XIII. Turing Reducibility Yuxi Fu BASICS, Shanghai Jiao Tong - - PowerPoint PPT Presentation

• XIII. Turing Reducibility

Yuxi Fu

BASICS, Shanghai Jiao Tong University

The problem with m-reduction is that it imposes too strong a restriction on the use of a result obtained by revoking a subroutine.

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Synopsis

• 1. Relative Computability
• 2. Turing Reduction
• 3. Jump Operator
• 4. Use Principle
• 5. Modulus Lemma and Limit Lemma

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• 1. Relative Computability

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Computation with Oracle

Suppose O is a total unary function. Informally a function f is computable relative to O, or O-computable, if f can be computed by an algorithm that is effective in the usual sense, except from time to time during computation f is allowed to consult the oracle function O. If f is computable in O, the degree of undecidability of f is no more than that of O.

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Partial Recursive Function with Oracle

Formally an O-partial recursive function f is constructed from the initial functions and O by substitution, primitive recursion and minimization.

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URM with Oracle

A URM with Oracle, URMO for short, can recognize a fifth kind of instruction, O(n), for every n ≥ 1. If O is the oracle function, then the effect of O(n) is to replace the content rn of Rn by O(rn).

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Turing Machine with Oracle

A Turing Machine with Oracle, TMO for short, has an additional read only oracle tape. If O is the oracle function, then the oracle tape is preloaded with the string of 0’s and 1’s that represents O. In the above definition it is convenient to restrict to those oracles that are characteristic functions.

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Numbering URMO Programs

We fix an effective enumeration of all URMO programs PO

0 , PO 1 , PO 2 , . . . .

It is important to notice that the G¨

• del number of an oracle

machine is independent of any specific oracle function.

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Notation and Terminology

PO

e is the e-th URMO.

φO,n

e

is the n-ary function O-computed by PO

e .

φO,1

e

is simplified to φO

e .

W O

e

is dom(φO

e ).

E O

e is rng(φO e ).

CO is the set of all O-computable functions.

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Relative Computability

Fact. (i) O ∈ CO. (ii) C ⊆ CO. (iii) If O is computable, then C = CO. (iv) CO is closed under substitution, recursion and minimalisation. (v) If ψ is a total function that is O-computable, then Cψ ⊆ CO.

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Relative S-m-n Theorem

Relative S-m-n Theorem. For all m, n ≥ 1 there is an injective primitive recursive (m+1)-ary function sm

n (e,

x) such that for each

• racle O the following holds:

φO,m+n

e

( x, y) ≃ φO,n

sm

n (e,

x)(

y). Notice that sm

n (e,

x) does not refer to O.

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Relative Enumeration Theorem

Relative Enumeration Theorem. For each n, the universal function ψO,n

U

for n-ary O-computable functions given by ψO,n

U

(e, x) ≃ φO,n

e

( x) is O-computable.

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Relative Recursion Theorem

Relative Recursion Theorem. Suppose f (y, z) is a total O-computable function. There is a primitive recursive function n(z) such that for all z φO,n

f (n(z),z)(

x) ≃ φO,n

n(z)(

x).

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Relative Theory

Once we have the three foundational theorems, we can do the recursion theory relative to an oracle function. A proof of a proposition relativizes if essentially it is also a proof of the relativized proposition.

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O-Recursive Set and O-r.e. Set

A is O-recursive if its characteristic function cA is O-computable. A is O-r.e. if its partial characteristic function χA is O-computable.

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O-Recursive Set and O-r.e. Set

• Theorem. The following hold.

(i) A is O-recursive iff A and A are O-r.e. (ii) The following are equivalent.

◮ A is O-r.e. ◮ A = W O e

for some e.

◮ A = E O e for some e. ◮ A = ∅ or A is the range of a total O-computable function. ◮ For some O-decidable predicate R(x, y), x ∈ A iff ∃y.R(x, y).

(iii) K O def = {x | x ∈ W O

x } is O-r.e. but not O-recursive.

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Computability Relative to Set

Computability relative to a set A means computability relative to its characteristic function cA. We write φA

e for φcA e .

We say A-computability rather than cA-computability. We write f ≤T A to indicate that f is A-computable.

