Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
In mathematics, logic and computer science, type theory is any of several formal systems that can serve as alternatives to naive set theory, or the study of such formalisms in general...
In computer programming, a data type is a classification identifying one of various types of data, such as floating-point, integer, or Boolean, that determines the possible values for that type; the operations that can be done on values of that type; the meaning of the data; and the way values of...
that allows only one value (and thus can hold no information). The carrier (underlying set) associated with a unit type can be any singleton set. There is an isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
between any two such sets, so it is customary to talk about the unit type and ignore the details of its value. One may also regard the unit type as the type of 0-tuple
Tuple
In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...
In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...
of no types.
The unit type is the terminal object in the category
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
of types and typed functions. It should not be confused with the zero or bottom type
Bottom type
In type theory, a theory within mathematical logic, the bottom type is the type that has no values. It is also called the zero or empty type, and is sometimes denoted with falsum .A function whose return type is bottom cannot return any value...
In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X...
in this category.
The unit type is implemented in most functional programming languages. The void type
Void type
The void type, in several programming languages derived from C and Algol68, is the type for the result of a function that returns normally, but does not provide a result value to its caller. Usually such functions are called for their side effects, such as performing some task or writing to their...
that is used in some imperative programming languages serves some of its functions, but because its carrier set is empty, there are some limitations (as detailed below).
A programming language is an artificial language designed to communicate instructions to a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine and/or to express algorithms precisely....
s provide a unit type to specify the result type of a function with the sole purpose of causing a side effect
Side effect (computer science)
In computer science, a function or expression is said to have a side effect if, in addition to returning a value, it also modifies some state or has an observable interaction with calling functions or the outside world...
, and the argument type of a function that does not require arguments.
Haskell is a standardized, general-purpose purely functional programming language, with non-strict semantics and strong static typing. It is named after logician Haskell Curry. In Haskell, "a function is a first-class citizen" of the programming language. As a functional programming language, the...
, the unit type is called and its only value is also , reflecting the 0-tuple interpretation.
ML is a general-purpose functional programming language developed by Robin Milner and others in the early 1970s at the University of Edinburgh, whose syntax is inspired by ISWIM...
Standard ML is a general-purpose, modular, functional programming language with compile-time type checking and type inference. It is popular among compiler writers and programming language researchers, as well as in the development of theorem provers.SML is a modern descendant of the ML...
), the type is called unit but the value is written as .
In Scala, the unit type is called Unit and its only value is written as .
Common Lisp, commonly abbreviated CL, is a dialect of the Lisp programming language, published in ANSI standard document ANSI INCITS 226-1994 , . From the ANSI Common Lisp standard the Common Lisp HyperSpec has been derived for use with web browsers...
the type named NULL is a unit type which has one value, namely the symbol NIL. NIL itself is used as the name of the bottom type
Bottom type
In type theory, a theory within mathematical logic, the bottom type is the type that has no values. It is also called the zero or empty type, and is sometimes denoted with falsum .A function whose return type is bottom cannot return any value...
.
The unit type is useful even in functions without side effects if the programming language supports algebraic data types. Any nullary data constructor is effectively isomorphic with the unit type. One can solve the semipredicate problem
Semipredicate problem
In computer programming, a semipredicate problem occurs when a subroutine intended to return a useful value can fail, but the signalling of failure uses an otherwise valid return value. The problem is that the caller of the subroutine cannot tell what the result means in this case.- Example :The...
(the problem of distinguishing between a "normal" return value of a function and an "error") elegantly in such a language, by encoding the "error" as the unit type. In Haskell, the polymorphic type
Parametric polymorphism
In programming languages and type theory, parametric polymorphism is a way to make a language more expressive, while still maintaining full static type-safety. Using parametric polymorphism, a function or a data type can be written generically so that it can handle values identically without...
