# Getting some (algorithmic) SAT-isfaction

How can you possibly solve a mission-critical problem with millions of variables—when the worst-case computational complexity of every known algorithm for that problem is exponential in the number of variables?

SAT (Satisfiability) solvers have seen dramatic orders-of-magnitude performance gains for many problems through algorithmic improvements over the last couple of decades or so. The SAT problem—finding an assignment of Boolean variables that makes a given Boolean expression true—represents the archetypal NP-complete problem and in the general case is intractable.

However, for many practical problems, solutions can be found very efficiently by use of modern methods. This “killer app” of computer science, as described by Donald Knuth, has applications to many areas, including software verification, electronic design automation, artificial intelligence, bioinformatics, and planning and scheduling.

Its uses are surprising and diverse, from running billion dollar auctions to solving graph coloring problems to computing solutions to Sudoku puzzles. As an example, I’ve included a toy code below that uses SMT, a relative of SAT, to find the English language suffix rule for regular past tense verbs (“-ed”) from data.

When used as a machine learning method, SAT solvers are quite different from other methods such as neural networks. SAT solvers can for some problems have long or unpredictable runtimes (though MAXSAT can sometimes relax this restriction), whereas neural networks have essentially fixed inference cost (though looping agent-based models do not).

On the other hand, answers from SAT solvers are always guaranteed correct, and the process is interpretable; this is currently not so for neural network-based large language models.

To understand better how to think about this difference in method capabilities, we can take a lesson from the computational science community. There, it is common to have a well-stocked computational toolbox of both slow, accurate methods and fast, approximate methods.

In computational chemistry, ab initio methods can give highly accurate results by solving Schrödinger’s equation directly, but only scale to limited numbers of atoms. Molecular dynamics (MD), however, relies more on approximations, but scales efficiently to many more atoms. Both are useful in different contexts. In fact, the two methodologies can cross-pollenate, for example when ab initio calculations are used to devise force fields for MD simulations.

A lesson to take from this is, it is paramount to find the best tool for the given problem, using any and all means at one’s disposal.

The following are some of my favorite general references on SAT solvers:

It would seem that unless P = NP, commonly suspected to be false, the solution of these kinds of problems for any possible input is hopelessly beyond reach of even the world’s fastest computers. Thankfully, many of the problems we care about have an internal structure that makes them much more solvable (and likewise for neural networks). Continued improvement of SAT/SMT methods, in theory and implementation, will greatly benefit the effective solution of these problems.

### A toy example: find the English past tense suffix rule using Z3

```import csv
import z3

def char2int(c): return ord(c) - ord('a')

def int2char(i): return chr(i + ord('a'))

# Access the language data from the file.
with open('eng_cols.txt', newline='') as csvfile:
table = [row for row in reader]

nrow, ncol = len(table), len(table[0])

# Identify which columns of input table have stem and targeted word form.
stem_col, form_col = 0, 1

# Calculate word lengths.
nstem = [len(table[i][stem_col]) for i in range(nrow)]
nform = [len(table[i][form_col]) for i in range(nrow)]

# Length of suffix being sought.
ns = 2

# Initialize optimizer.
solver = z3.Optimize()

# Define variables to identify the characters of suffix; add constraints.
var_suf = [z3.Int(f'var_suf_{i}') for i in range(ns)]

for i in range(ns):
solver.add(z3.And(var_suf[i] >= 0, var_suf[i] < 26))

# Define variables to indicate whether the given word matches the rule.
var_m = [z3.Bool(f'var_m_{i}') for i in range(nrow)]

# Loop over words.
for i in range(nrow):

# Constraint on number of characters.
constraints = [nform[i] == nstem[i] + ns]

# Constraint that the form contains the stem.
for j in range(nstem[i]):
constraints.append(
table[i][stem_col][j] == table[i][form_col][j]
if j < nform[i] else False)

# Constraint that the end of the word form matches the suffix.
for j in range(ns):
constraints.append(
char2int(table[i][form_col][nform[i]-1-j]) == var_suf[j]
if j < nform[i] else False)

# var_m[i] is the "and" of all these constraints.

# Seek suffix that maximizes number of matches.
count = z3.Sum([z3.If(var_m[i], 1, 0) for i in range(nrow)])
solver.maximize(count)

# Run solver, output results.
if solver.check() == z3.sat:
model = solver.model()
suf = [model[var_suf[i]] for i in range(ns)]
print('Suffix identified: ' +
''.join(list([int2char(suf[i].as_long())
for i in range(ns)]))[::-1])
print('Number of matches: ' + str(model.evaluate(count)) +
' out of ' + str(nrow) + '.')

var_m_values = [model[var_m[i]] for i in range(nrow)]

print('Matches:')
for i in range(nrow):
if var_m_values[i]:
print(table[i][stem_col], table[i][form_col])
```