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• 2. Turing Reduction

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Turing Reducibility

A set A is Turing reducible to B, or is recursive in B, notation A ≤T B, if cA ≤T B. The sets A, B are Turing equivalent, notation A ≡T B, if A ≤T B and B ≤T A. A <T B if A ≤T B and B ≤T A.

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Turing Completeness

An r.e. set C is (Turing) complete if A ≤T C for every r.e. set A.

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Property of Turing Reducibility

Fact. (i) ≤T is reflexive and transitive. (ii) ≡T is an equivalence relation. (iii) If A ≤m B then A ≤T B. (iv) A ≡T A for all A. (v) If A is recursive, then A ≤T B for all B. (vi) If B is recursive and A ≤T B, then A is recursive. (vii) If A is r.e. then A ≤T K.

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Turing Degree, or Degree of Unsolvability

The equivalence class dT(A) = {B | B ≡T A} is called the (Turing) degree of A. Let D be the set of all Turing degrees. D is an upper semi-lattice.

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Turing Degree

The set of Turing degrees is ranged over by a, b, c, . . .. a ≤ b iff A ≤T B for some A ∈ a and B ∈ b. a < b iff a ≤ b and a = b.

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Turing Degree

• Theorem. Every pair a, b have a unique least upper bound.

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Recursive Degree and Recursively Enumerable Degree

A degree containing a recursive set is called a recursive degree. A degree containing an r.e. set is called an r.e. degree. Theorem. (i) There is precisely one recursive degree 0, which consists of all the recursive sets and is the unique minimal degree. (ii) Let 0′ be the degree of K. Then 0 < 0′ and 0′ is the maximum of all r.e. degrees.

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Post’s Question

In his 1944 paper, Post raised the following question: ∃a. 0 < a < 0′ ?

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• 3. Jump Operator

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Relative Recursive Enumerability

A set A is recursively enumerable in B if χA ≤T B.

• Lemma. A is r.e. in B iff A is r.e. in B.
• Lemma. A ≤T B iff both A and A are r.e. in B.
• Lemma. Suppose B is recursively enumerable in C. If C ≤T D,

then B is recursively enumerable in D. We say that a is recursively enumerable in b if some A ∈ a is recursively enumerable in some B ∈ b.

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Jump Operator

The jump K A of A, notation A′, is defined by A′ = {x | x ∈ W A

x }.

The n-th jump: A(0) = A, A(n+1) = (A(n))′.

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Jump Theorem. The following hold: (i) A′ is r.e. in A. (ii) A ≤T A′ ≤T A. (in fact A, A ≤1 A′)

Proof.

(i) Given x calculate φA

x (x). If φA x (x) ↓ then output 1.

(ii) Using the Relative S-m-n Theorem one constructs an injective primitive recursive function s(x) such that φA

s(x)(y) =

y, if x ∈ A(or x / ∈ A); ↑,

• therwise.

(1) Clearly x ∈ A iff s(x) ∈ A′. Hence A, A ≤1 A′. Had A′ ≤T A, one would be able to construct an A-recursive function that is different from any A-recursive function, which is a contradiction.

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Jump Theorem. The following hold: (iii) A is r.e. in B iff A ≤1 B′. (iv) A ≤T B iff A′ ≤1 B′. Consequently A ≡T B iff A′ ≡1 B′.

Proof.

(iii) Suppose A is r.e. in B. Using the Relative S-m-n Theorem,

• ne gets an injective recursive function s(x) such that

φB

s(x)(y) ≃ if x ∈ A then y else ↑ .

Clearly x ∈ A iff s(x) ∈ B′. Hence A ≤1 B′. Conversely if d : A ≤1 B′ then χA(x) can be B-computed by “if φB

d(x)(d(x)) ↓ then 1 else ↑”.

(iv) This follows from (i,ii,iii) immediately.

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Beyond R.E. Degree

The jump of a, notation a′, is defined by dT(A′) for some A ∈ a. By definition 0′ is the jump of 0. Hence the infinite hierarchy 0 < 0′ < 0′′ < 0′′′ < . . . < 0(n) < . . . . Notice that 0 = dT(∅) and 0(n) = dT(∅(n)).

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• 4. Use Principle

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String as Subset

A finite string σ of 0’s and 1’s can be seen as an initial segment of a characteristic function. We write σ ⊂ A if σ ⊂ cA when both are treated as graphs. Let |σ| denote the length of σ. Let A↾x be {y ∈ A | y < x}. Similarly one defines σ↾x.

