Examples

 

The objective of this page is to introduce the reader to 5 examples that illustrate the type of conversations that are discussed in this site. If for whatever reasons (better be good ones!) you cannot read them all, I suggest that you do not miss the last one. It consists of a very long conversation with Bart on a well familiar mathematical topic, with a twist! In any case these are the examples discussed in this page.

• Example 1: My life story
• Example 2: Global min and max
• Example 3: Maximin decision rule
• Example 4: Knapsack problem
• Example 5: Towers of Hanoi puzzle
• Example 6: 1 + 2 + · · · + n = ?

Although the topics discussed here are technical (mathematical) in nature, the reader does not have to be technically trained/educated in order to understand and appreciate the practical implications that these examples identify with regard to the potential use of AI chatbots such as ChatGPT and Bard.

Before we examine the examples, let us see what ChatGPT and Bard have to say about the role that examples can and should play in the assessment of the capabilities and limitations of AI chatbots. So I posed this question to both ChatGPT and Bard

The Question

What role should examples play in the assessment of capabilities and limitation of chatbots such as ChatGPT and Bard? Can you suggest 2-3 specific examples? (<240 words)

ChatGPT’s answer

Bard’s answer

I’ll take ChatGPT’s and Bard’s tips into consideration in planning the conversations to be discussed in this site. Typically, the first question I pose in a conversation is deliberately designed to prepare the ground for further questions.

My favorite conversations are those where I try, not always successfully, to help the chatbot to correct or improve his/her answer.

Now back to our examples.

Example 1: My life story

I use this example to illustrate a number of things. First, chatbots ability to compose essays on rather abstract topics. Second, to illustrate the similarities and differences between the way ChatGPT and Bard respond to queries, and express themselves.

From my perspective, this example also illustrates some of the difficulties chatbots sometimes experience discussing issues related to … chatbots. This is why in this example I ask the ChatGPT and Bard to compose two essays on the “My life story” motif:

• My life
• My life as a chatbot

It is interesting that both ChatGPT and Bard addressed the first motif as … humans! In other words, they wrote the essay presuming that “My life” refers to a human being’s life.

Example 2: Global min and max

In this example I pose a very simple technical (mathematical) question regarding the concepts global minimum and global maximum:

Please provide three examples of a real-valued function on the real line, having the property that its global minimum and global maximum are attained at the same point. (<250 words)

I was greatly surprised from the answers. This gave me an opportunity to raise relevant follow-up questions. Interestingly, both ChatGPT and Bard had great difficulties handling this topic. I wonder why.

Example 3: Maximin decision rule

In this example I test ChatGPT’s and Bard’s understanding of the decision rule on which Wald’s famous maximin paradigm is based. The initial query ask them to determine which one, if any, of the following four optimization models is a Wald-type maximin model, assuming that [katex]X[/katex] denotes the decision space and [katex]U[/katex] denotes the uncertainty space:

[katex display=true]
\begin{align}
\text{Model 1:} \ \ \ \ &\max_{x\in X} \ \{f(x): g(x,u) \in \mathcal{K}, \forall u\in U\}\\
\text{Model 2:} \ \ \ \ &\max_{x\in X}\min_{u\in U}\ \{f(x,u): g(x,u) \in \mathcal{K}, \forall u\in U\}\\
\text{Model 3:} \ \ \ \ &\max_{x\in X}\ \{f(x,u): g(x,u) \in \mathcal{K}\}\ \ , \ \ u\in U\\
\text{Model 4:} \ \ \ \ &\max_{u\in U}\min_{x\in X} \ \{f(x,u): g(x,u) \in \mathcal{K}, \forall u\in U\}\\
\end{align}
[/katex]

Typically the answer is only partially correct, so there is plenty of scope here for follow-up queries/questions.

I should point out that both ChatGPT and Bard are familiar with [katex]\color{teal}\large \LaTeX[/katex] notation, so it is easy to use mathematical notation in the conversation. More on this technical issue in the Conversations page and Guide page.

Example 4: Knapsack problem

I use this generic optimization problem as a framework for testing certain mathematical modeling skills of chatbots. I may pose a question such as this:

You have a long bookcase consisting of single shelf, and a pile of books. Your mission is to maximize the total value of the books you put on the shelf. Assume that you know the length of the shelf, and the width and value ($) of each book. The question is then: which books should you put on the shelf? Write down a mathematical formulation of this problem.

Usually, the chatbots provide a solid, correct answer to this question. Furthermore, they are also very good in formulating a (correct) dynamic programming (DP) functional equation for the problem. But … they find it very difficult to deal with certain modified versions of the problem.

In my conversations with them on this topic I try to help them in the formulation of DP functional equations for this class of problems and also use this opportunity to test their DP modeling skills in general.

Example 5: Towers of Hanoi puzzle

In this example I discuss with ChatGPT and Bard dynamic programming functional equations for solving this famous puzzle.

You can experiment with this interactive model of the puzzle right here:

Details regarding how to interact with this model are discussed in Example 5.

• Tip:
  To solve the instance presented, click the solve button.
  To slow down the movement of disks, increase the delay.
  To see the log of the solution procedure, tick the Report check box.

Example 6: 1+2+ · · · + n = ?

