:
4836:
4833:
4820:
4816:
4812:
4808:
4804:
4793:record types
4787:
4772:
4767:
4763:
4759:
4754:
4749:
4731:
4728:
4677:
4672:
4667:
4662:
4645:language (a
4619:linear logic
4615:Linear types
4613:
4604:Linear types
4587:
4575:Dependent ML
4556:
4288:
4208:
4203:
4135:
4130:
2563:Intersection
2046:(resp. term
2042:by the type
1771:
1760:
1736:
1728:polymorphism
1727:
1725:
1699:
1693:
1690:Gilad Bracha
1687:
1671:
1625:
1613:
1609:
1583:
1495:
1483:
1475:
1466:
1462:
1460:
1455:
1453:
1438:weakly typed
1437:
1433:
1431:
1415:
1411:
1372:
1353:
1340:
1334:
1332:
1285:
1280:interpreters
1277:
1269:
1261:Dependent ML
1253:
1246:
1208:
1201:
1190:
1171:
1155:
1124:
1112:
1108:
1100:
1088:late binding
1079:
1077:
1055:
1039:
1032:
1012:
1000:
996:
992:
986:
975:
964:
956:
952:weakly typed
951:
947:
941:
933:compile time
928:
926:
908:
893:compile time
884:
870:
860:, for which
854:
841:
821:
816:compile-time
812:Optimization
811:
806:
795:
785:
769:
765:
759:
754:
750:
746:
740:
734:
732:
717:
712:
706:
700:
694:
690:
687:dependencies
668:
661:
653:
605:
603:
598:
592:
586:
583:Fundamentals
565:polymorphism
556:
554:
534:
520:
516:
505:
498:
483:
472:
452:
444:compile time
437:
426:
406:interpreters
399:
384:
353:
336:
327:
321:
259:Intersection
161:Type systems
160:
141:
132:
113:
83:
77:October 2010
74:
55:
7119:Type theory
7078:Type theory
7073:Type system
6923:Bottom type
6870:Option type
6811:generalized
6697:Long double
6642:Fixed point
6130:Bracha, G.
6091:. Microsoft
5878:. Microsoft
5795:C# in Depth
5776:Pierce 2002
5643:Pierce 2002
5619:Pierce 2002
5555:Type theory
5306:and a type
5203:and a type
4904:consistency
4868:similar to
4789:Existential
4751:Union types
4740:Union types
4639:Singularity
4610:Linear type
4579:undecidable
3040:Existential
2916:Polymorphic
2057:Type notion
1831:means that
1767:type theory
1743:abstraction
1450:Type safety
1399:metaclasses
1389:code since
1364:duck typing
1356:duck typing
1349:mock object
1273:Common Lisp
1178:Common Lisp
1127:downcasting
1104:downcasting
1092:downcasting
1035:downcasting
982:type safety
971:source code
967:type safety
885:type system
879:(as in the
851:Type errors
843:type safety
766:Abstraction
724:undecidable
691:type system
593:type system
589:type theory
569:Type theory
433:type errors
393:, or other
372:expressions
352:) to every
328:type system
223:Duck typing
175:Type safety
69:introducing
32:Type theory
7109:Data types
7098:Categories
6983:Empty type
6978:Type class
6928:Collection
6885:Refinement
6863:metaobject
6711:signedness
6570:Data types
6095:2013-07-17
6070:2013-07-17
6005:1581130953
5907:2021-07-07
5882:14 January
5856:1911/16900
5830:2021-03-28
5607:References
5456:equivalent
5417:x :=
5409:expression
5050:typability
5006:computable
4951:that adds
4746:Union type
1747:modularity
1636:JavaScript
1632:use strict
1300:version 10
1249:trade-offs
1068:See also:
1001:incomplete
950:f not, as
782:interfaces
770:modularity
720:inferences
620:such as a
587:Formally,
527:algorithms
490:parameters
475:C language
440:interfaces
395:data types
366:, such as
269:Refinement
218:structural
52:references
7058:Subtyping
7053:Interface
7036:metaclass
6988:Unit type
6958:Semaphore
6938:Exception
6843:Inductive
6833:Dependent
6798:Composite
6776:Character
6758:Reference
6655:Minifloat
6611:Bit array
6400:CiteSeerX
6176:CiteSeerX
5982:CiteSeerX
5964:CiteSeerX
5930:Microsoft
5678:604732680
5503:covariant
5483:subtyping
5354:τ
5314:τ
5294:Γ
5271:Γ
5251:τ
5211:τ
5191:Γ
5148:τ
5108:τ
5080:, a term
5068:Γ
5020:decidable
5015:System Fω
4507:→
4469:×
4355:×
4259:τ
4254:σ
4244:∑
4237:τ
4234:×
4228:σ
4186:τ
4181:σ
4171:∏
4164:τ
4161:→
4155:σ
4108:σ
4067:τ
4041:τ
4036:σ
4026:⋃
3997:τ
3992:σ
3982:⋃
3954:σ
3913:τ
3887:τ
3882:σ
3872:⋂
3843:τ
3838:σ
3828:⋂
3783:τ
3757:σ
3731:τ
3728:∩
3722:σ
3685:τ
3682:∩
3676:σ
3623:τ
3597:σ
3533:τ
3530:×
3524:σ
3487:τ
3484:×
3478:σ
3425:τ
3390:σ
3364:τ
3361:→
3355:σ
3318:τ
3315:→
3309:σ
3268:τ
3262:α
3259:μ
3253:α
3247:τ
3221:τ
3215:α
3212:μ
3184:τ
3178:α
3175:μ
3164:Recursive
3139:σ
3133:α
3127:τ
3101:τ
3095:α
3090:∃
3062:τ
3056:α
3051:∃
3015:σ
3009:α
3003:τ
2977:τ
2971:α
2966:∀
2938:τ
2932:α
2927:∀
2898:τ
2868:⟩
2865:τ
2856:⟨
2828:⟩
2825:τ
2816:⟨
2787:τ
2761:σ
2735:τ
2732:∪
2729:σ
2701:τ
2698:∪
2695:σ
2666:τ
2640:σ
2614:τ
2611:∩
2608:σ
2580:τ
2577:∩
2574:σ
2545:τ
2504:ι
2477:σ
2436:ι
2409:τ
2403:σ
2375:τ
2369:σ
2340:τ
2314:σ
2250:τ
2247:×
2244:σ
2216:τ
2213:×
2210:σ
2181:τ
2146:σ
2120:τ
2117:→
2114:σ
2086:τ
2083:→
2080:σ
2030:τ
1987:τ
1964:σ
1958:α
1952:τ
1859:σ
1851:has type
1819:σ
1790:τ
1784:σ
1751:subtyping
1726:The term
1655:TypeError
1465:(or just
1298:prior to
1290:(such as
997:decidable
917:empirical
774:low-level
751:with what
728:type safe
696:data type
642:character
538:data type
499:During a
494:variables
486:interface
410:compilers
376:functions
368:variables
244:Dependent
135:July 2016
7083:Variable
6973:Top type
6838:Equality
6746:physical
6723:Rational
6718:Interval
6665:bfloat16
6454:(2004).
6432:(2002).
5947:. InfoQ.
5824:Archived
5599:article.
5513:See also
4994:, while
4896:run-time
4870:System F
4807:of type
4211:, since
4138:, since
2069:Function
2063:Meaning
2060:Notation
1561:"%c
1192:DrRacket
1135:C++ RTTI
1131:subtypes
1080:type tag
937:run-time
883:). In a
826:compiler
730:manner.
671:subtypes
626:hardware
622:variable
616:or some
600:tension.
523:compiler
479:function
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