# The search for the perfect prompt

Anyone with more than casual experience with ChatGPT knows that prompt engineering is a thing. Minor or even trivial changes in a chatbot prompt can have significant effects, sometimes even dramatic ones, on the output [1]. For simple requests it may not make much difference, but for detailed requests it could matter a lot.

Industry leaders said they thought this would be a temporary limitation. But we are now a year and a half into the GPT-4 era, and it’s still a problem. And since the number of possible prompts has scaling that is exponential in the prompt length, it can sometimes be hard to find a good prompt given the task.

One proposed solution is to use search procedures to automate the prompt optimization / prompt refinement process. Given a base large language model (LLM) and an input (a prompt specification, commonly with a set of prompt/answer pair samples for training), a search algorithm seeks the best form of a prompt to use to elicit the desired answer from the LLM.

This approach is sometimes touted [2] as a possible solution to the problem. However, it is not without  limitations.

A main one is cost. With this approach, one search for a good prompt can take many, many trial-and-error invocations of the LLM, with cost measured in dollars compared to the fraction of a cent cost of a single token of a single prompt. I know of one report of someone who does LLM prompting with such a tool full time for his job, at cost of about \$1,000/month (though, for certain kinds of task, one might alternatively seek a good prompt “template” and reuse that across many near-identical queries, to save costs).

This being said, it would seem that for now (depending on budget) our best option for difficult prompting problems is to use search-based prompt refinement methods. Various new tools have come come out recently (for example, [3-6]). The following is a report on some of my (very preliminary) experiences with a couple of these tools.

### PromptAgent

The first is PromptAgent [5]. It’s a research code available on GitHub. The method is based on Monte Carlo tree search (MCTS), which tries out multiple chains of modification of a seed prompt and pursues the most promising. MCTS can be a powerful method, being part of the AlphaGo breakthrough result in 2016.

I ran one of the PromptAgent test problems using GPT-4/GPT-3.5 and interrupted it after it rang up a couple of dollars in charges. Looking at the logs, I was somewhat amazed that it generated long detailed prompts that included instructions to the model for what to pay close attention to, what to look out for, and what mistakes to avoid—presumably based on inspecting previous trial prompts generated by the code.

Unfortunately, PromptAgent is a research code and not fully productized, so it would take some work to adapt to a specific user problem.

### DSPy

DSPy [6] on the other hand is a finished product available for general users. DSPy is getting some attention lately not only as a prompt optimizer but also more generally as a tool for orchestrating multiple LLMs as agents. There is not much by way of simple examples for how to use the code. The website does have an AI chatbot that can generate sample code, but the code it generated for me required significant work to get it to behave properly.

I ran with the MIPRO optimizer which is most well-suited to prompt optimization. My experience with running the code was that it generated many random prompt variations but did not do in-depth prompt modifications like PromptAgent. PromptAgent does one thing, prompt refinement, and must do it well, unlike DSPy which has multiple uses. DSPy would be well-served to have implemented more powerful prompt refinement algorithms.

