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. Computability Theory, by Y. Fu XIII. Turing Reducibility 1 / 44

Synopsis 1. Relative Computability 2. Turing Reduction 3. Jump Operator 4. Use Principle 5. Modulus Lemma and Limit Lemma Computability Theory, by Y. Fu XIII. Turing Reducibility 2 / 44

1. Relative Computability Computability Theory, by Y. Fu XIII. Turing Reducibility 3 / 44

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 . Computability Theory, by Y. Fu XIII. Turing Reducibility 4 / 44

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. Computability Theory, by Y. Fu XIII. Turing Reducibility 5 / 44

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 r n of R n by O ( r n ). Computability Theory, by Y. Fu XIII. Turing Reducibility 6 / 44

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. Computability Theory, by Y. Fu XIII. Turing Reducibility 7 / 44

Numbering URMO Programs We fix an effective enumeration of all URMO programs P O 0 , P O 1 , P O 2 , . . . . It is important to notice that the G¨ odel number of an oracle machine is independent of any specific oracle function. Computability Theory, by Y. Fu XIII. Turing Reducibility 8 / 44

Notation and Terminology P O e is the e -th URMO. φ O , n is the n -ary function O -computed by P O e . e φ O , 1 is simplified to φ O e . e W O is dom ( φ O e ). e E O e is rng ( φ O e ). C O is the set of all O -computable functions. Computability Theory, by Y. Fu XIII. Turing Reducibility 9 / 44

Relative Computability Fact . (i) O ∈ C O . (ii) C ⊆ C O . (iii) If O is computable, then C = C O . (iv) C O is closed under substitution, recursion and minimalisation. (v) If ψ is a total function that is O -computable, then C ψ ⊆ C O . Computability Theory, by Y. Fu XIII. Turing Reducibility 10 / 44

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 s m n ( e , � x ) such that for each oracle O the following holds: y ) ≃ φ O , n φ O , m + n ( � x , � x ) ( � y ) . e s m n ( e , � Notice that s m n ( e , � x ) does not refer to O . Computability Theory, by Y. Fu XIII. Turing Reducibility 11 / 44

Relative Enumeration Theorem Relative Enumeration Theorem . For each n , the universal function ψ O , n for n -ary O -computable functions given by U ψ O , n x ) ≃ φ O , n ( e , � ( � x ) e U is O -computable. Computability Theory, by Y. Fu XIII. Turing Reducibility 12 / 44

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 φ O , n x ) ≃ φ O , n f ( n ( z ) , z ) ( � n ( z ) ( � x ) . Computability Theory, by Y. Fu XIII. Turing Reducibility 13 / 44

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. Computability Theory, by Y. Fu XIII. Turing Reducibility 14 / 44

O -Recursive Set and O -r.e. Set A is O -recursive if its characteristic function c A is O -computable. A is O -r.e. if its partial characteristic function χ A is O -computable. Computability Theory, by Y. Fu XIII. Turing Reducibility 15 / 44

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 for some e . 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. Computability Theory, by Y. Fu XIII. Turing Reducibility 16 / 44

Computability Relative to Set Computability relative to a set A means computability relative to its characteristic function c A . e for φ c A We write φ A e . We say A -computability rather than c A -computability. We write f ≤ T A to indicate that f is A -computable. Computability Theory, by Y. Fu XIII. Turing Reducibility 17 / 44

2. Turing Reduction Computability Theory, by Y. Fu XIII. Turing Reducibility 18 / 44

Turing Reducibility A set A is Turing reducible to B , or is recursive in B , notation A ≤ T B , if c A ≤ 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 . Computability Theory, by Y. Fu XIII. Turing Reducibility 19 / 44

Turing Completeness An r.e. set C is (Turing) complete if A ≤ T C for every r.e. set A . Computability Theory, by Y. Fu XIII. Turing Reducibility 20 / 44

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 . Computability Theory, by Y. Fu XIII. Turing Reducibility 21 / 44

Turing Degree, or Degree of Unsolvability The equivalence class d T ( 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. Computability Theory, by Y. Fu XIII. Turing Reducibility 22 / 44

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 . Computability Theory, by Y. Fu XIII. Turing Reducibility 23 / 44

Turing Degree Theorem . Every pair a , b have a unique least upper bound. Computability Theory, by Y. Fu XIII. Turing Reducibility 24 / 44

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. Computability Theory, by Y. Fu XIII. Turing Reducibility 25 / 44

Post’s Question In his 1944 paper, Post raised the following question: ∃ a . 0 < a < 0 ′ ? Computability Theory, by Y. Fu XIII. Turing Reducibility 26 / 44

3. Jump Operator Computability Theory, by Y. Fu XIII. Turing Reducibility 27 / 44

Relative Recursive Enumerability A set A is recursively enumerable in B if χ A ≤ 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 . Computability Theory, by Y. Fu XIII. Turing Reducibility 28 / 44

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 ) ) ′ . = Computability Theory, by Y. Fu XIII. Turing Reducibility 29 / 44

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 φ A x ( x ). If φ A x ( x ) ↓ then output 1. (ii) Using the Relative S-m-n Theorem one constructs an injective primitive recursive function s ( x ) such that � y , if x ∈ A ( or x / ∈ A ); φ A s ( x ) ( y ) = (1) ↑ , otherwise . 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. Computability Theory, by Y. Fu XIII. Turing Reducibility 30 / 44