Maybe is predefined for this purpose. Here Nothing is isomorphic to the unit type:
In programming languages and type theory, an option type or maybe type is a polymorphic type that represents encapsulation of an optional value; e.g. it is used as the return type of functions which may or may not return a meaningful value when they're applied...
C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system....
C++ is a statically typed, free-form, multi-paradigm, compiled, general-purpose programming language. It is regarded as an intermediate-level language, as it comprises a combination of both high-level and low-level language features. It was developed by Bjarne Stroustrup starting in 1979 at Bell...
Java is a programming language originally developed by James Gosling at Sun Microsystems and released in 1995 as a core component of Sun Microsystems' Java platform. The language derives much of its syntax from C and C++ but has a simpler object model and fewer low-level facilities...
, void expresses the empty type. The unit type in C would be struct {}, but an empty struct is forbidden by the C language specification. Instead void is used in a manner that simulates some, but not all, of the properties of the unit type, as detailed below. Like most imperative languages, C allows functions that do not return a value; these are specified as having the void return type. Such functions are called procedures in other imperative languages like Pascal
Pascal (programming language)
Pascal is an influential imperative and procedural programming language, designed in 1968/9 and published in 1970 by Niklaus Wirth as a small and efficient language intended to encourage good programming practices using structured programming and data structuring.A derivative known as Object Pascal...
, where a syntactic, instead of type-system distinction, is made between functions and procedures.
Difference in calling convention
The first notable difference between a true unit type and the void type is that the unit type may always be the type of the argument to a function, but the void type cannot be the type of an argument in C, despite the fact that it may appear as the sole argument in the list. This problem is best illustrated by the following program, which is a compile-time error in C:
void f(void) {}
void g(void) {}
int main(void)
{
f(g); // compile-time error here
return 0;
}
This issue does not arise in most programming practice in C, because since the void type carries no information, it is useless to pass it anyway; but it may arise in generic programming
Generic programming
In a broad definition, generic programming is a style of computer programming in which algorithms are written in terms of to-be-specified-later types that are then instantiated when needed for specific types provided as parameters...
Templates are a feature of the C++ programming language that allow functions and classes to operate with generic types. This allows a function or class to work on many different data types without being rewritten for each one....
, where void must be treated differently from other types. In C++ however, empty classes are allowed, so it is possible to implement a real unit type; the above example becomes compilable as:
(For brevity, we're not worried in the above example whether the_unit is really a singleton
Singleton pattern
In software engineering, the singleton pattern is a design pattern used to implement the mathematical concept of a singleton, by restricting the instantiation of a class to one object. This is useful when exactly one object is needed to coordinate actions across the system...
In software engineering, the singleton pattern is a design pattern used to implement the mathematical concept of a singleton, by restricting the instantiation of a class to one object. This is useful when exactly one object is needed to coordinate actions across the system...
for details on that issue.)
Difference in storage
The second notable difference is that the void type, being empty, can never be stored in a record type, i.e. in a struct or a class in C/C++. In contrast, the unit type can be stored in records in functional programming languages, i.e. it can appear as the type of a field; the above implementation of the unit type in C++ can also be stored. While this may seem a useless feature, it does allow one for instance to elegantly implement a set
Set (computer science)
In computer science, a set is an abstract data structure that can store certain values, without any particular order, and no repeated values. It is a computer implementation of the mathematical concept of a finite set...
In computer science, an associative array is an abstract data type composed of a collection of pairs, such that each possible key appears at most once in the collection....
to the unit type; in the absence of a unit type, one can still implement a set this way by storing some dummy value of another type for each key.
In Generics
In Java Generics, type parameters must be reference types. The wrapper type Void is often used when a unit type parameter is needed. Although the Void type can never have any instances, it does have one value, null (like all other reference types), so it acts as a unit type. In practice, any other non-instantiable type, e.g. Math, can also be used for this purpose, since they also have exactly one value, null.
public static Void f(Void x) { return null; }
public static Void g(Void x) { return null; }
public static void main(String[] args)
{
f(g(null));
}
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