### Conclusion

I would wholeheartedly agree that it doesn’t seem right for an LLM would be so dependent on the wording of a prompt. Hopefully, future LLMs, with training on more data and other improvements, will do a better job without need for such lengthy trial-and-error processes.

### References

[1]  “Quantifying Language Models’ Sensitivity to Spurious Features in Prompt Design or: How I learned to start worrying about prompt formatting,” https://openreview.net/forum?id=RIu5lyNXjT

[3]  “Evoke: Evoking Critical Thinking Abilities in LLMs via Reviewer-Author Prompt Editing,” https://openreview.net/forum?id=OXv0zQ1umU

[4] “Large Language Models as Optimizers,” https://openreview.net/forum?id=Bb4VGOWELI

[5] “PromptAgent: Strategic Planning with Language Models Enables Expert-level Prompt Optimization,” https://openreview.net/forum?id=22pyNMuIoa

[6] “DSPy: Compiling Declarative Language Model Calls into State-of-the-Art Pipelines,” https://openreview.net/forum?id=sY5N0zY5Od

# Thinking by playing around

Richard Feynman’s Nobel Prize winning discoveries in quantum electrodynamics were partly inspired by his randomly observing a spinning dinner plate one day in the cafeteria. Paul Feyerabend said regarding science discovery, “The only principle that does not inhibit progress is: anything goes” (within relevant ethical constraints, of course).

Ideas can come from anywhere, including physical play. Various books can improve creative discovery skills, like George Pólya’s How to Solve It, Isaac Watts’ Improvement of the Mind, W. J. J. Gordon’s Synectics, and methodologies like mind mapping and C-K theory, to name a few. Many software products present themselves as shiny new tools promising help. However, we are not just disembodied minds interacting with a computer, but instead integrated beings with reasoning integrated with memories, life history, emotions and multisensory input and interaction with the world The tactile is certainly a key avenue of learning, discovering, understanding.

Fidget toys are popular. Different kinds of toys have different semiotics with respect to how they interplay with our imaginations. Like Legos. Structured, like Le Corbusier-style architecture, or multidimensional arrays or tensors, or the snapping together of many software components with well-defined interfaces, with regular scaling from the one to many. Or to take a much older example, Tinkertoys—the analogy of the graph, interconnectedness, semi-structured but composable, like DNA or protein chains, or complex interrelated biological processes, or neuronal connections, or the wild variety within order of human language.

As creative workers, we seek ideas from any and every place to help us in what we do. The tactile, the physical, is a vital place to look.

# Choosing a Computer Language for a Project

Julia. Scala. Lua. TypeScript. Haskell. Go. Dart. Various computer languages new and old are sometimes proposed as better alternatives to mainstream languages. But compared to mainstream choices like Python, C, C++ and Java (cf. Tiobe Index)—are they worth using?

Certainly it depends a lot on the planned use: is it a one-off small project, or a large industrial-scale software application?

Yet even a one-off project can quickly grow to production-scale, with accompanying growing pains. Startups sometimes face a growth crisis when the nascent code base becomes unwieldy and must be refactored or fully rewritten (or you could do what Facebook/Meta did and just write a new compiler to make your existing code base run better).

The scope of different types of software projects and their requirements is so incredibly diverse that any single viewpoint from experience runs a risk of being myopic and thus inaccurate for other kinds of projects. With this caveat, I’ll share some of my own experience from observing projects for many dozens of production-scale software applications written for leadership-scale high performance computing systems. These are generally on a scale of 20,000 to 500,000 lines of code and often require support of mathematical and scientific libraries and middleware for build support, parallelism, visualization, I/O, data management and machine learning.

