VIII. Recursive Set Yuxi Fu BASICS, Shanghai Jiao Tong University - PowerPoint PPT Presentation

VIII. Recursive Set Yuxi Fu BASICS, Shanghai Jiao Tong University

Decision Problem, Predicate, Number Set The following emphasizes the importance of number set: ⇔ Decision Problem Predicate on Number ⇔ Set of Number A central theme of recursion theory is to look for sensible classification of number sets. Classification is often done with the help of reduction. Computability Theory, by Y. Fu VIII. Recursive Set 1 / 35

Synopsis 1. Reduction 2. Recursive Set 3. Undecidability 4. Rice Theorem Computability Theory, by Y. Fu VIII. Recursive Set 2 / 35

1. Reduction Computability Theory, by Y. Fu VIII. Recursive Set 3 / 35

Reduction between Problems A reduction is a way of defining a solution to a problem with the help of a solution to another problem. In recursion theory we are only interested in reductions that are computable. Computability Theory, by Y. Fu VIII. Recursive Set 4 / 35

Reduction There are several ways of reducing a problem to another. The differences between different reductions from A to B consists in the manner and the extent to which information about B is allowed to settle questions about A . Computability Theory, by Y. Fu VIII. Recursive Set 5 / 35

Many-One Reduction The set A is many-one reducible, or m-reducible, to the set B if there is a total computable function f such that x ∈ A iff f ( x ) ∈ B for all x . We shall write A ≤ m B or more explicitly f : A ≤ m B . If f is injective, then it is a one-one reducibility, denoted by ≤ 1 . Computability Theory, by Y. Fu VIII. Recursive Set 6 / 35

An Example Suppose G is a finite graph and k is a natural number. 1. The Independent Set Problem (IndSet) asks if there are k vertices of G with every pair of which unconnected. 2. The Clique Problem asks if there is a k -complete subgraph of G . There is a simple one-one reduction from IndSet to Clique. Computability Theory, by Y. Fu VIII. Recursive Set 7 / 35

Many-One Reduction 1. ≤ m is reflexive and transitive. 2. A ≤ m B iff A ≤ m B . 3. A ≤ m ω iff A = ω ; A ≤ m ∅ iff A = ∅ . 4. ω ≤ m A iff A � = ∅ ; ∅ ≤ m A iff A � = ω . Computability Theory, by Y. Fu VIII. Recursive Set 8 / 35

m-Degree 1. A ≡ m B if A ≤ m B ≤ m A . (many-one equivalence) 2. A ≡ 1 B if A ≤ 1 B ≤ 1 A . (one-one equivalence) 3. d m ( A ) = { B | A ≡ m B } is the m-degree represented by A . Computability Theory, by Y. Fu VIII. Recursive Set 9 / 35

m-Degree The set of m-degrees is ranged over by a , b , c , . . . . a ≤ m b iff A ≤ m B for some A ∈ a and B ∈ b . a < m b iff a ≤ m b and b �≤ m a . Computability Theory, by Y. Fu VIII. Recursive Set 10 / 35

The Structure of m-Degree Proposition . The m-degrees form a distributive lattice. Computability Theory, by Y. Fu VIII. Recursive Set 11 / 35

Recursive Permutation A recursive permutation is one-one recursive function. A is recursively isomorphic to B , written A ≡ B , if there is a recursive permutation p such that p ( A ) = B . Computability Theory, by Y. Fu VIII. Recursive Set 12 / 35

Recursive Invariance A property of sets is recursively invariant if it is invariant under all recursive permutations. ◮ ‘ A is infinite’ is a recursively invariant property. ◮ ‘2 ∈ A ’ is not recursively invariant. Computability Theory, by Y. Fu VIII. Recursive Set 13 / 35

Myhill Isomorphism Theorem Myhill Isomorphism Theorem (1955). A ≡ B iff A ≡ 1 B . Proof. The idea is to construct effectively the graph of an isomorphic function h by two simultaneous symmetric inductions: h 0 ⊆ h 1 ⊆ h 2 ⊆ h 4 ⊆ . . . ⊆ h i ⊆ . . . such that h = � i ∈ ω h i . At stage z + 1 = 2 x + 1, if h z ( x ) is defined, do nothing. Otherwise enumerate { f ( x ) , f ( h − 1 z ( f ( x ))) , . . . } until a number y not in rng ( h z ) is found. Let h z +1 ( x ) = y . Computability Theory, by Y. Fu VIII. Recursive Set 14 / 35

The Restriction of m-Reduction Suppose G is a finite directed weighted graph and m is a number. ◮ The Hamiltonian Circle Problem (HC) asks if there is a circle in G whose overall weight is no more than m . ◮ The Traveling Sales Person Problem TSP asks for the overall weight of a circle with minimum weight if there are circles. TSP can be reduced to HC. The reduction is not m-reduction. Computability Theory, by Y. Fu VIII. Recursive Set 15 / 35

2. Recursive Set Computability Theory, by Y. Fu VIII. Recursive Set 16 / 35

Definition of Recursive Set Let A be a subset of ω . The characteristic function of A is given by � 1 , if x ∈ A , c A ( x ) = 0 , if x / ∈ A . A is recursive if c A ( x ) is computable. Computability Theory, by Y. Fu VIII. Recursive Set 17 / 35

