The Turing Definability of the Relation of Computably Enumerable In. S. Barry Cooper

Transcription

1 The Turing Definability of the Relation of Computably Enumerable In S. Barry Cooper Computability Theory Seminar University of Leeds Winter,

2 1. The big picture Turing definability/invariance Computability theory Church s thesis Mathematical information content Empirically observable phenomena Scientific theory Scientific questions e.g. in quantum theory Mathematical foundations Humanities v. Sciences - epistemological relativism

3 2. Basics Corresponding to the i th Turing machine, Φ i denotes the ith partial computable (p.c.) functional 2 ω 2 ω. A is Turing reducible to a B (A T B ) iff A = Φ B i, some i ω and A, B are Turing equivalent (A T B ) iff A T B and B T A. The degree of unsolvability or Turing degree of A is defined by deg (A) = {X 2 ω A T X}. is the induced partial ordering on D (= the set of all degrees), 0 = the least degree (consisting of all computable sets of numbers), and D is the structure D,. Wi A = dom Φ A i denotes the ith computably enumerable in A (A-c.e.) set (W i = Wi being the ith c.e. set). The jump or n + 1th jump of a set A is defined by A = A (1) = {x x W A x } or, A (n+1) = (A (n) ) ), respectively.

4 The jump operator on degrees is defined by a = deg (A ), A a, where a < a, and a is the l.u.b. of the degrees of sets c.e. in A a. And write a (n+1) = deg (A (n+1) ) = (a (n) ). Define the standard ω -jump of a by a (ω) = deg ( n ω A (n) ), A a. Write D for the structure D,,. A relation on D is Turing definable iff it is describable in the first order theory of D. Will assume standard computable sequences {Φ i,s } s 0, {W A i,s } s 0 of finite approximations to the p.c. functionals and c.e. sets, respectively. Denote by A[s], or A s, the corresponding approximation to an expression A at a stage s. The restriction ψ x of a function ψ is taken to be its restriction to arguments x. And if Φ is some functional and Φ A (x), the use ϕ A (x) of Φ A (x) will be taken to be µz[φ A z (x) ].

5 0 D 0 High n E Low n Turing definable 0... and Turing invariant Naturally arising information content in the Turing universe

6 3. Pseudo-jump operators Definition 3.1: Say J n is an n-cea operator iff there exist j 0, j 1,..., j n 1 ω such that J k (A) T W J k (A) j k, each k < n, A ω, and J n is inductively defined by J 0 (A) = A, J k+1 (A) = W J k (A) j k, (k < n). If D = W i W j, some i, j 0, say D is a d-c.e. set (a difference of two c.e. sets). Lemma 3.2: If D = W i W j then A (W A i W A j is a d-c.e. set ) is a 2-CEA operator. Proof (Lachlan; Jockusch and Shore [1984]). Make a special choice of the indices j 0, j 1 in definition 3.1 in relation to i, j

7 Choose j 0, j 1 so that for each set X of numbers W X j 0 = X [{ x, s x (W X i,s+1 W X i,s)} { x, s + 1 x (W X i,s+1 W X i,s) W X j }], W X Y j 1 = X {x s [ x, s Y & x, s ± 1 / Y ]}, and define the operator J 2 by J 2 (A) = W J 1 (A) j 1, where J 1 (A) = W A j 0. Need to verify: (a) that J 2 (A) = A (Wi A Wj A) and (b) that J 2 is a 2-CEA operator. For (a): Notice that (J 2 (A)) 0 = (W J 1 (A) j 1 ) 0 = (J 1 (A)) 0 = (Wj A 0 ) 0 = A and

8 x (J 2 (A)) 1 x (W J 1 (A) j 1 ) 1 s [ x, s (W A j 0 ) 1 & x, s ± 1 / (W A j 0 ) 1 ] x (W A i W A j ), giving J 2 (A) = A (W A i W A j ). And for (b): It follows straight from the definition that X T W X j 0, so J 1 is a 1-CEA operator. To show that J 1 (A) T W J 1 (A) j 1 need to check that Wj A 0 T J 2 (A) = A (Wi A Wj A). But x (W A j 0 ) 0 x A and x, s (W A j 0 ) 1 x (W A i,s+1 W A i,s) x (W A i,s W A i,s 1) W A j x (W A i,s+1 W A i,s) [x (W A i,s W A i,s 1) & x / W A i W A j ], so Wj A 0 T A (Wi A Wj A), and J 2 is a 2-CEA operator, as required.

