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Semantics vs. Syntax vs. Computations  Machine Models For Type2 . . .
 JOURNAL OF COMPUTER AND SYSTEM SCIENCE
, 1997
"... This paper investigates analogs of the KreiselLacombeShoenfield Theorem in the context of the type2 basic feasible functionals. We develop a direct, polynomialtime analog of effective operation in which the time boundingon computations is modeled after Kapron and Cook's scheme for their basic po ..."
Abstract

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This paper investigates analogs of the KreiselLacombeShoenfield Theorem in the context of the type2 basic feasible functionals. We develop a direct, polynomialtime analog of effective operation in which the time boundingon computations is modeled after Kapron and Cook's scheme for their basic polynomialtime functionals. We show that if P = NP, these polynomialtime effective operations are strictly more powerful on R (the class of recursive functions) than the basic feasible functions. We also consider a weaker notion of polynomialtime effective operation where the machines computing these functionals have access to the computations of their procedural parameter, but not to its program text. For this version of polynomialtime effective operations, the analog of the KreiselLacombeShoenfield is shown to holdtheir power matches that of the basic feasible functionals on R.
Kleene’s Amazing Second Recursion Theorem (Extended Abstract)
"... This little gem is stated unbilled and proved (completely) in the last two lines of §2 of the short note Kleene (1938). In modern notation, with all the hypotheses stated explicitly and in a strong form, it reads as follows: Theorem 1 (SRT). Fix a set V ⊆ N, and suppose that for each natural number ..."
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This little gem is stated unbilled and proved (completely) in the last two lines of §2 of the short note Kleene (1938). In modern notation, with all the hypotheses stated explicitly and in a strong form, it reads as follows: Theorem 1 (SRT). Fix a set V ⊆ N, and suppose that for each natural number n ∈ N = {0, 1, 2,...}, ϕ n: N n+1 ⇀ V is a recursive partial function of (n + 1) arguments with values in V so that the standard assumptions (1) and (2) hold with {e}(⃗x) = ϕ n e (⃗x) = ϕ n (e, ⃗x) (⃗x = (x1,..., xn) ∈ N n). (1) Every nary recursive partial function with values in V is ϕ n e for some e. (2) For all m, n, there is a recursive (total) function S = S m n: N m+1 → N such that {S(e, ⃗y)}(⃗x) = {e}(⃗y, ⃗x) (e ∈ N, ⃗y ∈ N m, ⃗x ∈ N n). Then, for every recursive, partial function f(e, ⃗y, ⃗x) of (1+m+n) arguments with values in V, there is a total recursive function ˜z(⃗y) of m arguments such that