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Use Function

We write φA

e,s(x) = y if ◮ e, x, y < s; ◮ PA s (x) outputs y in < s steps; ◮ only numbers < s are tested for membership of A.

The use function u(A; e, x, s) is “1 + the maximum number tested for membership of A during the computation of φA

e,s(x)” if

φA

e,s(x) ↓; and u(A; e, x, s) is 0 if φA e,s(x) ↑.

The use function u(A; e, x) is u(A; e, x, s) for some s such that u(A; e, x, s) ↓.

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Use Function

φσ

e,s(x) and φA↾u e,s (x) are defined accordingly.

φσ

e (x) = y if ∃s.φσ e,s(x) = y.

We shall also use notations like W A

e,s, W σ e,s, W σ e .

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Use Principle

• Theorem. The following hold:

(i) φA

e (x) = y implies ∃s.∃σ ⊂ A.φσ e,s(x) = y.

(ii) φσ

e,s(x) = y implies ∀t ≥ s.∀τ ⊇ σ.φτ e,t(x) = y.

(iii) φσ

e (x) = y implies ∀A ⊇ σ.φA e (x) = y.

The Use Principle implies the following (φA

e,s(x) = y ∧ A↾u = B↾u) ⇒ φB e,s(x) = y,

where u = u(A; e, x, s).

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• 5. Modulus Lemma and Limit Lemma

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Degrees ≤T 0′

We are mainly interested in degrees ≤T 0′, and particularly in the r.e. degrees. We provide some alternative characterizations of such degrees.

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Modulus of Convergence

• 1. A sequence {fs(x)}s∈ω of total functions is recursive if there is a

recursive function f (s, x) such that fs(x) = f (s, x) for all s, x.

• 2. The sequence {fs(x)}s∈ω converges pointwise to f (x), notation

f = lims fs, if for each x, fs(x) = f (x) for all but finitely many s.

• 3. A modulus of convergence for the sequence {fs(x)}s∈ω is a

function m(x) such that fs(x) = f (x) for all s ≥ m(x).

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Modulus Lemma. Suppose A is r.e. and f ≤T A. Then there are (1) a recursive sequence {fs}s∈ω such that (2) f = lims fs and (3) a modulus m of {fs}s∈ω that is recursive in A.

Proof.

Suppose A is r.e. and f = φA

e . Let As = Wi,s for some Wi = A.

Define a converging family {fs}sω by fe(x) = φAs

e,s(x),

if φAs

e,s(x) ↓,

0,

• therwise.

Clearly {fs}s∈ω is a recursive sequence. Define m by m(x) = µs.∃z ≤ s.(φAs↾z

e,s (x) ↓ ∧ As↾z = A↾z).

Now m is A-recursive, and by Use Principle is a modulus since φAs↾z

e,s (x) = φA↾z e,s (x) = φA e (x) = f (x) for s ≥ m(x).

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Limit Lemma. f ≤T A′ iff there is an A-recursive sequence {fs}s∈ω such that f = lims fs.

Proof.

Suppose f ≤T A′. Since A′ is r.e. in A, the A-recursive sequence {fs}s∈ω exists by Relative Modulus Lemma. Suppose f = lims fs for an A-recursive sequence {fs}s∈ω. Define Ax = {s | ∃t.(s ≤ t ∧ ft(x) = ft+1(x))}, which is finite. Now m(x) = µs.(s / ∈ Ax) is Turing equivalent to B = {s, x | s ∈ Ax}, which is r.e. in A. Hence m ≡T B ≤T A′. Conclude that f ≤T A′ since f (x) = fm(x)(x).

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Constructing Degrees below 0′

• Corollary. A function f has degree ≤ 0′ (meaning f ≤T ∅′) iff

f = lims fs for some recursive sequence {fs}s∈ω.

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Constructing R.E. Degrees

• Corollary. A function f has r.e. degree iff f is the limit of a

recursive sequence {fs}s∈ω that has a modulus m ≤T f .

Proof.

If f ≡T A for some r.e. set A, then by Modulus Lemma m ≤T A ≡T f . Suppose f = lims fs with modulus m ≤T f . As in the proof of the Limit Lemma, f ≤T m.

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