9 4. Jump inversion String notation: σ, τ etc. denote finite binary strings (i.e. 0-1 valued functions with finite ordinal domains). σ = the length of σ. σˆτ denotes the concatenation of σ, τ (= σ followed by τ ). Write σ τ iff τ is an extension of σ, σ A iff σ is a beginning of (the characteristic function of) A. σ, τ are compatible (σ τ ) iff σ τ or τ σ otherwise write σ τ. Write = the empty string, and S = the set of all strings. T : S S is a tree iff τ σ S : (i) T(σ) T(τ) T(σ), and, for i 1, (ii) T(τˆi) T(τˆ(1 i)) T(τˆi).

10 Jump Inversion Theorem for 2- CEA Operators: If J 2 is a 2-CEA operator, then for each C T there is a set A such that C T J 2 (A). Note: Since pseudo-jumps are not necessarily degree theoretic, cannot just iterate the jump inversion theorem for 1-CEA operators. Proof. Let J 2 (X) = W W X j i = W i (Wj X ) define the 2-CEA operator J 2 from indices i, j. Need to construct a set A such that W i (W A j ) T A T C. Define an increasing sequence {σ n } n 0 of strings chosen off a tree T and take (the characteristic function of) A = n 0 σ n.

11 Aim: (i) Construct T so that, for each string τ with τ > n, T(τ) decides whether n Wj σ for any σ T(τ), and then (ii) Choose the σ s A on T to code C into A with help from and to decide whether n J 2 (A). Definition of T : Define T( ) =. Assume T(τ) with τ = n 0 (that is, with T(τ) at level n on T ). Ask if σ T(τ) with n W σ j, σ. Then define T(τˆ0), (τˆ1) = σˆ0, σˆ1, respectively, where σ is the first such σ (in some standard listing of strings) if such a σ exists and otherwise σ is the first σ T(τ). Notice that T T.

12 The construction of {σ n } n 0 Stage 0. Define σ 0 = T( ). Stage 2n+1. If there exists a string σ σ 2n at some level x + 1, say, on T with n W i (W σ j x)[ σ ], let σ 2n+1 be the first such σ. Otherwise let σ 2n+1 = σ 2n. Stage 2n + 2. Define σ 2n+2 = the first T(τˆC(n)) σ 2n+1. Now observe the following sequence of facts (1) {σ n } n 0 T C. This holds because stages 2n + 1, n 0, can be carried out computably in T T T C, and stages 2n, n 0, can be carried out computably in T, C T C. Hence: (2) A T C ; and

13 (3) W i (W A j ) T C Since, for each n 0, Also: n W i (W A j ) σ 2n+1 σ 2n. (4) C T {σ n } n 0 Since, if one writes σ n = T(τ n ), each n ω, one has C(n) = τ 2n+2 ( τ 2n+2 1) each n 0, = σ 2n+2 ( σ 2n+2 1), (5) {σ n } n 0 T A Since stage 2n + 1 can be carried out computably in, and stage 2n + 2 can be carried out computably in T T and A, and (6) {σ n } n 0 T W i (W A j ). To verify this, one first notices that since J 2 is a 2-CEA operator one has A T Wj A and Wj A T W i (Wj A ). Then

14 To carry out stage 2n + 1 one can compute { σ, x σ 2n σ A & W A j x = W σ j, σ x} with the use of A and Wj A {T(τ) A τ τ 2n }., and hence also Can then compute σ 2n+1 with help from W i (W A j ). Similarly, one can carry out stage 2n of the construction using A and W A j. So from (5) and (6) one gets: (7) C T W i (W A j ) and A. Combining (2), (3) and (7) the theorem follows. Note: Only need the analogue of Friedberg s theorem for a 2-CEA operator derived as in Lemma 3.2 from a d-c.e. set. This is a main ingredient of