Here are some of the main issues regarding choice of programming languages and compilers for these codes:

1. Language and compiler sustainability. While the lifetime of computing systems is measured in years, the lifetime of an application code base can sometimes be measured in decades. Is the language it is written in likely to survive and be well-supported long into the future? For example, Fortran, though still used and frequently supported, is is a less common language thus requiring special effort from vendors, with fewer developer resources than more popular languages. Is there a diversity of multiple compilers from different providers to mitigate risk? A single provider means a single point of failure, a high risk; what happens if the supplier loses funding? Are the language and compilers likely to be adaptable for future computer hardware trends (though sometimes this is hard to predict)? Is there a large customer base to help ensure future support? Similarly, is there an adequate pool of available programmers deeply skilled in the language? Does the language have a well-featured standard library ecosystem and good support for third-party libraries and frameworks? Is there good tool support (debuggers, profilers, build tools)?

2. Related to this is the question of language governance. How are decisions about future updates to the language made? Is there broad participation from the user community and responsiveness to their needs? I’ve known members of the C++ language committee; from my limited experience they seem very reasonable and thoughtful about future directions of the language. On the other hand, some standards have introduced features that scarcely anyone ever uses—a waste of time and more clutter for the standard.

3. Productivity. It is said that programmer productivity is limited by the ability of a few lines of code to express high level abstractions that can do a lot with minimal syntax. Does the language permit this? Does the language standard make sense (coherent, cohesive) and follow the principle of least surprise? At the same time, the language should not engulf what might better be handled modularly by a library. For example, a matrix-matrix product that is bound up with the language might be highly productive for simple cases but have difficulty supporting the many variants of matrix-matrix product provided for example by the NVIDIA CUTLASS library. Also, in-language support for CUDA GPU operations, for example, would make it hard for the language not to lag behind in support of the frequent new releases of CUDA.

4. Strategic advantage. The 10X improvement rule states that an innovation is only worth adopting if it offers 10X improvement compared to existing practice according to some metric . If switching to a given new language doesn’t bring significant improvement, it may not be worth doing. This is particularly true if there is an existing code base of some size. A related question is whether the new language offers an incremental transition path for an existing code to the new language (in many cases this is difficult or impossible).

5. Performance (execution speed). Does the language allow one to get down to bare-metal performance rather than going through costly abstractions or an interpreter layer? Are the features of the underlying hardware exposed for the user to access? Is performance predictable? Can one get a sense of the performance of each line of code just by inspection, or is this occluded by abstractions or a complex compilation process? Is the use of just-in-time compilation or garbage collection unpredictable, which could be a problem for parallel computing wherein unexpected “hangs” can be caused by one process unexpectedly performing one of these operations? Do the compiler developers provide good support for effective and accurate code optimization options? Have results from standardized non-cherry-picked benchmarks been published (kernel benchmarks, proxy apps, full applications)?

Early adopters provide a vibrant “early alert” system for new language and tool developments that are useful for small projects and may be broadly impactful. Python was recognized early in the scientific computing community for its potential complementary use with standard languages for large scientific computations. When it comes to planning large-scale software projects, however, a range of factors based on project requirements must be considered to ensure highest likelihood of success.

# Hallucinations of AI Science Models

AlphaFold 2, FourCastNet and CorrDiff are exciting. AI-driven autonomous labs are going to be a big deal [1]. Science codes now use AI and machine learning to make scientific discoveries on the world’s most powerful computers [2].

It’s common practice for scientists to ask questions about the validity, reliability and accuracy of the mathematical and computational methods they use. And many have voiced concerns about the lack of explainability and potential pitfalls of AI models, in particular deep neural networks (DNNs) [3].

The impact of this uncertainty varies highly according to project. Science projects that are able to easily check AI-generated results against ground truth may not be that concerned. High-stakes projects like design of a multimillion dollar spacecraft with high project risks may ask about AI model accuracy with more urgency.

## Neural network accuracy

Understanding of the properties of DNNs is still in its infancy, with many as-yet unanswered questions. However, in the last few years some significant results have started to come forth.

