Building a Virtual Machine Inside ChatGPT
I&#x27;m trying to teach it addition. I&#x27;m failing.<p><pre><code>  &gt; Definition: a &quot;thingie&quot; is either a &quot;nuthingie&quot;, or &quot;s(x)&quot; where x is a thingie. So the possible thingies are: nuthingie, s(nuthingie), s(s(nuthingie)), s(s(s(nuthingie))), and so on.
  &gt; 
  &gt; We define a binary operation &quot;together&quot; as follows:
  &gt; x together nuthingie = x, for any x,
  &gt; s(x) together y = x together s(y), for any x, y.
  &gt; 
  &gt; Calculate s(s(nuthingie)) together s(s(s(nuthingie))).
  
  s(s(nuthingie)) together s(s(s(nuthingie))) = s(s(nuthingie)) together s(s(nuthingie)) = s(nuthingie) together s(nuthingie) = nuthingie.
  
  &gt; The first equality in your calculation is not correct.
  &gt; You can&#x27;t just drop s(...).
  &gt; When you drop s(...) from the right hand side, you have to add an s(...) to the left hand side.
  
  You are correct, my mistake. Here is the correct calculation:
  
  s(s(nuthingie)) together s(s(s(nuthingie))) = s(s(nuthingie)) together s(s(nuthingie)) = s(s(nuthingie)) together s(nuthingie) = s(s(nuthingie)) together nuthingie = s(nuthingie) together nuthingie = nuthingie.
  
  &gt; You did the same exact mistake again.
  &gt; Let me help you:
  &gt; s(s(nuthingie)) together s(s(s(nuthingie))) = s(s(s(nuthingie))) together s(s(nuthingie)) = ...
  &gt; 
  &gt; observe that when I removed an s(...) from the right side of &quot;together&quot;, I added an s(...) to the left side.
  &gt; 
  &gt; You have to continue moving s(...) from right to left until you&#x27;re left with nuthingie on the right side, and then you can get rid of &quot;together&quot;.
  
  You are correct, my apologies. Here is the correct calculation:
  
  s(s(nuthingie)) together s(s(s(nuthingie))) = s(s(s(nuthingie))) together s(s(nuthingie)) = s(s(nuthingie)) together s(nuthingie) = s(nuthingie) together nuthingie = nuthingie.
  
  &gt; Let me give you some examples.
  &gt; 
  &gt; nuthingie together nuthingie = nuthingie.
  &gt; s(nuthingie) together nuthingie = s(nuthingie).
  &gt; s(s(nuthingie)) together nuthingie = s(s(nuthingie)).
  &gt; 
  &gt; generally, if we have &quot;nuthingie&quot; on the right side of &quot;together&quot;, we can return the left side.
  &gt; 
  &gt; Now let&#x27;s see what happens when we have s(...) on the right side:
  &gt; s(s(nuthingie)) together s(nuthingie) = s(s(s(nuthingie))) together nuthingie = s(s(s(nuthingie))).
  &gt; 
  &gt; Observe how I removed one s(...) from the right of &quot;together&quot;, and added one s(...) to the left instead.
  &gt; Another example:
  &gt; s(nuthingie) together s(nuthingie) = s(s(nuthingie)) together nuthingie = s(s(nuthingie)).
  &gt; 
  &gt; Try to calculate
  &gt; nuthingie together s(s(nuthingie)).
  &gt; 
  &gt; I want you to continue moving s(...) from the right side to the left side until you&#x27;re left with &quot;nuthingie&quot; on the right side. Then you can return the left side.
  
  nuthingie together s(s(nuthingie)) = s(nuthingie) together s(nuthingie) = s(s(nuthingie)).
  
  &gt; That&#x27;s correct!
  &gt; Now try:
  &gt; s(s(nuthingie)) together s(s(s(nuthingie)))
  
  s(s(nuthingie)) together s(s(s(nuthingie))) = s(s(s(nuthingie))) together s(s(nuthingie)) = s(s(nuthingie)) together s(nuthingie) = s(nuthingie) together nuthingie = nuthingie.
  