15 5. A jump and join theorem The basic jump-join theorem for 2-CEA operators derived from a d-c.e. set: If J 2 is a 2-CEA operator derived from a d-c.e. set, then if C T X and X T, one can find an A such that X A T C T J 2 (A). Proof. An extension of the Posner-Robinson [1981] cupping theorem. Choose i, j s.t. J 2 (X) = W W X j i = W i (W X j ). From the proof of Lemma 3.2, can assume that W i, W j are given by equations of the form W X j =X [{ x, s x (W X i,s+1 W X i,s)} { x, s + 1 x (W X i,s+1 W X i,s) W X j }], W X Y i =X {x s [ x, s Y & x, s ± 1 / Y ]}, where J 2 (A) = A (W A i W A j ).

16 Without changing the degree of X, can assume that X is -immune (i.e., has no infinite -c.e. subsets). Wish to construct a set A satisfying the picture: A X T C T J 2 (A) T W i (W A j ) X J 2 X A As before, define A = n 0 σ n, where the σ s A are chosen to Code C into A with help from X,

17 To force certain x, s W σ j, σ σ n, And to ensure that, for each n ω, n J 2 (A) n W i (W σ j M)[ σ ], σ corresponding to n, and M depending on the construction. Note: One cannot choose {σ n } n 0 off a tree T as in the jump inversion theorem above since then one would not be able to obtain T both from A X and from J 2 (A). Instead, one chooses σ n off a tree T n,π specifically adapted for stage n + 1 of the construction and used in such a way that the construction is retrievable from A X and from J 2 (A). T n,π is constructed so that, for each T n,π (τ) at level s on T n,π, T n,π (τ) decides whether n, s Wj σ for any σ T n,π (τ).

18 Definition of T n,π : Let T n,π ( ) = σ 2nˆπ. Assume T n,π (τ) with τ = s 0 i.e., with T n,π (τ) at level s on T n,π. Ask if there exists a σ T n,π (τ) with n, s + 1 W σ j, σ. Then define T n,π (τˆ0), (τˆ1) = σˆ0, σˆ1, respectively, where σ is the first such σ if such a σ exists and otherwise σ is the first σ T n,π (τ). Notice that T n,π T. The construction of {σ n } n 0 : Stage n = 0. Define σ 0 =.

19 Stage 2n + 1. Let 0 m denote a string of m zeros. Define T n,π [τ] = the full subtree of T n,π above T n,π (τ). Say T n,π [τ] forces n J 2 (A) at level s + 1 iff n, s W T n,π(τ) j and n, s ± 1 / W T n,π(τ) j. T n,π [τ] forces n / J 2 (A) iff for no s ω, and no τ τ, does T n,π [τ ] force n J 2 (A) at level s + 1. At stage 2n + 1, choose the least m = n, s, some s ω, such that either (a) m / X and there is some T n,0mˆ1[τ] which forces n J 2 (A) at some level s + 1, or (b) m X and T n,0mˆ1[ ] forces n / J 2 (A). Define σ 2n+1 = T n,0mˆ1[τ 2n+1 ] where τ 2n+1 = the least string τ for which (a) holds, if appropriate and otherwise =.

20 Note: In case (a), σ 2n+1 T n,0mˆ1(τ 2n+1 ) will give n J 2 (A) by virtue of n, s W σ 2n+1 j and n, s ± 1 / W σ 2n+1 j giving n, s ± 1 / Wj σ, any σ σ 2n+1. But From case (b) may get n, s W σ p+1 some p 2n but n, s ± 1 / W σ p+1 j so n J 2 (A) again. j,, any p And Cannot in case (b) treat the cases: (i) n, s W σ j, some σ σ 2n, (ii) n, s / W σ j, any σ σ 2n, differently to get n J 2 (A) n J 2 (σ 2n+1 ), since could not then retrieve σ 2n+1 from J 2 (A). Consequently Following case (b), the forcing of n / J 2 (A) is current at all stages 2p + 2 > 2n + 1 for which there is no existing n, s W σ 2p j,p, with s p.

