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Variables bound at the top level of a program are technically free variables within the terms to which they are bound but are often treated specially because they can be compiled as fixed addresses. Similarly, an identifier bound to a recursive function is also technically a free variable within its own body but is treated specially.
All other variables are called free. For example, in the expression λy.x x y, y is a bound variable and x is a free variable. Also a variable is bound by its nearest abstraction. In the following example the single occurrence of x in the expression is bound by the second lambda: λx.y (λx.z x).
A quantifier has a scope, and an occurrence of a variable x is free if it is not within the scope of a quantification for that variable. Thus in ((,)) (,) the occurrence of both x and y in C(y, x) is free, while the occurrence of x and y in B(y, x) is bound (i.e. non-free).
Variable (names) that have already been matched to formal parameter variable are said to be bound. All other variables in the expression are called free . For example, in the following expression y is a bound variable and x is free: λ y . x x y {\displaystyle \lambda y.x\ x\ y} .
Free and bound variables of a formula need not be disjoint sets: in the formula P(x) → ∀x Q(x), the first occurrence of x, as argument of P, is free while the second one, as argument of Q, is bound. A formula in first-order logic with no free variable occurrences is called a first-order sentence.
Free variables and bound variables A random variable is a kind of variable that is used in probability theory and its applications. All these denominations of variables are of semantic nature, and the way of computing with them ( syntax ) is the same for all.
bound variables are distinct from free variables, and all binders bind variables not already in scope. In the general context of an inductive definition, it is not possible to apply α-conversion as needed to convert an inductive definition using the convention to one where the convention is not used, because a variable may appear in both a ...
The scope of a logical connective occurring within a formula is the smallest well-formed formula that contains the connective in question. [2] [6] [8] The connective with the largest scope in a formula is called its dominant connective, [9] [10] main connective, [6] [8] [7] main operator, [2] major connective, [4] or principal connective; [4] a connective within the scope of another connective ...