A fruitful approach to analyzing DNNs is to see them as function approximators (which, of course, they are). One can study how accurately DNNs approximate a function representing some physical phenomenon in a domain (for example, fluid density or temperature).

The approximation error can be measured in various ways. A particularly strong measure is “sup-norm” or “max-norm” error, which requires that the DNN approximation be accurate at every point of the target function’s domain (“uniform approximation”). Some science problems may have a weaker requirement than this, such as low RMS or 2-norm error. However, it’s not unreasonable to ask about max-norm approximation behaviors of numerical methods [4,5].

An illuminating paper by Ben Adcock and Nick Dexter looks at this problem [6]. They show that standard DNN methods applied even to a simple 1-dimensional problem can result in “glitches”: the DNN as a whole matches the function well but at some points totally misapproximates the target function. For a picture that shows this, see [7].

Other mathematical papers have subsequently shed light on these behaviors. I’ll try to summarize the findings below, though the actual findings are very nuanced, and many details are left out here. The reader is advised to refer to the respective papers for exact details.

The findings address three questions: 1) how many DNN parameters are required to approximate a function well? 2) how much data is required to train to a desired accuracy? and 3) what algorithms are capable of training to the desired accuracy?

## How many neural network weights are needed?

How large does the neural network need to be for accurate uniform approximation of functions? If tight max-norm approximation requires an excessively large number of weights, then use of DNNs is not computationally practical.

Some answers to this question have been found—in particular, a result 1 is given in [8, Theorem 4.3; cf. 9, 10]. This result shows that the number of neural network weights required to approximate an arbitrary function to high max-norm accuracy grows exponentially in the dimension of the input to the function.

This dependency on dimension is no small limitation, insofar as this is not the dimension of physical space (e.g., 3-D) but the dimension of the input vector (such as the number of gridcells), which for practical problems can be in the tens [11] or even millions or more.

Sadly, this rules out the practical use of DNN for some purposes. Nonetheless, for many practical applications of deep learning, the approximation behaviors are not nearly so pessimistic as this would indicate (cp. [12]). For example, results are more optimistic:

• if the target function has a strong smoothness property;
• if the function is not arbitrary but is a composition of simpler functions;
• if the training and test data are restricted to a (possibly unknown) lower dimensional manifold in the high dimensional space (this is certainly the case for common image and language modeling tasks);
• if the average case behavior for the desired problem domain is much better than the worst case behavior addressed in the theorem;
• The theorem assumes multilayer perceptron and ReLU activation; other DNN architectures may perform better (though the analysis is based on multidimensional Taylor’s theorem, which one might conjecture applies also to other architectures).
• For many practical applications, very high accuracy is not a requirement.
• For some applications, only low 2-norm error is sufficient, (not low max-norm).
• For the special case of physics informed neural networks (PINNs), stronger results hold.

Thus, not all hope is lost from the standpoint of theory. However, certain problems for which high accuracy is required may not be suitable for DNN approximation.

## How much training data is needed?

Assuming your space of neural network candidates is expressive enough to closely represent the target function—how much training data is required to actually find a good approximation?

A result 2 is given in [13, Theorem 2.2] showing that the number of training samples required to train to high max-norm accuracy grows, again, exponentially in the dimension of the input to the function.

The authors concede however that “if additional problem information about [the target functions] can be incorporated into the learning problem it may be possible to overcome the barriers shown in this work.” One suspects that some of the caveats given above might also be relevant here. Additionally, if one considers 2-norm error instead of max-norm error, the data requirement grows polynomially rather than exponentially, making the training data requirement much more tractable. Nonetheless, for some problems the amount of data required is so large that attempts to “pin down” the DNN to sufficient accuracy become intractable.

## What methods can train to high accuracy?

The amount of training data may be sufficient to specify a suitable neural network. But, will standard methods for finding the weights of such a DNN be effective for solving this difficult nonconvex optimization problem?

