Compatibility: Equivalence and Subtyping
A type-checker for a statically typed language must verify that the type of any expression is consistent with the type expected by the context in which that expression appears. For instance, in an assignment statement of the form x := e, the inferred type of the expression e must be consistent with the declared or inferred type of the variable x. This notion of consistency, called compatibility, is specific to each programming language.
If the type of e and the type of x are the same and assignment is allowed for that type, then this is a valid expression. In the simplest type systems, therefore, the question of whether two types are compatible reduces to that of whether they are equal (or equivalent). Different languages, however, have different criteria for when two type expressions are understood to denote the same type. These different equational theories of types vary widely, two extreme cases being structural type systems, in which any two types are equivalent that describe values with the same structure, and nominative type systems, in which no two syntactically distinct type expressions denote the same type (i.e., types must have the same "name" in order to be equal).
In languages with subtyping, the compatibility relation is more complex. In particular, if A is a subtype of B, then a value of type A can be used in a context where one of type B is expected, 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 ad hoc polymorphism in a language may also have implications for type compatibility.
Read more about this topic: Type System