This article relies largely or entirely on a single source. (February 2018)
Substructural type systems are a family of type systems analogous to substructural logics where one or more of the structural rules are absent or only allowed under controlled circumstances. Such systems are useful for constraining access to system resources such as files, locks and memory by keeping track of changes of state that occur and preventing invalid states.
Several type systems have emerged by discarding some of the structural rules of exchange, weakening, and contraction:
The explanation for affine type systems is best understood if rephrased as "every occurrence of a variable is used at most once".
Linear type systems allow references but not aliases. To enforce this, a reference goes out of scope after appearing on the right-hand side of an assignment, thus ensuring that only one reference to any object exists at once. Note that passing a reference as an argument to a function is a form of assignment, as the function parameter will be assigned the value inside the function, and therefore such use of a reference also causes it to go out of scope.
A linear type system is similar to C++'s unique_ptr class, which behaves like a pointer but can only be moved (i.e. not copied) in an assignment. Although the linearity constraint is checked at compile time, dereferencing an invalidated unique_ptr causes undefined behavior at run-time.
The single-reference property makes linear type systems suitable as programming languages for quantum computation, as it reflects the no-cloning theorem of quantum states. From the category theory point of view, no-cloning is a statement that there is no diagonal functor which could duplicate states; similarly, from the combinator point of view, there is no K-combinator which can destroy states. From the lambda calculus point of view, a variable x can appear exactly once in a term.
Linear type systems are the internal language of closed symmetric monoidal categories, much in the same way that simply typed lambda calculus is the language of Cartesian closed categories. More precisely, one may construct functors between the category of linear type systems and the category of closed symmetric monoidal categories.
Affine types are a version of linear types allowing to discard (i.e. not use) a resource, corresponding to affine logic. An affine resource can only be used once, while a linear one must be used once.
Relevant types correspond to relevant logic which allows exchange and contraction, but not weakening, which translates to every variable being used at least once.
Ordered types correspond to noncommutative logic where exchange, contraction and weakening are discarded. This can be used to model stack-based memory allocation (contrast with linear types which can be used to model heap-based memory allocation). Without the exchange property, an object may only be used when at the top of the modelled stack, after which it is popped off resulting in every variable being used exactly once in the order it was introduced.
The following programming languages support linear or affine types: