X. Creative Set Yuxi Fu BASICS, Shanghai Jiao Tong University - PowerPoint PPT Presentation

X. Creative Set Yuxi Fu BASICS, Shanghai Jiao Tong University

Quotation from Post The terminology ‘creative set’ was introduced by E. Post in Recursively Enumerable Sets of Positive Integers and their Decision Problems. Bulletin of American Mathematical Society , 1944. “. . . every symbolic logic is incomplete and extensible relative to the class of propositions”. “The conclusion is inescapable that even for such fixed, well-defined body of mathematical propositions, mathematical thinking is, and must remain, essentially creative.” Computability Theory, by Y. Fu X. Creative Set 1 / 30

What are the Most Difficult Semi-Decidable Problems? We know that K is the most difficult semi-decidable problem. What is then the m-degree d m ( K )? What is an r.e. set C s.t. A ≤ m C for every r.e. set A ? Computability Theory, by Y. Fu X. Creative Set 2 / 30

What are the Most Difficult Semi-Decidable Problems? An r.e. set is very difficult if it is very non-recursive. An r.e. set is very non-recursive if its complement is very non-r.e.. A set is very non-r.e. if it is easy to distinguish it from any r.e. set. These sets are creative respectively productive. Computability Theory, by Y. Fu X. Creative Set 3 / 30

Synopsis 1. Productive Set 2. Creative Set 3. The Lattice of m-Degrees Computability Theory, by Y. Fu X. Creative Set 4 / 30

1. Productive Set Computability Theory, by Y. Fu X. Creative Set 5 / 30

Suppose W x ⊆ K . Then x ∈ K \ W x . So x witnesses the strict inclusion W x � K . In other words the identity function is an effective proof that K differs from every r.e. set. Computability Theory, by Y. Fu X. Creative Set 6 / 30

Productive Set A set A is productive if there is a total computable function p such that whenever W x ⊆ A , then p ( x ) ∈ A \ W x . The function p is called a productive function for A . A productive set is not r.e. by definition. Computability Theory, by Y. Fu X. Creative Set 7 / 30

Example 1. K is productive. 2. { x | c / ∈ W x } is productive. 3. { x | c / ∈ E x } is productive. 4. { x | φ x ( x ) � = 0 } is productive. Computability Theory, by Y. Fu X. Creative Set 8 / 30

Example Suppose A = { x | φ x ( x ) � = 0 } . By S-m-n Theorem one gets a primitive recursive function p ( x ) such that φ p ( x ) ( y ) = 0 if and only if φ x ( y ) is defined. Then p ( x ) ∈ W x ⇔ p ( x ) / ∈ A . So if W x ⊆ A we must have p ( x ) ∈ A \ W x . Thus p is a productive function for A . Computability Theory, by Y. Fu X. Creative Set 9 / 30

Productive Set Lemma . If A ≤ m B and A is productive, then B is productive. Proof. Suppose r : A ≤ m B and p is a production function for A . By applying S-m-n Theorem to φ x ( r ( y )), one gets a primitive recursive function k ( x ) such that W k ( x ) = r − 1 ( W x ). Then rpk is a production function for B . Computability Theory, by Y. Fu X. Creative Set 10 / 30

Productive Set Theorem . Suppose that B is a set of unary computable functions with f ∅ ∈ B and B � = C 1 . Then B = { x | φ x ∈ B} is productive. Proof. Suppose g / ∈ B . Consider the function f defined by � g ( y ) , if x ∈ W x , f ( x , y ) ≃ ↑ , if x / ∈ W x . By S-m-n Theorem there is a primitive recursive function k ( x ) such that φ k ( x ) ( y ) ≃ f ( x , y ). ∈ W x iff φ k ( x ) = f ∅ iff φ k ( x ) ∈ B iff k ( x ) ∈ B . Clearly x / Hence k : K ≤ m B . Computability Theory, by Y. Fu X. Creative Set 11 / 30

Property of Productive Set Lemma . Suppose that g is a total computable function. Then there is a primitive recursive function p such that for all x , W p ( x ) = W x ∪ { g ( x ) } . Proof. Using S-m-n Theorem, take p ( x ) to be a primitive recursive function such that � 1 , if y ∈ W x ∨ y = g ( x ) , φ p ( x ) ( y ) ≃ ↑ , otherwise . We are done. Computability Theory, by Y. Fu X. Creative Set 12 / 30

Property of Productive Set Theorem . A productive set contains an infinite r.e. subset. Proof. Suppose p is a production function for A . Take e 0 to be some index for ∅ . Then p ( e 0 ) ∈ A by definition. By the Lemma there is a primitive recursive function k such that for all x , W k ( x ) = W x ∪ { p ( x ) } . Apparently { e 0 , . . . , k n ( e 0 ) , . . . } is r.e. Consequently { p ( e 0 ) , . . . , p ( k n ( e 0 )) , . . . } is a r.e. subset of A , which must be infinite by the definition of k . Computability Theory, by Y. Fu X. Creative Set 13 / 30

Productive Function via a Partial Function Proposition . A set A is productive iff there is a partial recursive function p such that ∀ x . ( W x ⊆ A ⇒ ( p ( x ) ↓ ∧ p ( x ) ∈ A \ W x )) . (1) Proof. Suppose p is a partial recursive function satisfying (1). Let s be a primitive recursive function such that � y , p ( x ) ↓ ∧ y ∈ W x , φ s ( x ) ( y ) ≃ ↑ , otherwise . A productive function q can be defined by running p ( x ) and p ( s ( x )) in parallel and stops when either terminates. Computability Theory, by Y. Fu X. Creative Set 14 / 30

Productive Function Made Injective Proposition . A productive set has an injective productive function. Proof. Suppose p is a productive function of A . Let W h ( x ) = W x ∪ { p ( x ) } . Clearly W x ⊆ A ⇒ W h ( x ) ⊆ A . (2) Define q (0) = p (0). ◮ If p ( x +1) , ph ( x +1) , . . . , ph x +1 ( x +1) are pairwise distinct, let q ( x +1) be the smallest one not in { q (0) , . . . , q ( x ) } . ◮ Otherwise we can let q ( x +1) be µ y . y / ∈ { q (0) , . . . , q ( x ) } . This is fine since W x �⊆ A due to (2). It is easily seen that q is an injective production function for A . Computability Theory, by Y. Fu X. Creative Set 15 / 30

Myhill’s Characterization of Productive Set Theorem . (Myhill, 1955) A is productive iff K ≤ 1 A iff K ≤ m A . K ≤ 1 A implies K ≤ m A , which in turn implies “ A is productive”. Computability Theory, by Y. Fu X. Creative Set 16 / 30

Proof Suppose p is a productive function for A . Define � 0 , if z = p ( x ) and y ∈ K , f ( x , y , z ) ≃ ↑ , otherwise . By S-m-n Theorem there is an injective primitive recursive function s ( x , y ) such that φ s ( x , y ) ( z ) ≃ f ( x , y , z ) . By definition, � { p ( x ) } , if y ∈ K , W s ( x , y ) = ∅ , otherwise . Computability Theory, by Y. Fu X. Creative Set 17 / 30

Proof By Recursion Theorem there is an injective primitive recursive function n ( y ) such that W s ( n ( y ) , y ) = W n ( y ) for all y . So � { p ( n ( y )) } , if y ∈ K , W n ( y ) = ∅ , otherwise . We claim that K ≤ m A . y ∈ K ⇒ W n ( y ) = { p ( n ( y )) } ⇒ p ( n ( y )) / ∈ A . y / ∈ K ⇒ W n ( y ) = ∅ ⇒ p ( n ( y )) ∈ A . By the previous theorem we may assume that p is injective. So the reduction function p ( n ( )) is injective. Conclude K ≤ 1 A . Computability Theory, by Y. Fu X. Creative Set 18 / 30

2. Creative Set Computability Theory, by Y. Fu X. Creative Set 19 / 30

Creative Set A set A is creative if it is r.e. and its complement A is productive. Intuitively a creative set A is effectively non-recursive in the sense that the non-recursiveness of A , hence the non-recursiveness of A , can be effectively demonstrated. Computability Theory, by Y. Fu X. Creative Set 20 / 30

Creative Set 1. K is creative. 2. { x | c ∈ W x } is creative. 3. { x | c ∈ E x } is creative. 4. { x | φ x ( x ) = 0 } is creative. Computability Theory, by Y. Fu X. Creative Set 21 / 30

Creative Set Theorem . Suppose that A ⊆ C 1 and let A = { x | φ x ∈ A} . If A is r.e. and A � = ∅ , N , then A is creative. Proof. Suppose A is r.e. and A � = ∅ , N . If f ∅ ∈ A , then A is productive by a previous theorem. This is a contradiction. So A is productive by the same theorem. Hence A is creative. Computability Theory, by Y. Fu X. Creative Set 22 / 30

Creative Set The set K 0 = { x | W x � = ∅} is creative. It corresponds to the set A = { f ∈ C 1 | f � = f ∅ } . Computability Theory, by Y. Fu X. Creative Set 23 / 30

Creative Sets are m-Complete Theorem . (Myhill, 1955) C is creative iff C is m-complete iff C is 1-complete iff C ≡ K . Computability Theory, by Y. Fu X. Creative Set 24 / 30

3. The Lattice of m-Degrees Computability Theory, by Y. Fu X. Creative Set 25 / 30

What Else? Q : In the world of recursively enumerable sets, is there anything between the recursive sets and the creative sets? Computability Theory, by Y. Fu X. Creative Set 26 / 30

What Else? Q : In the world of recursively enumerable sets, is there anything between the recursive sets and the creative sets? A : There is plenty. Computability Theory, by Y. Fu X. Creative Set 26 / 30

Trivial m-Degrees 1. o = {∅} . 2. n = { N } . 3. o ≤ m a provided a � = n . 4. n ≤ m a provided a � = o . Computability Theory, by Y. Fu X. Creative Set 27 / 30

Nontrivial m-Degrees 5. The recursive m-degree 0 m consists of all the nontrivial recursive sets. 6. An r.e. m-degree contains only r.e. sets. 7. The maximum r.e. m-degree d m ( K ) is denoted by 0 ′ m . Computability Theory, by Y. Fu X. Creative Set 28 / 30

The Distributive Lattice of m-Degrees . . . 0 ′ ❅ � m ❅ � ❅ � c ❅ � ❅ � b ❅ � a ❅ � ❅ � 0 m The m-degrees ordered by ≤ m form a distributive lattice. Computability Theory, by Y. Fu X. Creative Set 29 / 30