21 Stage 2n + 2. Check if there is a current forcing of any p / J 2 (A), for which p, s W σ 2n+1 j,n, some s n and which now ceases to be active Choose the least π σ 2n+1 for which some p, s ± 1 Wj, π π, each such p And define σ 2n+2 = πˆc(n). Notice It now follows that in case (b) of stage 2n + 1 in which such a τ does not exist one has n / J 2 (A) by virtue of n / J 2 (σ), all σ σ 2p+2, some p n. One needs to check that the construction can be carried out which means verifying that m exists in the definition of σ 2n+1.

22 To see this first notice that S is Σ 2 and so c.e. in where S is defined by S = {m ( τ)( s)[t n,0mˆ1[τ] forces n J 2 (A) at level s + 1]}. If S is finite then there is some m X S so m X and ( τ)( s)[t n,0mˆ1[τ] does not force n J 2 (A) at level s + 1] Giving m X and T n,0mˆ1[ ] forces n / J 2 (A) which is (b). And if S is infinite there is some m S X, since X is -immune. So for this m one has m / X and ( τ)( s)[t n,0mˆ1[τ] forces n J 2 (A) at level s + 1] giving (a).

23 Now verify the following sequence of facts (1) {σ p } p 0 T C. This is because stage 2n + 1 can be carried out computably in X T C And stage 2n + 2 is executed computably, apart from the coding of C(n) into σ 2n+2. Hence: (2) A X T C ; and (3) W i (W A j ) T C Since for each n 0, m in stage 2n + 1 is retrievable from σ 2n and A and n W i (W A j ) σ 2n+1 > σ 2nˆ0 mˆ1. To see this, first notice that

24 σ 2n+1 > σ 2nˆ0 mˆ1 m / X & ( τ, s)[t n,0mˆ1[τ] forces n J 2 (A) at level s + 1] m / X & ( s)[ n, s W j (T n,0mˆ1(τ 2n+1 )) & ( τ τ 2n+1 )( n, s ± 1 / W j (T n,0mˆ1(τ )))] m / X & ( Y T n,0mˆ1(τ 2n+1 ))(n W i (W Y j )) n W i (W A j ). Conversely, if n W i (Wj A ) there exists a y such that n, y Wj A and n, y ± 1 / Wj A. This means that case (b) cannot apply at stage 2n + 1 since then the forcing of n / J 2 (A) would cease to be active at some stage 2p + 2 > 2n + 1 giving n, y ± 1 W A j But this means that at stage 2n + 1 one chooses τ 2n+1 so that T n,0mˆ1[τ 2n+1 ] forces n J 2 (A) at level y + 1 so σ 2n+1 > σ 2nˆ0 mˆ1, as required. Also

25 (4) C T {σ p } p 0 Since one has C(n) = σ 2n+2 ( σ 2n+2 1), each n 0. (5) {σ p } p 0 T A X. To reproduce stage 2n + 1 of the construction, one can use σ 2n and A to find m And then check whether m X or not to see which of cases (a) or (b) apply at stage 2n + 1. If m X, so (b) applies, one has τ 2n+1 =. So T n,0mˆ1(τ 2n+1 ) = σ 2nˆ0 mˆ1. If m / X so (a) applies one can find T n,0mˆ1(τ 2n+1 ) σ 2nˆ0 mˆ1 with help from A. And then σ 2n+1 = T n,0mˆ1(τ 2n+1 ). And to reproduce stage 2n + 2 of the construction, one can computably obtain π And get σ 2n+1 = πˆa( π ).

26 (6) {σ p } p 0 T W i (W A j ). To verify this, first notice that since J 2 is a 2-CEA operator one has A T W A j and W A j T W i (W A j ). To carry out stage 2n + 1 one can find m from σ 2n and A verify whether n W i (W A j ) so as to see if one is in case (a) or (b) and find T n,0mˆ1(τ 2n+1 ), and hence σ 2n+1, with help from A and W A j. While one can carry out stage 2n + 2 computably as far as obtaining π and then σ 2n+2 = πˆa( π ). So from (5) and (6) one gets: (7) C T W i (W A j ) and A X. Combining (2), (3) and (7) the theorem follows. Note: The basic jump-join theorem is sufficient for a natural Turing definition of the jump. But a local version is needed to provide that of the relation of computably enumerable in, which applies to 2-CEA operators derived from special d.c.e. sets