Double Recursion Theorems

More documents

Recommendations

Info

I. Productivity and <strong>Double</strong> Productivity 125(3) t 6 B =>• u»fc (0 = {0(&(0»&(0)}»(4) t ^ B =» w faw = 0.Proof. Let A and J0 be r.e. sets. Let Mi(x,y, £1,22) t> e the r.e.relation, y € A A x = 5(21,22) an d l et M%(x, y,z\,22) be the r.e.relation, y € -B A x = g(zi,z 2 ). Applying Theorem 2.3, Chapter 9,to MI and MI we obtain recursive functions 0i(y) and suc hthat for all i,(a) w^lW = x : (i € A A x = 0(&(t)»&(0))>(b) o^2(i) = z:(i€BAx = flf(^i(t),02(*)))-(1) and (2) foUow from (a); (3) and (4) foUow from (b).Now suppose («,/?) is weakly doubly co-productive under g(x, y).Let (A, B) be a disjoint pair of r.e. sets that we wish to reduce toas(a,f3). Take recursive functions z(y)m the abovelemma. We show that g((t>i(y),2(y)) reduces (A,B) to (a,/?).1. Suppose y e A. Then W0i( y ) = {s(^i(y),$2(y))}- Also we havey ; = a and MJ = f3 U {p(i,j)}' ^9(^1 j) & a ) then /? U {
126 Chapter X. Productivity and <strong>Double</strong> Productivitywhich is a contradiction. This proves (2). Condition (3) is provedby a symmetric argument.Theorem 3. If (a,/3) is E.I.—or even weakly E.I.—and a and ftare r.e., then (a,/3) is D.U.Proof. Suppose a and ft are r.e. and (a,/9) is weakly E.I. underk(x,y). Let g(x,y} = k(y,x). Then it is obvious that (/3,a) isweakly E.I. under g(x>y). That is, for all x and y, (1) if uj x = ft andu> y = a, then g(x,y) £ ft\ (2) if u> x = ft and u y = a U {g(x, y)}, thend(x,y) € ft; (3) if u x = ft U {g(x,y}} and u y = a, then g(x,y) £ a.By the iteration theorem we can take recursive functions t\(y] andt-i(y} such that for all z,We show that (a,/3) is weakly doubly co-productive under the functionh(x,y) = g(t!(x),t 2 (y)).1. Suppose Ui = u>j = 0. Then u> tl (j) = ft and w t2 (j) = a, sog(ti(i),h(j)) i ft U a. Hence h(i,j) 2(y) as in Step 1 and show that the function g(
Page 4 and 5:
Recursion Theory forMetamathematics
Page 6 and 7:
To Blanche
Page 11 and 12:
This page intentionally left blank
Page 13 and 14:
xiiContentsII Undecidability and Re
Page 15 and 16:
xivContents§2. Weak Double Product
Page 17 and 18:
This page intentionally left blank
Page 19 and 20:
2 Chapter 0. Prerequisitesinfinite
Page 21 and 22:
4 Chapter 0. PrerequisitesMore spec
Page 23 and 24:
6 Chapter 0. PrerequisitesA, then t
Page 25 and 26:
8 Chapter 0. PrerequisitesOne also
Page 27 and 28:
In G.I.T. we gave a proof of the fo
Page 29 and 30:
12 Chapter 0. PrerequisitesTheorem
Page 31 and 32:
14 Chapter 0. Prerequisitesk ^ k +
Page 33 and 34:
16 Chapter 0. Prerequisitesthe prov
Page 35 and 36:
18 Chapter 0. PrerequisitesTheorem
Page 37 and 38:
20 Chapter 0. Prerequisitesshowed t
Page 39 and 40:
22 Chapter 0. PrerequisitesEh(n,h}
Page 41 and 42:
Chapter IRecursive Enumerability an
Page 43 and 44:
26 Chapter I. Recursive Enumerabili
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
Page 53 and 54:
Page 55 and 56:
Chapter IIUndecidability and Recurs
Page 57 and 58:
40 Chapter II. Undecidability and R
Page 59 and 60:
Page 61 and 62:
44 Chapter II. UndecidabiHty and Re
Page 63 and 64:
Page 65 and 66:
Chapter IIIIndexingFor the remainin
Page 67 and 68:
50 Chapter III. Indexingsynonymous
Page 69 and 70:
52 Chapter III. IndexingTheorem 2
Page 71 and 72:
54 Chapter III. Indexingsuch that f
Page 73 and 74:
56 Chapter III. Indexingsive functi
Page 75 and 76:
Chapter IVGenerative Sets and Creat
Page 77 and 78:
60 Chapter IV. Generative Sets and
Page 79 and 80:
Page 81 and 82:
Page 83 and 84:
Page 85 and 86:
68 Chapter V. Double Generativity a
Page 87 and 88:
Page 89 and 90:
Page 91 and 92: 74 Chapter V. Double Generativity a
Page 99 and 100: Chapter VIUniversal and Doubly Univ
Page 101 and 102: 84 Chapter VI. Universal and Doubly
Page 107 and 108: 90 Chapter VII. Shepherdson Revisit
Page 109 and 110: 92 Chapter VII. Shepherdson Revisit
Page 111 and 112: Chapter VIIIRecursion TheoremsWe ha
Page 113 and 114: 96 Chapter VIII. Recursion Theorems
Page 115 and 116: 98 Chapter VIII. Recursion Theorems
Page 117 and 118: 100 Chapter VIII. Recursion Theorem
Page 123 and 124: 106 Chapter IX. Symmetric and Doubl
Page 137 and 138: Chapter XProductivity and Double Pr
Page 139 and 140: 122 Chapter X. Productivity and Dou
Page 141: 124 Chapter X. Productivity and Dou
Page 145 and 146: Chapter XIThree Special TopicsThe t
Page 147 and 148: 130 Chapter XI. Three Special Topic
Page 149 and 150: 132 Chapter XL Three Special Topics
Page 157 and 158: 140 Chapter XL Three Special Topics
Page 159 and 160: 142 Chapter XII. Uniform Godelizati
Page 175 and 176: This page intentionally left blank
Page 177 and 178: 160 References[14] Raymond M. Smull
Page 179 and 180: 162 Indexdouble universality, 76dou
show all

Double Recursion Theorems

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?