Handling tree structures — recursive SPs, nested sets, recursive CTEs
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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 sensibleclassification 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 thehelp of a solution to another problem.
In recursion theory we are only interested in reductions that arecomputable.
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Reduction
There are several ways of reducing a problem to another.
The differences between different reductions from A to B consistsin the manner and the extent to which information about B isallowed 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 ifthere 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 kvertices 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 6= ∅; ∅ ≤m A iff A 6= 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 6≤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 arecursive 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 allrecursive permutations.
I ‘A is infinite’ is a recursively invariant property.
I ‘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 isomorphicfunction 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−1z (f (x))), . . .} until a number ynot 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.
I The Hamiltonian Circle Problem (HC) asks if there is a circlein G whose overall weight is no more than m.
I The Traveling Sales Person Problem TSP asks for the overallweight 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 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.
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Theorem. An infinite set is recursive iff it is the range of a totalincreasing computable function.
Proof.Suppose A is recursive and infinite. Then A is range of theincreasing 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 computablefunction 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(x̃) is decidable if its characteristic function cM(x̃)given by
cM(x̃) =
{1, if M(x̃) holds,0, if M(x̃) 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 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}.
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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.
I To show that A is undecidable, it suffices to construct anm-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 functionk(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 recursivefunction 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 DecisionProblems. 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∅ 6∈ 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 recursivefunction 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 totalcomputable 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 byrestricting 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 classesof programmes.
All non-trivial semantic problems are algorithmically undecidable.
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