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A.Miller Recursion Theorem 1 The Recursion Theorem and Infinite Sequences Arnold W. Miller 1 Abstract 8 0 In this paper we use the Recursion Theorem to show the exis- 0 tence of various infinite sequences and sets. Our main result is 2 that there is an increasing sequence e < e < ··· such that 0 1 n W = {e } for every n. Similarly, we prove that there ex- a en n+1 J ists an increasing sequence such that W = {e ,e ,...} for en n+1 n+2 4 every n. We call a nonempty computably enumerable set A self- 1 constructing if We = A for every e ∈ A. We show that every ] nonempty computable enumerable set which is disjoint from an O infinitecomputablesetisone-oneequivalent toaself-constructing L . set. h t a Kleene’s Recursion Theorem says the following: m For any computable function f there exists an e with ψ = ψ . [ e f(e) 1 In this Theorem hψ : e ∈ ωi is a standard computable numbering of all e v partial computable functions. For example, ψ might be the partial function 7 e 9 computed by the eth Turing machine. The number e is referred to as a fixed 0 point for f and this theorem is also called the Fixed Point Theorem. 2 For a proof of Kleene’s Theorem see any of the standard references, . 1 Cooper [1], Odifreddi [3], Rogers [4], or Soare [6]. See especially Smullyan [5] 0 8 for many variants and generalizations of the fixed point theorem. The Recur- 0 sion Theorem applies to all acceptable numberings (in the sense of Rogers, : v see Odifreddi [3] p.215-221). All natural enumerations are acceptable. We i X use W to denote the domain of ψ and hence hW : e ∈ ωi is a uniform e e e r computable listing of all computably enumerable sets. a The proof of the Recursion Theorem is short but tricky. It can be uni- formized to yield what is called the Recursion Theorem with Parameters. The proof also yields an infinite computable set of fixed points by using the Padding Lemma. See Soare [6] pages 36-37. We will use the following version of the Recursion Theorem with Param- eters which includes a uniform use of the Padding Lemma: 1 Mathematics Subject Classification 2000: 03E25; 03E99 Keywords: Recursion Theorem, fixed points. A.Miller Recursion Theorem 2 Lemma 1 For any computable function f : ω×ω → ω there is a computable function h : ω → ω such that W is an infinite computable set for every x h(x) and for every y ∈ W we have that ψ = ψ . h(x) f(x,y) y The proof is left to the reader. It is an exercise (see Miller [2] section 13) to show (using Smullyan’s double recursion theorem or more particularly its n-ary generalization, see [5] Chapter IX) that for any n > 0 there is a sequence e < e < ··· < e 0 1 n such that W = {e } for i < n and W = {e }. It occurred to us to ask if ei i+1 en 0 it would be possible to have an infinite sequence like this. We show that it is. Here is our main result: Theorem 1 There is a strictly increasing sequence: e < e < ··· < e < ··· 0 1 n such that W = {e } for every n. en n+1 Proof We use W to denote the set of all y < s such that ψ (y) converges in e,s e less than s-steps. We use hx,yi to denote a pairing function, a computable bijection from ω2 to ω, e.g., hx,yi = 2x(2y +1)−1. Let q(e,x) be a computable function such that for all x and e: {y} if (∃s∃y ∈ W y > x) and hs,yi is the least such pair W = e,s q(e,x) (cid:26) ∅ otherwise. Such a q is constructed by a standard argument using the s-m-n or Parame- terization Theorem. To see this one defines a partial computable function θ as follows: 0 if (∃s∃y ∈ W y > x) and hs,yi is the least such pair θ(e,x,y) = e,s (cid:26) ↑ otherwise. A.Miller Recursion Theorem 3 The uparrow stands for a