Free and Bound Variables
The abstraction operator, λ, is said to bind its variable wherever it occurs in the body of the abstraction. Variables that fall within the scope of a lambda are said to be bound. All other variables are called free. For example in the following expression y is a bound variable and x is free: λy.x x y. Also note that a variable binds to its "nearest" lambda. In the following expression one single occurrence of x is bound by the second lambda: λx.y (λx.z x)
The set of free variables of a lambda expression, M, is denoted as FV(M) and is defined by recursion on the structure of the terms, as follows:
- FV(x) = {x}, where x is a variable
- FV(λx.M) = FV(M) \ {x}
- FV(M N) = FV(M) ∪ FV(N)
An expression that contains no free variables is said to be closed. Closed lambda expressions are also known as combinators and are equivalent to terms in combinatory logic.
Read more about this topic: Lambda Calculus, Formal Definition
Famous quotes containing the words free, bound and/or variables:
“I expect that any day now, I will have said all I have to say; I’ll have used up all my characters, and then I’ll be free to get on with my real life.”
—Anne Tyler (b. 1941)
“I shall never be a heretic; I may err in dispute, but I do not wish to decide anything finally; on the other hand, I am not bound by the opinions of men.”
—Martin Luther (1483–1546)
“Science is feasible when the variables are few and can be enumerated; when their combinations are distinct and clear. We are tending toward the condition of science and aspiring to do it. The artist works out his own formulas; the interest of science lies in the art of making science.”
—Paul Valéry (1871–1945)