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.

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Synopsis

• 1. Reduction
• 2. Recursive Set
• 3. Undecidability
• 4. Rice Theorem

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• 1. Reduction

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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.

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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.

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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.

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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.

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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.

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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. dm(A) = {B | A ≡m B} is the m-degree represented by A.

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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.

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The Structure of m-Degree

• Proposition. The m-degrees form a distributive lattice.

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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.

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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.

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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: h0 ⊆ h1 ⊆ h2 ⊆ h4 ⊆ . . . ⊆ hi ⊆ . . . such that h =

i∈ω hi.

At stage z + 1 = 2x + 1, if hz(x) is defined, do nothing. Otherwise enumerate {f (x), f (h−1

z (f (x))), . . .} until a number y

not in rng(hz) is found. Let hz+1(x) = y.

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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.

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• 2. Recursive Set

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Definition of Recursive Set

Let A be a subset of ω. The characteristic function of A is given by cA(x) = 1, if x ∈ A, 0, if x / ∈ A. A is recursive if cA(x) is computable.

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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.

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• 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) = µy(y ∈ A), f (n + 1) = µy(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).

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• 3. Undecidability

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Unsolvable Problem

A decision problem f : ω → {0, 1} is solvable if it is computable. It is unsolvable if it is not solvable.

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Undecidable Predicate

A predicate M( x) is decidable if its characteristic function cM( x) given by cM( x) = 1, if M( x) holds, 0, if M( x) does not hold. is computable. It is undecidable if it is not decidable.

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Non-recursive ⇔ Unsolvable ⇔ Undecidable

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Some Important Undecidable Sets

K = {x | x ∈ Wx}, K0 = {π(x, y) | x ∈ Wy}, K1 = {x | Wx = ∅}, Fin = {x | Wx is finite}, Inf = {x | Wx is infinite}, Con = {x | φx is total and constant}, Tot = {x | φx is total}, Cof = {x | Wx is cofinite}, Rec = {x | Wx is recursive}, Ext = {x | φx is extensible to a total recursive function}.

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• Fact. K is undecidable.

Proof.

If K were recursive, the characteristic function c(x) = 1, if x ∈ Wx, 0, if x / ∈ Wx, would be computable. Let m be an index for g(x) = 0, if c(x) = 0, ↑, if c(x) = 1. Then m ∈ Wm iff c(m) = 0 iff m / ∈ Wm.

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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.

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• Fact. There is a computable function h such that both Dom(h)

and Ran(h) are undecidable.

Proof.

Define h(x) = x, if x ∈ Wx, ↑, if x / ∈ Wx. Clearly x ∈ Dom(h) iff x ∈ Wx iff x ∈ Ran(h).

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• Fact. Both Tot and {x | φx ≃ λz.0} are undecidable.

Proof.

Consider the function f defined by f (x, y) = 0, if x ∈ Wx, ↑, if x / ∈ Wx. By S-m-n Theorem there is an injective primitive recursive function k(x) such that φk(x)(y) ≃ f (x, y). It is clear that k : K ≤1 Tot and k : K ≤1 {x | φx ≃ λ.0}.

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• Fact. Both {x | c ∈ Wx} and {x | c ∈ Ex} are undecidable.

Proof.

Consider the function f defined by f (x, y) = y, if x ∈ Wx, ↑, if x / ∈ Wx. By S-m-n Theorem there is some injective primitive recursive function k(x) such that φk(x)(y) ≃ f (x, y). It is clear that k is a one-one reduction from K to both {x | c ∈ Wx} and {x | c ∈ Ex}.

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• Fact. The predicate ‘φx(y) is defined’ is undecidable.
• Fact. The predicate ‘φx ≃ φy’ is undecidable.

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• 4. Rice Theorem

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Henry Rice Classes of Recursively Enumerable Sets and their Decision

• Problems. Transactions of the American Mathematical Society,

77:358-366, 1953.

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Rice Theorem (1953). If ∅ B C, then {x | φx ∈ B} is not recursive.

Proof.

Suppose f∅ ∈ B and g ∈ B. Let f be defined by f (x, y) = g(y), if x ∈ Wx, ↑, if x / ∈ Wx. By S-m-n Theorem there is some injective primitive recursive function k(x) such that φk(x)(y) ≃ f (x, y). It is clear that k is a one-one reduction from K to {x | φx ∈ B}.

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Applying Rice Theorem

Assume that f (x) ≃ φx(x) + 1 could be extended to a total computable function say g(x). Let e be an index of g(x). Then φe(e) = g(e) = f (e) = φe(e) + 1. Contradiction. So we may use Rice Theorem to conclude that Ext = {x | φx is extensible to a total recursive function} is not recursive. Comment: Not every partial recursive function can be obtained by restricting a total recursive function.

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Remark on Rice Theorem

Rice Theorem deals with programme independent properties. It talks about classes of computable functions rather than classes

• f programmes.

All non-trivial semantic problems are algorithmically undecidable.

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