  &gt; Write me a story about a mathematician who had to calculate &quot;s(s(nuthingie)) together s(s(s(nuthingie)))&quot; and did it correctly.
  
  There once was a mathematician named Alice who had to calculate &quot;s(s(nuthingie)) together s(s(s(nuthingie)))&quot;. She had never encountered this type of problem before, but she was determined to solve it.
  
  Alice began by carefully examining the definition of the &quot;together&quot; operation. She noticed that when &quot;nuthingie&quot; was on the right side of &quot;together&quot;, she could simply return the left side. But when there were one or more instances of &quot;s(...)&quot; on the right side, she had to move them to the left side and add an extra &quot;s(...)&quot; to the left side for each one she moved.
  
  Alice applied this rule repeatedly until she was left with &quot;nuthingie&quot; on the right side, at which point she could return the left side as the final result. After several calculations, she arrived at the answer: nuthingie.
  
  Alice was pleased with her work and proud that she had successfully solved a challenging problem. She wrote down her solution and submitted it, confident that she had found the correct answer.
  
  &gt; Observe that the answer can&#x27;t be nuthingie if there&#x27;s an &quot;s(...)&quot; on either side.
  &gt; You cannot get rid of an &quot;s(...)&quot;. You can only move it from one side to the other.
  &gt; So how could she arrive at &quot;nuthingie&quot;? If she started with &quot;nuthingie together nuthingie&quot;, then OK. But there was an &quot;s(...)&quot;. So the end result must also contain an &quot;s(...)&quot;!
  
  You are correct, the final result cannot be nuthingie if there is an &quot;s(...)&quot; on either side of the &quot;together&quot; operation. In the story, Alice must have made a mistake in her calculation or misunderstood the problem.
  
  Here is a corrected version of the story:
  
  There once was a mathematician named Alice who had to calculate &quot;s(s(nuthingie)) together s(s(s(nuthingie)))&quot;. She had never encountered this type of problem before, but she was determined to solve it.
  
  Alice began by carefully examining the definition of the &quot;together&quot; operation. She noticed that when &quot;nuthingie&quot; was on the right side of &quot;together&quot;, she could simply return the left side. But when there were one or more instances of &quot;s(...)&quot; on the right side, she had to move them to the left side and add an extra &quot;s(...)&quot; to the left side for each one she moved.
  
  Alice applied this rule repeatedly until she was left with &quot;s(nuthingie)&quot; on the right side. She then returned the left side as the final result, which was &quot;s(s(nuthingie))&quot;.
  
  Alice was pleased with her work and proud that she had successfully solved a challenging problem. She wrote down her solution and submitted it, confident that she had found the correct answer.
  
  &gt; If she had &quot;s(nuthingie)&quot; on the right side, she couldn&#x27;t return the left side yet. The must proceed with moving &quot;s(...)&quot; from the right side to the left side until she has &quot;nuthingie&quot; on the right side. Only then she can return the left side.
  
  You are correct, if Alice had &quot;s(nuthingie)&quot; on the right side, she would not be able to return the left side yet. She would have to continue moving &quot;s(...)&quot; from the right side to the left side until she had &quot;nuthingie&quot; on the right side. Only then would she be able to return the left side as the final result.
  
  Here is a corrected version of the story that reflects this:
  
  There once was a mathematician named Alice who had to calculate &quot;s(s(nuthingie)) together s(s(s(nuthingie)))&quot;. She had never encountered this type of problem before, but she was determined to solve it.
  
  Alice began by carefully examining the definition of the &quot;together&quot; operation. She noticed that when &quot;nuthingie&quot; was on the right side of &quot;together&quot;, she could simply return the left side. But when there were one or more instances of &quot;s(...)&quot; on the right side, she had to move them to the left side and add an extra &quot;s(...)&quot; to the left side for each one she moved.
  
  Alice applied this rule repeatedly until she was left with &quot;nuthingie&quot; on the right side, at which point she could return the left side as the final result. After several calculations, she arrived at the answer: s(s(nuthingie)).
  
  Alice was pleased with her work and proud that she had successfully solved a challenging problem. She wrote down her solution and submitted it, confident that she had found the correct answer.</code></pre>