Inside
The Question
It is only natural for visitors to this site, the public in general, and perhaps others, to ask the following seemingly simple question:
What could have possibly motivated a 78 years old retired academic to create a website dedicated to commentary of his conversations with chatbots about topics related to dynamic programming and decision-making under severe uncertainty?
The Answer
Well, these days it is very tempting to seek advise on this matter from AI robots, so let us pose this intriguing question to the two chatbots featuring prominently in this site. First, ChatGPT:
Being the very person in question, I can categorically say that these answers are not bad at all, except that they do not mention the following very important reason that motivated me to embark on this challenging project:
I am more than a bit skeptic about the ability of the current state of the art in AI to deal successfully and reliably with subtle, but important, mathematical modeling issues. Conversations with chatbots about these issues could give me a much clearer picture of capabilities and limitations of the current state of the art in AI on this front. I am particularly interested in checking the similarities and differences between the way humans and chatbots address certain modeling issues.
Practically speaking, I hope that my conversations with chabots will shade some new light on subtle mathematical modeling issues in the area of dynamic programming and decision-making under severe uncertainty. I plan to report on lessons learned from this experiment in my books on these topics. The following example illustrates this point.
The Knapsack Problem
This famous problem is discussed in details in some of the conversations I had with ChatGPT and Bard. Here we discuss two very simple versions of the problem, both instances of the generic 0-1 version of the problem.
Book shelving problem: Version 1.
You have a book shelve with one shelve and a pile of books of various widths. Your task is to put as many books as possible on the shelve. Assume that the length of the shelve and the widths of the books are known.
Book shelving problem: Version 2.
You have a book shelve whose length is known and a pile of books of various widths. Your task is to put as many books as possible on the shelve, making sure that there is no gap on the shelve. Assume that the length of the shelve and the widths of the books are known.
These two versions of the problem can be formulated mathematically as follows:
[katex display=true]
\begin{align}
{\color{blue} {\mathrm{Version\ 1:}}}\ \ &\max_{x_{1},\dots,x_{n}} \left\{\sum_{i=1}^{n}x_{i}: \sum_{i=1}^{n}w_{i} \ {\color{blue}\le}\ L,\ \ x_{i}=\{0,1\}, i=1,\dots,n\right\}\\[1mm]
{\color{red} {\mathrm{Version\ 2:}}}\ \ &\max_{x_{1},\dots,x_{n}} \left\{\sum_{i=1}^{n}x_{i}: \sum_{i=1}^{n}w_{i} \ {\color{red}=}\ L, \ \ x_{i}=\{0,1\},\ \ i=1,\dots,n\right\}
\end{align}
[/katex]
In these formulations
[katex display=true]
\begin{align}
L & = \text{ length of the shelve},\\[2mm]
w_{i}& = \text{ width of book $i$}, \ \ i=1,2,\dots,n, \\[2mm]
x_{i}& = \text{ binary decision regarding book $i$, namely}\\[2mm]
& = \begin{cases}
0 & \ \ , \ \ \text{ do not select book $i$}\\
1 & \ \ , \ \ \text{ select book $i$}\\
\end{cases}\ \ , \ \ i=1,2,\dots,n.\\
\end{align}
[/katex]
I plan to explore in detail these problems and their many formulations in my conversations with ChatGPT and Bard. I am particularly interested to check if these chatbots experience the same difficulties that Humans (e.g. human students) have dealing with dynamic programming formulations of these problems.
Here it is sufficient to point out that the mathematical formulations of the two versions are identical except that in Version 1 the functional constraint is a “≤ type” constraint whereas in Version 2 the functional constraint is a “= type” constraint.
My experience over many years of teaching has been that both undergraduate and graduate (human) students have serious difficulties formulating valid dynamic programming functional equations for Version 2 of the problem. Do chatbots exhibit similar difficulties?
Well, my preliminary experiments indicates in no uncertain terms that both ChatGPT and Bard find it difficult to deal with dynamic programming formulations of Version 2.
What does this mean?
An interesting question that I plan to explore in my conversations.
Remark
In my early conversations with ChatGPT (March 2023) and Bard (April 2023) I posed rather “simple” “mathematical questions”, and was extremely surprised, and greatly disappointed, from the answers displayed on the screen. Here is an example of such a long conversation with Bard.
A long conversation with Bard.
This rather long conversation was conducted on April 27, 2023, sometime past 3PM, Melbourne time. As you’ll see, some of Bard’s answers were surprising.
I could not possibly accept this! So …
There is more …
In short then, in addition to the reasons discussed above by ChatGPT and Bard, the main reason that motivated me to embark on this project is the desire to understand the capabilities and limitations of chatbots ability to deal with mathematical modeling issues of the type encountered in the areas of Dynamic Programming and Decision-making Under Severe Uncertainty.
Are these capabilities and limitations similar to those associated with Humans? If not, in what ways are they different?
My conversations with ChatGPT and Bard, presented and analyzed in this site were inspired by my desire to address such questions.
Moshe
Melbourne
April 27, 2023
Appendix: Chatbots and mathematical notation
[katex display=true]\large x=\frac{-b\pm \sqrt{b^{2}-4ac}^{\ }}{2a}[/katex]
Hide/show the Appendix
It should be pointed out that conversing with chatbots such as ChatGPT and Bard on mathematically oriented topics whose treatment requires the use of mathematical notation is not that difficult because these chatbots are “familiar” with [latex]\large {\color{teal} \LaTeX}[/latex] notation and its various imitations (e.g. MathJax and KaTeX)
For instance, here is a very short chat with ChatGPT.
