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Substitutability is a principle in object-oriented programming. It states 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., objects of type S may be substituted for objects of type T) without altering any of the desirable properties of that program (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 entitled 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:
In the same paper, Liskov and Wing detailed their notion of behavioral subtyping in an extension of Hoare logic, which bears a certain resemblance with Bertrand Meyer's Design by Contract in that it considers the interaction of subtyping with pre- and postconditions.
Liskov's notion of a behavioral subtype defines a notion of substitutability for mutable 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 trivially 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 these, there are a number of behavioral conditions that the subtype must meet. 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. 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 language that supports 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.
A typical example that violates LSP is a Square class that derives from a Rectangle class, assuming getter and setter methods exist for both width and height. The Square class always assumes that the width is equal with the height. If a Square object is used in a context where a Rectangle is expected, unexpected behavior may occur because the dimensions of a Square cannot (or rather should not) be modified independently. This problem cannot be easily fixed: if we can modify the setter methods in the Square class so that they preserve the Square invariant (i.e., keep the dimensions equal), then these methods will weaken (violate) the postconditions for the Rectangle setters, which state that dimensions can be modified independently. Violations of LSP, like this one, may or may not be a problem in practice, depending on the postconditions or invariants that are actually expected by the code that uses classes violating LSP. Mutability is a key issue here. If Square and Rectangle had only getter methods (i.e., they were immutable objects), then no violation of LSP could occur.