A recent paper [14] from Max Tegmark’s group empirically studies DNN training to high accuracy. They find that as the input dimension grows, training to very high accuracy with standard stochastic gradient descent methods becomes difficult or impossible.

They also find second order methods perform much better, though these are more computationally expensive and have some difficulty also when the dimension is higher. Notably, second order methods have been used effectively for DNN training for some science applications [15]. Also, various alternative training approaches have been tried to attempt to stabilize training; see, e.g., [16].

## Prospects and conclusions

Application of AI methods to scientific discovery continues to deliver amazing results, in spite of lagging theory. Ilya Sutskever has commented, “Progress in AI is a game of faith. The more faith you have, the more progress you can make” [17].

Theory of deep learning methods is in its infancy. The current findings show some cases for which use of DNN methods may not be fruitful, Continued discoveries in deep learning theory can help better guide how to use the methods effectively and inform where new algorithmic advances are needed.

## Footnotes

1 Suppose the function to be approximated takes d inputs and has the smoothness property that all nth partial derivatives are continuous (i.e., is in Cn(Ω) for compact Ω). Also suppose a multilayer perceptron with ReLU activation functions must be able to approximate any such function to max-norm no worse than ε. Then the number of weights required is at least a fixed constant times (1/ε)d/(2n).

2 Let F be the space of all functions that can be approximated exactly by a broad class of ReLU neural networks. Suppose there is a training method that can recover all these functions up to max-norm accuracy bounded by ε. Then the number of training samples required is at least a fixed constant times (1/ε)d.

## References

[1] “Integrated Research Infrastructure Architecture Blueprint Activity (Final Report 2023),” https://www.osti.gov/biblio/1984466.

[2] Joubert, Wayne, Bronson Messer, Philip C. Roth, Antigoni Georgiadou, Justin Lietz, Markus Eisenbach, and Junqi Yin. “Learning to Scale the Summit: AI for Science on a Leadership Supercomputer.” In 2022 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW), pp. 1246-1255. IEEE, 2022, https://www.osti.gov/biblio/2076211.

[3] “Reproducibility Workshop: The Reproducibility Crisis in ML‑based Science,” Princeton University, July 28, 2022, https://sites.google.com/princeton.edu/rep-workshop.

[4] Wahlbin, L. B. (1978). Maximum norm error estimates in the finite element method with isoparametric quadratic elements and numerical integration. RAIRO. Analyse numérique, 12(2), 173-202, https://www.esaim-m2an.org/articles/m2an/pdf/1978/02/m2an1978120201731.pdf

[5] Kashiwabara, T., & Kemmochi, T. (2018). Maximum norm error estimates for the finite element approximation of parabolic problems on smooth domains. https://arxiv.org/abs/1805.01336.

[6] Adcock, Ben, and Nick Dexter. “The gap between theory and practice in function approximation with deep neural networks.” SIAM Journal on Mathematics of Data Science 3, no. 2 (2021): 624-655, https://epubs.siam.org/doi/10.1137/20M131309X.

[7] “Figure 5 from The gap between theory and practice in function approximation with deep neural networks | Semantic Scholar,” https://www.semanticscholar.org/paper/The-gap-between-theory-and-practice-in-function-Adcock-Dexter/156bbfc996985f6c65a51bc2f9522da2a1de1f5f/figure/4

[8] Gühring, I., Raslan, M., & Kutyniok, G. (2022). Expressivity of Deep Neural Networks. In P. Grohs & G. Kutyniok (Eds.), Mathematical Aspects of Deep Learning (pp. 149-199). Cambridge: Cambridge University Press. doi:10.1017/9781009025096.004, https://arxiv.org/abs/2007.04759.