The exact meaning of the popular mathematical symbol · · · appearing in expressions such as [katex] 1 + 2 + \cdots + n[/katex] depends not only on the context in which it is used, but also on the way individuals (humans and chatbots alike) interpret it. Informally, it often means “and so on” in the context of which it is clear what “and so on” means. So there is no ambiguity what [katex]\cdots[/katex] means in this expression:
[katex display=true]\begin{align}
1 + 2 + 3 + \cdots + n =\ ?
\end{align}[/katex]
if [katex]n[/katex] is a positive integer. In fact, it is well known that in that case,
[katex display=true]\begin{align}
1 + 2 + 3 + \cdots + n & = \sum_{k=1}^{n} k\\
& = \frac{(1+n)n}{2}.
\end{align}[/katex]

Its meaning is also clear in the context of these expressions:
[katex display=true]\begin{align}
f(a,n):&= 1 + \frac{1}{a} + \frac{1}{a^{2}} + \frac{1}{a^{3}} + \cdots + \frac{1}{a^{n}}\\[4mm]
g(a):&= 1 + \frac{1}{a} + \frac{1}{a^{2}} + \frac{1}{a^{3}} + \cdots
\end{align}[/katex]

It may take a while though for people who are not familiar with the Fibonacci sequence to determine the meaning of [katex]\cdots[/katex] in the context of this expression:
[katex display=true]\begin{align}
S :=\{377, 610, 987, 1597, 2584, 4181 + \cdots\}
\end{align}[/katex]

Given all of this, what is the meaning of [katex]\cdots[/katex] in the context of this very simple looking expression:

[katex display=true]\begin{align}
s = 1 + 2+ 3 + \cdots + 9.78
\end{align}[/katex]

The issue here is that because 9.78 is not a positive integer, some may argue that this sum is not well defined. One obvious interpretation of this expression is that since [katex]9.78= 9 + 0.78[/katex], we have

[katex display=true]\begin{align}
s = 1 + 2+ 3 + \cdots + 9.78 &= 1 + 2 + 3 +\cdots + 9 +0.78\\
&=0.78 + 1 + 2 + 3 +\cdots + 9
\end{align}[/katex]

Note that here there is no ambiguity what [katex]\cdots[/katex] means, hence we conclude

[katex display=true]\begin{align}
s = 1 + 2+ 3 + \cdots + 9.78 &= \overbrace{\ 1 + 2 + 3 +\cdots + 9\ } +0.78\\
&=45 + 0.78\\
& = 45.78.
\end{align}[/katex]

On the other hand, if we write [katex]9.78 = 0.78+9[/katex] we are back in square one, as [katex] 1+ 2+ 3 + \cdots + 0.78[/katex] is ambiguous. Example 6 is based on long conversations I had with ChatGPT and Bard on the meaning of the expression [katex]1+2+3 + \cdots + A[/katex] in cases where [katex]A[/katex] is not an integer.

Some readers may wonder what is the motivation for posing such a question to chatbots in the first place. In fact, how often do we pose such questions to students?

I should therefore stress that in my conversations with ChatGPT and Bard I sometimes, in fact often, deliberately pose questions that are a bit ambiguous, or even ill-defined from a strict mathematical point of view. I do this because one of the important objectives of this project is to check how chatbots deal with such “strange” questions, the answer to which is unlikely to appear in the training database of the chatbots.

For the benefit of readers who did not have the please of conversing with chatbots, the following conversations illustrate the kind of things you can discover about chatbots by asking them “strange” questions. First, here is the Bard conversation (\cdots is the Latex code for [katex]\cdots[/katex]):

And here is ChatGPT’s.

Both answers are problematic and therefore in Example 6 I discuss with ChatGPT and Bard a number of technical issues related to their interpretation of the meaning of [katex]\cdots[/katex] is the context of the problem discussed above.

To illustrate, here is a short discussion with Bard on this issue.

In Example 6 I examine certain aspects of Bard’s answer that deserve attention. Here it suffices to say that apparently the issue of “the first k real numbers” occupies the minds of humans as well. For instance, see the discussion on this issue in Quora.

Practicalities

For a number of practical reasons, in this project it is not convenient to display the conversations with ChatGPT and Bard graphically, as done above, by the popular cut-and-paste procedure, namely by pasting screenshots of these conversations in the WordPress pages. What we do here is edit the display of the conversations slightly to better suit our needs in this project.

The main editorial modifications are as follows:

• Replacing ChatGPT’s and Bard’s icons with decorated texts, as shown below.
• Using `proper’ mathematical notation wherever it is convenient and practical.
• Making the layout more compact, space-wise.
• Adding a time-stamp to the question/query.

With regard to the last item, the reader is advised that as far as the conversations discussed in this site are concerned, the chatbots response to queries in practically instantaneous.

The following examples illustrate these modifications in action. The modifications are provisional and may change in the future. Suggestions on this matter will be greatly appreciated.

The first example illustrate a case where the same query is posed to the two Chatbots. Here C is short for ChatGPT and B is short for Bard. The color scheme does not represent the preferences of the respective chatbots and may change.

The second example illustrates the use of conventional mathematical notation in the text. Here is the screenshot of the original conversation with ChatGPT.

And here is modified version:

At present I use the [katex]\large \KaTeX[/katex] plugin to display mathematical expressions in the modified versions of my conversations with ChatGPT and Bard.

What’s next?

Good question!

If this is your first visit to the site, then I suggest that you surf to one of the examples discussed above. Here are the links:

• Example 1: My life story
• Example 2: Global min and max
• Example 3: Maximin decision rule
• Example 4: Knapsack problem
• Example 5: Sequential vs multistage decision models
• Example 6: 1 + 2 + · · · + n = ?

Alternatively, you may wish to surf to the Conversations page where the real action is.