Double Generativity and Complete Effective Inseparability

1993 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Recursive Function ◽  
Disjoint Pair ◽  
Recursion Theorem ◽  
Trivial Fact

§1. Complete Effective Inseparability. A disjoint pair (A1,A2) is by definition recursively inseparable if no recursive superset of A1 is disjoint from A2. This is equivalent to saying that for any disjoint r.e. supersets ωi and ωj of A1 and A2, the set ωi is not the complement of ωj —in other words, there is a number n outside both ωi and ωj. The disjoint pair (A1,A2) is called effectively inseparable—abbreviated E.I.—if there is a recursive function δ(x, y)—called an E.I. function for (A1, A2)—such that for any numbers i and j such that A1⊆ ωi and A2Í ωj. with ωi being disjoint from ωj, the number d(i , j) is outside both a;,- and ωj. We shall call a disjoint pair (A1, A2) completely E.I. if there is a recursive function δ(x, y)—which we call a complete E.I. function for (A1, A2)—such that for any numbers i and j, if A1⊆ ωi and A2Í ωj, then δ(i , j) Í ωi ↔ d(i , j)Í ωj (in other words, d(i, j) is either inside or outside both sets ωi and ωj.). [If ωi and ωj happen to be disjoint, then, of course, d(i, j) is outside both ωi and ωj, so any complete E.I. function for (A1,A2) is also an E.I. function for (A1,A2) In a later chapter, we will prove the non-trivial fact that if (A1, A2) is E.I. and A1 and A2 are both r.e., then (A1,A2) is completely E.I. [The proof of this uses the result known as the double recursion theorem, which we will study in Chapter 9.] Effective inseparability has been well studied in the literature. Complete effective inseparability will play a more prominent role in this volume—especially in the next few chapters. Proposition 1. (1) If (A1,A2) is completely E.I., then so is (A2,A1) —in fact, if d(x,y) is a complete E.I. function for (A1,A2), then d(y,x) is a complete E.I. function for (A2, A1).

On size vs. efficiency for programs admitting speed-ups

1971 ◽  
Vol 36 (1) ◽  
pp. 21-27 ◽  
Author(s):  
John Helm ◽  
Paul Young
Keyword(s):  
Computational Complexity ◽  
Recursive Function ◽  
Recursion Theory ◽  
Typical Application ◽  
Recursion Theorem ◽  
Efficient Program ◽  
Do So

Since the publication in 1967 of the two papers [1] and [2] by Manuel Blum, the techniques and results of “pure” recursion theory, particularly the recursion theorem and priority methods, have come to play an increasingly important role in studies of computational complexity. This paper gives a typical application of the recursion theorem with a fairly intricate diagonalization to answer a question raised by Blum in [3]. Roughly, we prove the existence of functions which have the property that if we are given any program for computing the function and want to pass to a program which computes the function much more efficiently, then we can only do so at the expense of obtaining a much larger program: the function which describes the necessary increase in the size of the more efficient program must grow more rapidly than any recursive function.

Recursion Theorems

1993 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Recursive Function ◽  
Function Theory ◽  
Recursion Theorem

We have proved that the complement of every completely productive set (in other words, every generative set) is universal, and this was enough to establish Theorem A of Chapter 6. In Chapter 10 we will prove Myhill’s stronger result that the complement of every productive set is universal. For this proof, we will need the recursion theorem of this chapter. Recursion theorems (which can be stated in many forms) have profound applications in recursive function theory and metamathematics, and we shall devote considerable space to their study. To illustrate their rather startling nature, consider the following mathematical “believe-it-or-not’s”: Which of the following propositions, if true, would surprise you? . . . 1. There is a number n such that ωn = ωn+1.

