The Shape of Code

This post is a overview of the techniques I use when working with LLMs as a mathematics assistant to derive equations for the aggregate behavior of a collection of software processes. Much of the following could well apply to non-software processes, but my experience is software based.

Any analysis starts with one or more questions/problems and the environment within which any answer is likely to be applicable. Possible questions include:

1. Expected number of lines of code, LOC, in the next release of a program,
2. number of distinct statement sequences that can be written using if-statements and assignment statements,
3. fraction of a program’s functions/methods that have not been modified after fixing all reported faults.

Prompting an LLM with a software related question is likely to result in it giving a software related answer summarising statements made on blogs and perhaps findings from various research papers. These summaries may, or may not, contain the desired answer.

Obtaining a mathematical answer requires that a mathematical question be asked. The software question has to be reframed in mathematical terms. This requires some creativity, lateral thinking, and prior experience is very helpful.

For instance, for question 1, expected lines of code could be reframed as a recurrence relation, such as , where: is the LOC in the ‘th release, and , are constants. An LLM can solve this to relation to give an equation for based on , , and (the size of the first release).

A more sophisticated set of recurrence relations could include the number of developers working on the project, along the probability distribution of the number of LOC they produce per day/week/month/release, and an equation for the expected time between releases.

Expressing question 2 in mathematical terms requires some lateral thinking. One possibility is to treat each line of code as a step on a 2-D lattice. An assignment statement occupying a line is one step down the page, while an if-statement occupies both a line and is (usually) indented to the right of the previous statement at the start, and indented to the left when it ends. Paths through a lattice are a well studied problem, with lots of existing mathematics for LLMs to have been trained on. This reframing of the question was good enough for me to be able to shepherd ChatGPT towards deriving an answer to the question. My pre-LLM research for answering a related question helped.

Creating a mathematical description of a question requires a lot of hard thinking (at least it does for me), and is an iterative process. If you are lucky, a good enough mapping to a starting formula is found, and the mathematics appears in textbooks, e.g., question 1. For other questions, lateral thinking may produce a mapping to a well researched area within some mathematical niche, e.g., question 2.

Creating a correct mathematical specification of the question is essential. Get this specification wrong, and any final equation will be the answer to a different question than the one intended. Mathematicians are used to describing problems in mathematical terms, while non-mathematicians (like me, and I suspect many readers) are likely to make mistakes and under/over specify the problem.

What can be done to check whether an LLM has interpreted the question in the way intended?

LLM chain-of-thought output provides the required feedback about how it interprets the question given. Some LLMs provide chain-of-thought output of their interpretation of the information contained in the prompt (e.g., Deepseek and Kimi), while others (e.g., until recently both ChatGPT and Grok; both have improved in the last few months) provide none, i.e., they are silent for a long time before giving an answer (which is can be useful for double-checking the output from other LLMs, but is otherwise of little use).

The following discussion is based on the process I used to obtain an answer to question 3.

The number of ways of placing some number of balls in some number of boxes is an established mathematical area of study that can be mapped to question 3. Functions are treated as empty boxes and fault reports as balls that are randomly placed in these boxes.

The initial mathematical question contains the minimum of constraints, and successive questions added constraints to better mimic the characteristics of source code and fault reports.

The following was my starting question:

$m$ identical boxes can each hold a maximum of $L$ balls,
and there are $b$ identical balls,
balls are uniformly at randomly placed in non-full boxes,
where $L < m$ and $m*L > b$ and 
$b$ can have a similar size as $m$
What is the expected number of boxes that do not contain a ball

The mathematics is enclosed in $ characters. LLMs support LaTeX input and output mathematics using LaTeX.

The first two lines specify the structure of the system. It’s important to specify that the balls are identical, otherwise the LLM has to decide whether they are, or are not, identical (non-identical has a very different solution).

