Answer
See explanation
Work Step by Step
Below are **sample derivations** within the MIU‐system (as presented in Hofstadter’s *Gödel, Escher, Bach*) showing how to obtain the strings **MIUI** and **MUIIU** from the initial axiom **MI** using the five standard production rules:
1. **Axiom**: \(MI\) is in the system.
2. **Rule 1** (Append \(U\)): If a string ends in \(I\), you may add \(U\) at the end.
3. **Rule 2** (Double‐after‐\(M\)): If your string is \(M x\), you may replace it with \(M x x\) (i.e.\ duplicate everything after the initial \(M\)).
4. **Rule 3** (\(III \to U\)): If a string contains the substring \(III\), you may replace that substring by \(U\).
5. **Rule 4** (\(UU \to \epsilon\)): If a string contains the substring \(UU\), you may remove that substring.
We label these rules for clarity (the exact numbering can vary by text).
---
## Derivation of **MIUI**
We start from the **axiom** \(MI\).
1. **\(MI\)** is in the system by definition.
2. Apply **Rule 2** (double‐after‐\(M\)) to \(MI\). The part after \(M\) is \(I\), so duplicating \(I\) gives
\[
MII.
\]
So \(MII\) is in the system.
3. Apply **Rule 2** again to \(MII\). Now the part after \(M\) is \(II\). Doubling \(II\) gives \(IIII\). So we get
\[
MIIII.
\]
4. Apply **Rule 3** (\(III \to U\)) to \(MIIII\). We see the substring \(\text{M}[\text{III}]I\). Replace \(III\) by \(U\):
\[
M\,[III]\,I \;\longrightarrow\; M\,U\,I \;=\; MIUI.
\]
Hence **\(MIUI\)** is in the MIU‐system.
---
## Derivation of **MUIIU**
Again, we begin with **\(MI\)**.
1. **\(MI\)** is in the system (axiom).
2. By **Rule 2** (doubling after \(M\)), from \(MI\) we get
\[
MII.
\]
3. By **Rule 2** again, from \(MII\) we get
\[
MIIII.
\]
4. By **Rule 3** (\(III \to U\)), from \(MIIII\) we replace the substring \(III\) with \(U\):
\[
M\,[III]\,I \;\to\; MUI.
\]
So \(MUI\) is in the system.
5. Now \(MUI\) ends in \(I\), so by **Rule 1** (append \(U\) if it ends in \(I\)), we get
\[
MUIU.
\]
6. By **Rule 2** (double after \(M\)) on \(MUIU\): the part after \(M\) is \(UIU\). Doubling that yields \(UIUUIU\). So we get
\[
MUIUUIU.
\]
7. Finally, by **Rule 4** (\(UU \to \epsilon\)), observe the substring \(\dots U\,U \dots\). Specifically, \(MUIUUIU\) contains \(UI[UU]IU\). Removing \(UU\) gives \(UIIU\). Thus
\[
M\,UIUUIU \;\longrightarrow\; M\,UIIU.
\]
That is exactly **\(MUIIU\)**.
Therefore **\(MUIIU\)** is also in the MIU‐system.
---
### Conclusion
By starting with the axiom \(MI\) and applying the rules in a stepwise fashion, we have explicit derivations showing:
- **\(MIUI\)** is in the MIU‐system.
- **\(MUIIU\)** is in the MIU‐system.
These illustrate the **recursive** nature of the MIU‐system’s production rules.