Fact about Recursive Set Fact . If A is recursive then A is recursive. Fact . If A is recursive and B � = ∅ , ω , then A ≤ m B . Fact . If A , B are recursive and A , B , A , B are infinite then A ≡ B . Fact . If A ≤ m B and B is recursive, then A is recursive. Fact . If A ≤ m B and A is not recursive, then B is not recursive. Computability Theory, by Y. Fu VIII. Recursive Set 18 / 35

Theorem . An infinite set is recursive iff it is the range of a total increasing computable function. Proof. Suppose A is recursive and infinite. Then A is range of the increasing function f given by f (0) = µ y ( y ∈ A ) , f ( n + 1) = µ y ( y ∈ A and y > f ( n )) . The function is total, increasing and computable. Conversely suppose A is the range of a total increasing computable function f . Obviously y = f ( n ) implies n ≤ y . Hence y ∈ A ⇔ y ∈ Ran ( f ) ⇔ ∃ n ≤ y ( f ( n ) = y ). Computability Theory, by Y. Fu VIII. Recursive Set 19 / 35

3. Undecidability Computability Theory, by Y. Fu VIII. Recursive Set 20 / 35

Unsolvable Problem A decision problem f : ω → { 0 , 1 } is solvable if it is computable. It is unsolvable if it is not solvable. Computability Theory, by Y. Fu VIII. Recursive Set 21 / 35

Undecidable Predicate A predicate M ( � x ) is decidable if its characteristic function c M ( � x ) given by � 1 , if M ( � x ) holds , c M ( � x ) = 0 , if M ( � x ) does not hold . is computable. It is undecidable if it is not decidable. Computability Theory, by Y. Fu VIII. Recursive Set 22 / 35

Non-recursive ⇔ Unsolvable ⇔ Undecidable Computability Theory, by Y. Fu VIII. Recursive Set 23 / 35

Some Important Undecidable Sets { x | x ∈ W x } , K = = { π ( x , y ) | x ∈ W y } , K 0 { x | W x � = ∅} , K 1 = = { x | W x is finite } , Fin Inf = { x | W x is infinite } , = { x | φ x is total and constant } , Con Tot = { x | φ x is total } , = { x | W x is cofinite } , Cof Rec = { x | W x is recursive } , = { x | φ x is extensible to a total recursive function } . Ext Computability Theory, by Y. Fu VIII. Recursive Set 24 / 35

Fact . K is undecidable. Proof. If K were recursive, the characteristic function � 1 , if x ∈ W x , c ( x ) = ∈ W x , 0 , if x / would be computable. Let m be an index for � 0 , if c ( x ) = 0 , g ( x ) = ↑ , if c ( x ) = 1 . Then m ∈ W m iff c ( m ) = 0 iff m / ∈ W m . Computability Theory, by Y. Fu VIII. Recursive Set 25 / 35

K is often used to prove undecidability result. ◮ To show that A is undecidable, it suffices to construct an m-reduction from K to A . Computability Theory, by Y. Fu VIII. Recursive Set 26 / 35

Fact . There is a computable function h such that both Dom ( h ) and Ran ( h ) are undecidable. Proof. Define � x , if x ∈ W x , h ( x ) = ↑ , if x / ∈ W x . Clearly x ∈ Dom ( h ) iff x ∈ W x iff x ∈ Ran ( h ). Computability Theory, by Y. Fu VIII. Recursive Set 27 / 35

Fact . Both Tot and { x | φ x ≃ λ z . 0 } are undecidable. Proof. Consider the function f defined by � 0 , if x ∈ W x , f ( x , y ) = ↑ , if x / ∈ W x . By S-m-n Theorem there is an injective primitive recursive function k ( x ) such that φ k ( x ) ( y ) ≃ f ( x , y ). It is clear that k : K ≤ 1 Tot and k : K ≤ 1 { x | φ x ≃ λ. 0 } . Computability Theory, by Y. Fu VIII. Recursive Set 28 / 35

Fact . Both { x | c ∈ W x } and { x | c ∈ E x } are undecidable. Proof. Consider the function f defined by � y , if x ∈ W x , f ( x , y ) = ↑ , if x / ∈ W x . By S-m-n Theorem there is some injective primitive recursive function k ( x ) such that φ k ( x ) ( y ) ≃ f ( x , y ). It is clear that k is a one-one reduction from K to both { x | c ∈ W x } and { x | c ∈ E x } . Computability Theory, by Y. Fu VIII. Recursive Set 29 / 35

Fact . The predicate ‘ φ x ( y ) is defined’ is undecidable. Fact . The predicate ‘ φ x ≃ φ y ’ is undecidable. Computability Theory, by Y. Fu VIII. Recursive Set 30 / 35

4. Rice Theorem Computability Theory, by Y. Fu VIII. Recursive Set 31 / 35

Henry Rice Classes of Recursively Enumerable Sets and their Decision Problems. Transactions of the American Mathematical Society, 77:358-366, 1953. Computability Theory, by Y. Fu VIII. Recursive Set 32 / 35