Substitutability is a principle in object-oriented programming stating that, in a computer program, if S is a subtype of T, then objects of type T may be replaced with objects of type S (i.e. an object of type T may be substituted with any object of a subtype S) without altering any of the desirable properties of T (correctness, task performed, etc.). More formally, the Liskov substitution principle (LSP) is a particular definition of a subtyping relation, called (strong) behavioral subtyping, that was initially introduced by Barbara Liskov in a 1987 conference keynote address titled Data abstraction and hierarchy. It is a semantic rather than merely syntactic relation because it intends to guarantee semantic interoperability of types in a hierarchy, object types in particular. Barbara Liskov and Jeannette Wing formulated the principle succinctly in a 1994 paper as follows:
Subtype Requirement: Let be a property provable about objects of type T. Then should be true for objects of type S where S is a subtype of T.
In the same paper, Liskov and Wing detailed their notion of behavioral subtyping in an extension of Hoare logic, which bears a certain resemblance to Bertrand Meyer's design by contract in that it considers the interaction of subtyping with preconditions, postconditions and invariants.
Liskov's notion of a behavioral subtype defines a notion of substitutability for objects; that is, if S is a subtype of T, then objects of type T in a program may be replaced with objects of type S without altering any of the desirable properties of that program (e.g. correctness).
Behavioral subtyping is a stronger notion than typical subtyping of functions defined in type theory, which relies only on the contravariance of argument types and covariance of the return type. Behavioral subtyping is provably undecidable in general: if q is the property "method for x always terminates", then it is impossible for a program (e.g. a compiler) to verify that it holds true for some subtype S of T, even if q does hold for T. Nonetheless, the principle is useful in reasoning about the design of class hierarchies.
Liskov's principle imposes some standard requirements on signatures that have been adopted in newer object-oriented programming languages (usually at the level of classes rather than types; see nominal vs. structural subtyping for the distinction):
In addition to the signature requirements, the subtype must meet a number of behavioral conditions. These are detailed in a terminology resembling that of design by contract methodology, leading to some restrictions on how contracts can interact with inheritance:
The rules on pre- and postconditions are identical to those introduced by Bertrand Meyer in his 1988 book Object-Oriented Software Construction. Both Meyer, and later Pierre America, who was the first to use the term behavioral subtyping, gave proof-theoretic definitions of some behavioral subtyping notions, but their definitions did not take into account aliasing that may occur in programming languages that support references or pointers. Taking aliasing into account was the major improvement made by Liskov and Wing (1994), and a key ingredient is the history constraint. Under the definitions of Meyer and America, a MutablePoint would be a behavioral subtype of ImmutablePoint, whereas LSP forbids this.
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