Question

Let f(x₁,...,xₙ) be computed by program P, and suppose that for some primitive recursive function g(x₁,...,xₙ), STP⁽ⁿ⁾(x₁,...,xₙ, # (P), g (x₁,...,xₙ))

is true for all x₁,..., xₙ Show that f(x₁,...,xₙ) is primitive recursive______

Answer 1

The function f(x₁,...,xₙ) is primitive recursive because it is computed by a program that terminates in a number of steps given by a primitive recursive function, satisfying the definition of primitive recursion.

Step-by-step explanation:

If a student asks whether a function f(x₁,...,xₙ) is primitive recursive given that it is computed by program P, and there exists a primitive recursive function g(x₁,...,xₙ) such that STP⁴⁹(x₁,...,xₙ, #(P), g(x₁,...,xₙ)) is true for all x₁,...,xₙ, we can affirm that f(x₁,...,xₙ) is indeed primitive recursive.

This is because the Statement of Program Termination (STP) being true implies that program P halts for all inputs after g(x₁,...,xₙ) steps. Since g is a primitive recursive function, and the number of steps that P takes is bounded by a primitive recursive function, this means that P itself embodies a primitive recursive process.

Thus, the function computed by P, which is f(x₁,...,xₙ), is guaranteed to halt with a value for all possible inputs within a fixed number of steps that is computable by a primitive recursive function, thereby satisfying the definition of a primitive recursive function.