# Exercises

```{admonition} Question 1
:class: question

Prove that $f: \set{0,1}^* \to {\mathbb{R}}$ is computable if and only if it is both lower and upper semi-computable.
```


```{admonition} Question 2
:class: question

Prove that a function $f: \set{0,1}^* \to {\mathbb{R}}$ is lower semicomputable if and only if its lower graph

$$G^-(f)=\set{(x,q)\in\set{0,1}^*\times\mathbb{Q}\mid q<f(x)}$$

is computably enumerable.
```

```{admonition} Question 3
:class: question
Prove that the real number 

$$\alpha_K=\sum_{k\in K}\frac{1}{(k+1)^2}$$

is lower semicomputable but not computable.
```

```{admonition} Question 4
:class: question

Prove that a probability measure $p$ on $\set{0,1}^*$ is computable if and only if it is lower semicomputable.
```

```{admonition} Question 5
:class: question

Prove that 

$$P=\set{i: \lvert {dom \varphi_i} \rvert \leq 3}$$

is computably enforceably.
```