computation that diverges, i.e, does not halt. By the s-m-n Theorem there is a computable q such that ψ (y) = θ(e,x,y) for all e,x,y. q(e,x) Using Lemma 1 let h be a computable function such that for every e, the set W is an infinite set of fixed points for q(e,·), i.e., W = W for all h(e) x q(e,x) x ∈ W . Let e be a fixed point for h, so W = W . h(e) e h(e) Let x be any element of W . Then x ∈ W so W = W = {y} e h(e) x g(x,e) where y > x and y ∈ W . Hence, starting with any e ∈ W we get an e 0 e infinite increasing sequence as required. QED Note that to obtain a sequence with W = {e } is trivial. Also note en+1 n that the sequence in Theorem 1 must be computable. This is not necessarily true for our next result: Theorem 2 There exists a computable strictly increasing sequence he : n < n ωi such that for every n W = {e : m > n}. en m Proof Using the s-m-n Theorem find q a computable function such that for every x and e: W = {max(W ) : s ∈ ω}\{0,1,...,x}. q(e,x) e,s As in the above proof, let h be a computable function such that for every e, the set W is an infinite set of fixed points for q(e,·), i.e., W = W h(e) x q(e,x) for all x ∈ W . Let e be a fixed point for h, so W = W . h(e) e h(e) Note that W is infinite and let e {e < e < e < ...} = {max(W ) : s ∈ ω}. 0 1 2 e,s For any x ∈ W = W we have that e h(e) W = W = {max(W ) : s ∈ ω}\{0,1,...,x}. x q(e,x) e,s Hence for any n we have that W = {e : m > n}. en m A.Miller Recursion Theorem 4 QED A variation on this theorem would be to get a computable strictly increasing sequence e < e < ··· such that 0 1 ψ (m) = e for every n,m < ω. en n+m+1 The proof of this is left as an exercise for the reader. Usually the first example given of an application of the Recursion The- orem is to prove that there exists an e such that W = {e}. We say that a e nonempty computably enumerable set A is self-constructing iff for all e ∈ A we have that W = A. So W = {e} is an example of a self-constructing set. e e Our next result shows there are many self-constructing sets. Theorem 3 For any nonempty computably enumerable set B the following are equivalent: 1. B is disjoint from an infinite computable set. 2. There is a computable permutation π of ω such that A = π(B) and A is self-constructing. Proof (2) → (1) Let E be an infinite computable set such that for every e ∈ E we have that W = ∅. Any self-constructing set is disjoint from E. e (1) → (2) Given any e consider the following computably enumerable set Q . Let e {c < c < c < ...} = {max(W ) : s ∈ ω} 0 1 2 e,s which may be finite or even empty. Then put Q = {c : n ∈ B}. e n By the s-m-n Theorem we can find a computable q such that for every e: W = Q . q(e) e A.Miller Recursion Theorem 5 By the Padding Lemma there is a computable h such that for every e the set W is infinite and for all x ∈ W we have that W = W . Now let e be h(e) h(e) x q(e) a fixed point for h so that W = W . Let A = Q . Then for all x ∈ A we h(e) e e have that W = A. So A is self-constructing. x To get π let D and E be two infinite pairwise disjoint computable sets disjoint from B. Take one-one computable enumerations of them: D = {d : n < ω} and E = {e : n < ω}. n n Note that W = W is infinite and let C be the infinite computable set: e h(e) C = {c < c < c < ...} = {max(W ) : s ∈ ω}. 0 1 2 e,s Since W = W is a set of fixed points, it is coinfinite and hence C ⊆ W e h(e) e is coinfinite. Take a one-one computable enumeration of the complement C of C: C = {c : n < ω}. n Now we can define π: d if n ∈ D 2n π(c ) =  d if n ∈ E n 2n+1  n otherwise  π(c ) = e . n n Note that π bijectively maps C to E and C to E. Furthermore if n ∈ B then π(c ) = n, and since A = {c : n ∈ B} we have that π(A) = B. n n QED As a corollary we get that there are self-constructing sets of each finite cardinality and there is a self-constructing set which is not computable, in fact, there is a creative self-constructing set. It is not hard to show that S = {e : W is self-constructing} e is Π0-complete. To see this first note that it is easy to show that S is Π0. 