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 6= ∅; ∅ ≤m A iﬀ A 6= 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. dm(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.

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.

I ‘A is inﬁnite’ is a recursively invariant property.

I ‘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:

h0 ⊆ h1 ⊆ h2 ⊆ h4 ⊆ ... ⊆ hi ⊆ ... S such that h = i∈ω hi .

At stage z + 1 = 2x + 1, if hz (x) is deﬁned, do nothing. −1 Otherwise enumerate {f (x), f (hz (f (x))),...} until a number y not in rng(hz ) is found. Let hz+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.

I The Hamiltonian Circle Problem (HC) asks if there is a circle in G whose overall weight is no more than m.

I 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 (x) = A 0, if x ∈/ A.

A is recursive if cA(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 6= ∅, 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) = 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).

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(ex) is decidable if its characteristic function cM (ex) given by 1, if M(ex) holds, cM (ex) = 0, if M(ex) 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

K = {x | x ∈ Wx },

K0 = {π(x, y) | x ∈ Wy },

K1 = {x | Wx 6= ∅},

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}.

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 , c(x) = x 0, if x ∈/ Wx ,

would be computable. Let m be an index for

0, if c(x) = 0, g(x) = ↑, if c(x) = 1.

Then m ∈ Wm iﬀ c(m) = 0 iﬀ m ∈/ Wm.

Computability Theory, by Y. Fu VIII. Recursive Set 25 / 35 K is often used to prove undecidability result.

I 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 , h(x) = x ↑, if x ∈/ Wx .

Clearly x ∈ Dom(h) iﬀ x ∈ Wx iﬀ x ∈ Ran(h).

Computability Theory, by Y. Fu VIII. Recursive Set 27 / 35 Fact. Both Tot and {x | φx ' λz.0} are undecidable. Proof. Consider the function f deﬁned by

0, if x ∈ W , f (x, y) = x ↑, 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}.

Computability Theory, by Y. Fu VIII. Recursive Set 28 / 35 Fact. Both {x | c ∈ Wx } and {x | c ∈ Ex } are undecidable. Proof. Consider the function f deﬁned by

y, if x ∈ W , f (x, y) = x ↑, 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 }.

Computability Theory, by Y. Fu VIII. Recursive Set 29 / 35 Fact. The predicate ‘φx (y) is deﬁned’ is undecidable.

Fact. The predicate ‘φx ' φy ’ 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 Rice Theorem (1953).

If ∅ ( B ( C, then {x | φx ∈ B} is not recursive. Proof. Suppose f∅ 6∈ B and g ∈ B. Let f be deﬁned by

g(y), if x ∈ W , f (x, y) = x ↑, 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}.

Computability Theory, by Y. Fu VIII. Recursive Set 33 / 35 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.

Computability Theory, by Y. Fu VIII. Recursive Set 34 / 35 Remark on Rice Theorem

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

All non-trivial semantic problems are algorithmically undecidable.

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