Implicit measurements of dynamic complexity properties and splittings of speedable sets

1999 ◽  
Vol 64 (3) ◽  
pp. 1037-1064 ◽  
Author(s):  
Michael A. Jahn
Keyword(s):  
Complexity Theory ◽  
Dynamic Complexity ◽  
Direct Construction ◽  
Computably Enumerable Set ◽  
Computably Enumerable Sets ◽  
Disjoint Pair ◽  
Tree Construction ◽  
Self Similar ◽  
Recursion Theorem ◽  
Computably Enumerable

AbstractWe prove that any speedable computably enumerable set may be split into a disjoint pair of speedable computably enumerable sets. This solves a longstanding question of J.B. Remmel concerning the behavior of computably enumerable sets in Blum's machine independent complexity theory. We specify dynamic requirements and implement a novel way of detecting speedability—by embedding the relevant measurements into the substage structure of the tree construction. Technical difficulties in satisfying the dynamic requirements lead us to implement “local” strategies that only look down the tree. The (obvious) problems with locality are then resolved by placing an isomorphic copy of the entire priority tree below each strategy (yielding a self-similar tree). This part of the construction could be replaced by an application of the Recursion Theorem, but shows how to achieve the same effect with a more direct construction.

Kleene's Amazing Second Recursion Theorem

2010 ◽  
Vol 16 (2) ◽  
pp. 189-239 ◽  
Author(s):  
Yiannis N. Moschovakis
Keyword(s):  
Natural Number ◽  
Recursive Function ◽  
Short Note ◽  
Partial Function ◽  
Modern Notation ◽  
Recursion Theorem ◽  
Image Position

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 (uniform) form, it reads as follows:Second Recursion Theorem (SRT). Fix a set V ⊆ ℕ, and suppose that for each natural number n ϵ ℕ = {0, 1, 2, …}, φn: ℕ1+n ⇀ V is a recursive partial function of (1 + n) arguments with values in V so that the standard assumptions (a) and (b) hold with.(a) Every n-ary recursive partial function with values in V is for some e.(b) For all m, n, there is a recursive function : Nm+1 → ℕ such that.Then, for every recursive, partial function f of (1+m+n) arguments with values in V, there is a total recursive function of m arguments such thatProof. Fix e ϵ ℕ such that and let .We will abuse notation and write ž; rather than ž() when m = 0, so that (1) takes the simpler formin this case (and the proof sets ž = S(e, e)).

J. C. Shepherdson. Algorithmic procedures, generalized Turing algorithms, and elementary recursion theory. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 285–308. - J. C. Shepherdson. Computational complexity of real functions. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 309–315. - A. J. Kfoury. The pebble game and logics of programs. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 317–329. - R. Statman. Equality between functionals revisited. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 331–338. - Robert E. Byerly. Mathematical aspects of recursive function theory. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 339–352.

1990 ◽  
Vol 55 (2) ◽  
pp. 876-878
Author(s):  
J. V. Tucker
Keyword(s):  
New York ◽  
Computational Complexity ◽  
Recursive Function ◽  
Function Theory ◽  
Recursion Theory ◽  
Foundations Of Mathematics ◽  
Real Functions ◽  
Pebble Game ◽  
Logics Of Programs

Nonrecursive tilings of the plane. I

1974 ◽  
Vol 39 (2) ◽  
pp. 283-285 ◽  
Author(s):  
William Hanf
Keyword(s):  
Turing Machine ◽  
Recursive Function ◽  
Positive Answer ◽  
Additional Restriction ◽  
Detailed History ◽  
Finite Set

A finite set of tiles (unit squares with colored edges) is said to tile the plane if there exists an arrangement of translated (but not rotated or reflected) copies of the squares which fill the plane in such a way that abutting edges of the squares have the same color. The problem of whether there exists a finite set of tiles which can be used to tile the plane but not in any periodic fashion was proposed by Hao Wang [9] and solved by Robert Berger [1]. Raphael Robinson [7] gives a more detailed history and a very economical solution to this and related problems; we will assume that the reader is familiar with §4 of [7]. In 1971, Dale Myers asked whether there exists a finite set of tiles which can tile the plane but not in any recursive fashion. If we make an additional restriction (called the origin constraint) that a given tile must be used at least once, then the positive answer is given by the main theorem of this paper. Using the Turing machine constructed here and a more complicated version of Berger and Robinson's construction, Myers [5] has recently solved the problem without the origin constraint.Given a finite set of tiles T1, …, Tn, we can describe a tiling of the plane by a function f of two variables ranging over the integers. f(i, j) = k specifies that the tile Tk is to be placed at the position in the plane with coordinates (i, j). The tiling will be said to be recursive if f is a recursive function.

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