Μ operator

In computability theory, the μ-operator, minimization operator, or unbounded search operator searches for the least natural number with a given property. Adding the μ-operator to the primitive recursive functions makes it possible to define all computable functions. Suppose that R(y, x1, ..., xk) is a fixed (k+1)-ary relation on the natural numbers. The μ-operator "μy", in either the unbounded or bounded form, is a "number theoretic function" defined from the natural numbers to the natural numbers. However, "μy" contains a predicate over the natural numbers, which can be thought of as a condition that evaluates to true when the predicate is satisfied and false when it is not. The bounded μ-operator appears earlier in Kleene (1952) Chapter IX Primitive Recursive Functions, §45 Predicates, prime factor representation as: "" (p. 225) Stephen Kleene notes that any of the six inequality restrictions on the range of the variable y is permitted, i.e. y < z, y ≤ z, w < y < z, w < y ≤ z, w ≤ y < z and w ≤ y ≤ z. "When the indicated range contains no y such that R(y) [is "true"], the value of the "μy" expression is the cardinal number of the range" (p. 226); this is why the default "z" appears in the definition above. As shown below, the bounded μ-operator "μyy