2 2 We can get a many-one reduction f of Tot =def {e : W = ω} e A.Miller Recursion Theorem 6 to S as follows. Fix an infinite self-constructing set A with one-to-one computable enumeration A = {a : n ∈ ω}. By the s-m-n Theorem construct a n computable f so that for every e: W = {a : n ∈ W }. f(e) n e Hence, e ∈ Tot iff f(e) ∈ S. Since Tot is Π0-complete (see Soare [6] page 2 66), S is too. References [1] Cooper, S. Barry Computability theory. Chapman & Hall/CRC, Boca Raton, FL, 2004. x+409 pp. ISBN: 1-58488-237-9 [2] Miller, Arnold W. Lecture Notes in Computability Theory. eprint Jan 08: http://www.math.wisc.edu/∼miller/res [3] Odifreddi, Piergiorgio Classical recursion theory. The theory of functions and sets of natural numbers. With a foreword by G. E. Sacks. Studies in Logic and the Foundations of Mathematics, 125. North-Holland Publishing Co., Amsterdam, 1989. xviii+668 pp. ISBN: 0-444-87295-7 [4] Rogers, Hartley, Jr. Theory of recursive functions and effec- tive computability. McGraw-Hill Book Co., New York-Toronto, Ont.- London 1967 xx+482 pp. [5] Smullyan, Raymond M. Recursion theory for metamathematics. OxfordLogicGuides, 22.TheClarendonPress, OxfordUniversity Press, New York, 1993. xvi+163 pp. ISBN: 0-19-508232-X [6] Soare, Robert I. Recursively enumerable sets and degrees. A study of computable functions and computably generated sets. Perspectives inMathematical Logic.Springer-Verlag, Berlin, 1987. xviii+437 pp. ISBN: 3-540-15299-7 Arnold W. Miller [email protected] http://www.math.wisc.edu/∼miller A.Miller Recursion Theorem 7 University of Wisconsin-Madison Department of Mathematics, Van Vleck Hall 480 Lincoln Drive Madison, Wisconsin 53706-1388 Appendix Electronic on-line version only 1 Appendix The appendix is not intended for final publication but for the on-line electronic version only. Theorem 1 There exists an infinite strictly increasing sequence he : n < ωi n such that for every n W = {e : m > n} en m and the sequence is not computable. Proof This is a combination of the proof of Theorem 2 and Theorem 3. Fix B a computably enumerable but not computable set. For any e let {c < c < c < ...} = {max(W ) : s ∈ ω} 0 1 2 e,s By the s-m-n Theorem find computable q such that for any x and e W = {c : n ∈ B and c > x}. q(e,x) n n Take h computable so that W is an infinite set of fixed points for q(e,·), h(e) i.e., x ∈ W implies W = W . Take e to be a fixed point for h, so h(e) x q(e,x) W = W . Then putting h(e) e {e : n < ω} = {c : n ∈ B} n n is not computable and have the required property. QED Theorem 2 There is a computable strictly increasing sequence e < e < ··· 0 1 such that ψ (m) = e for every n,m < ω. en n+m+1 Proof As in the proof of Theorem 2 given any e let {e < e < e < ...} = {max(W ) : s ∈ ω} 0 1 2 e,s Appendix Electronic on-line version only 2 and using the s-m-n Theorem find q a computable function such that for every x, m, and e: ψ (m) = e where n is minimal so that e ≥ x}. q(e,x) n+m+1 n Let h be a computable function such that for every e, the set W is an h(e) infinite set of fixed points for q(e,·), i.e., ψ = ψ for all x ∈ W . Let e x q(e,x) h(e) be a fixed point for h, so W = W . e h(e) Now for any n we have that e ∈ W = W and so ψ = ψ and n e h(e) en q(e,en) hence for every m: ψ (m) = e . en n+m+1 QED