QUESTION

Problem 9.

For A ⊆ Σ

∗ and n ∈ N, we define the n

th slice of A to be the language

An = {y ∈ Σ

∗

|<n, y=””> ∈ A} ,

where <n, y=””> = <sn, y=””> and s0, s1, . . . is the standard enumeration of Σ∗

.

Let C and D be classes of languages.

1. C parametrizes D (or C is universal for D) if there exists A ∈ C such that

D = {An|n ∈ N}.

2. D is C-countable if there exists A ∈ C such that D ⊆ {An|n ∈ N}.

(a) Prove: A class D of languages is countable if and only if D is P(Σ∗

)-countable.

(b) Prove that DEC is not DEC-countable.

Problem 10.

(a) Assume that C and D are sets of languages and g : C

onto −−→ D. Prove: if C is countable,

then D is countable.

(b) Prove: if C is a countable set of languages, then ∃C and ∀C are countable.

Problem 11. Prove that the class of countable languages (defined as CTBL in class) is a

σ-ideal on P(Σ∗

).

1

Problem 12 Prove all the inclusions in the infinite diagram.

∆0

1

⊇

Σ

0

1

⊆

Π0

1

⊆

⊇

∆0

2

⊇

Σ

0

2

⊆

Π0

2

⊆

⊇

∆0

3

⊇

Σ

0

3

⊆

Π0

3

⊆

⊇ .

.

.

Problem 13. Prove that there is a function g : N → N with the following properties.

(i) g is nondecreasing, i.e., g(n) ≤ g(n + 1) holds for all n ∈ N.

(ii) g is unbounded, i.e., for every m ∈ N there exists n ∈ N such that g(n) > m.

(iii) For every computable, nondecreasing, unbounded function f : N → N, f(n) > g(n)

holds for all but finitely many n ∈ N.

Problem 14. Prove that a partial function f : ⊆ Σ

∗ → Σ

∗

is computable if and only if its

graph

Gf = {<x, f(x)=””> | x ∈ dom f}

is c.e.

Problem 15. Let A ⊆ Σ

∗ be c.e., and let B be an infinite decidable subset of A. Prove: If

A is undecidable, then A B is undecidable.

Problem 16. Let A = L(U) be the universal c.e. language defined in class lectures, and let

B ⊆ Σ

∗

. Prove: If A ≤m B and Σ∗ A ≤m B, then B is neither c.e. nor co-c.e.</x,></sn,></n,></n,>

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