The third line specifies the process behavior. The phrase “uniformly at randomly” is the mathematical way of saying “the behavior is random with all possibilities being equally likely”. When I first started using LLMs, this phrase is not something I used. However, “uniformly at randomly” often appeared in LLM output, so I switched to using it (LLMs having been trained on a lot more maths than me).

Lines four and five specify relationships between variables. Sometimes constraints such as these reduce the space of possibilities, and lead to a more concise answer. These constraints specify that a function will not contain more reported faults than it has lines of code (which is not always true), and that a program will not have more reported faults than lines of code.

The last line is the question I want answered (Kimi response).

In practice functions don’t all have the same size. Most functions are short, with fewer longer ones. The number of functions containing a given number of lines has (roughly) a power law distribution. Adding this information to the problem gives:

There are $m$ boxes, $B_n$, $1 <= n <= m$ where the
number of boxes that can hold $L$ balls, $1 <= L <= T$,
is proportional to $L^{-b}$,
and there are $F$ identical balls,
balls are uniformly at randomly placed in non-full boxes,
where the number of balls $F$ is much less than
the available places to hold them.
What is the expected number of boxes that do not contain a ball

Boxes can now have different sizes, so they need to be labelled (i.e., $B_n$), with ball carrying capacity specified as a power law, i.e., $L^{-b}$.

The explicit constraints previously given are replaced by the general statement: “… the number of balls $F$ is much less than the available places to hold them.” This gives the LLM some flexibility about how to interpret the constraint.

LLMs make mistakes. I have seen them make a basic algebra mistake on one output line, followed by output that looks like penetrating insight (if a human had made it).

The chain-of-thought output reads like a derivation that a human would write (at least it does for Kimi, Deepseek, and recently GLM 5.1 from Z.ai). Checking the correctness of this derivation is necessary to gain confidence that the final answer is correct, or not.

This chain-of-thought often makes use a theorem or identity that is new to me. Kimi’s response to the updated prompt above made use of the polylogarithm function, which I had heard of, but knew nothing about.

When new to me maths is generated, Wikipedia is the first place I look. However, some Wikipedia maths articles appear to be written by mathematicians, who assume the reader already understands the topic, and simply summarise the relevant details; which is useless. Of course, one can always ask an LLM (a different one, so there is some cross-checking).

If the chain-of-thought looks correct, is the answer correct?

An LLM once give me an answer that was obviously wrong. It was an equation that could produce negative values, and in practice only positive values were possible. Each step of the chain-of-thought looked correct. It took me a while to spot that two disjoint assumptions in the LLM analysis combined to produce the incorrect answer.

I usually have an expectation of the behavior of any answer. Plotting the value of the equation given in an answer can show whether it follows the pattern of expected behavior.

Another way of gaining confidence in the answer is to give the prompt to multiple LLMs. Sometimes they all agree, and sometimes they disagree. The disagreement may be because the answer has been written in a slightly different form (e.g., summing a series from zero rather than one), or because the LLMs made slightly different assumptions. Comparing chain-of-thought will locate the points where assumptions diverge.

The third major iteration tried to address the observation that some functions are executed more often than others, and so are more likely to be involved in a fault report.

The specification was updated to include a preferential attachment component, with a box containing a ball having a higher probability of receiving a ball than one that did not contain a ball. The added text:

The equation in the answer was rather complicated (ChatGPT response). I have not checked this equation.

Most of the mathematically oriented questions/problems I have worked on have turned out to have uninteresting answers. Knowing this I can cross them off my list of things to think about. A few might lead to something interesting (e.g., fault prediction is starting to look like a waste of time), but need more work.

The answer checking process increases confidence that a particular answer is a solution to the question asked. It is possible that the specification of the question asked does not have a strong connection to reality.

My current first choice of LLMs for mathematical problems are Deepseek, Kimi, and recently GLM 5.1 (which has compute availability issues). This is primarily because they provide chain-of-thought output. In the last few months both ChatGPT and Grok have started providing more chain-of-thought output.

I usually start with one LLM to refine the question, and depending on progress later involve other LLMs to check and verify output.