[9] D. Yarotsky. Error bounds for approximations with deep ReLU networks. Neural Netw., 94:103–114, 2017, https://arxiv.org/abs/1610.01145

[10] I. Gühring, G. Kutyniok, and P. Petersen. Error bounds for approximations with deep relu neural networks in Ws,p norms. Anal. Appl. (Singap.), pages 1–57, 2019, https://arxiv.org/abs/1902.07896

[11] Matt R. Norman, “The MiniWeather Mini App,” https://github.com/mrnorman/miniWeather

[12] Lin, H.W., Tegmark, M. & Rolnick, D. Why Does Deep and Cheap Learning Work So Well?. J Stat Phys 168, 1223–1247 (2017). https://doi.org/10.1007/s10955-017-1836-5

[13] Berner, J., Grohs, P., & Voigtlaender, F. (2022). Training ReLU networks to high uniform accuracy is intractable. ICLR 2023, https://openreview.net/forum?id=nchvKfvNeX0.

[14] Michaud, E. J., Liu, Z., & Tegmark, M. (2023). Precision machine learning. Entropy, 25(1), 175, https://www.mdpi.com/1099-4300/25/1/175.

[15] Markidis, S. (2021). The old and the new: Can physics-informed deep-learning replace traditional linear solvers?. Frontiers in big Data, 4, 669097, https://www.frontiersin.org/articles/10.3389/fdata.2021.669097/full

[16] Bengio, Y., Lamblin, P., Popovici, D., & Larochelle, H. (2006). Greedy layer-wise training of deep networks. Advances in neural information processing systems, 19,  https://papers.nips.cc/paper_files/paper/2006/hash/5da713a690c067105aeb2fae32403405-Abstract.html

[17] “Chat with OpenAI CEO and Co-founder Sam Altman, and Chief Scientist Ilya Sutskever,” https://www.youtube.com/watch?v=mC-0XqTAeMQ&t=250s

# Experiences with Thread Programming in Microsoft Windows

Lately I’ve been helping a colleague to add worker threads to his GUI-based Windows application.

Thread programming can be tricky. Here are a few things I’ve learned along the way.

Performance. This app does compute-intensive work. It is helpful to offload this very compute-heavy work to a worker thread. Doing this frees the main thread to service GUI requests better.

Thread libraries. Windows has multiple thread libraries, for example Microsoft Foundation Class library threads and C++ standard library threads. It is hazardous to use different thread libraries in the same app. In the extreme case, different thread libraries, such as GOMP  vs. LOMP, used in resp. the GCC and LLVM compiler families, have different threading runtimes which keep track of threads in different ways. Mixing them in the same code can cause hazardous silent errors.

Memory fences are a thing. Different threads can run on different processor cores and hold variables in different respective L1 caches that are not flushed (this to maintain high performance). An update to a variable by one thread is not guaranteed to be visible to other threads without special measures. For example, one could safely transfer information using `::PostMessage` coupled with a handler function on the receiver thread. Or one could send a signal using an MFC `CEvent` on one thread and read its `Lock` on the other. Also, a thread launch implicitly does a memory fence, so that, at least then, the new thread is guaranteed to correctly see the state of all memory locations.

GUI access should be done from the master thread only, not a worker thread. Doing so can result in deadlock. A worker thread can instead `::PostMessage` to ask the master thread to do a GUI action.

Thread launch. By default `AfxBeginThread` returns a thread handle which MFC takes care of deleting when no longer needed. If you want to manage the life cycle of the handle yourself, you can do something like:

``````myWorkerThreadHandle = AfxBeginThread(myFunc, myParams,
``````

Joint use of a shared library like the DAO database library has hazards. One should beware of using the library to allocate something in one thread and deallocating in another, as this will likely allocate in a thread-local heap or stack instead of a shared thread-safe heap, this resulting in a crash.

Initialization. One should call `CoInitializeEx(NULL, COINIT_APARTMENTTHREADED)` and `AfxDaoInit()` (if using DAO) at thread initialization on both master and worker threads, and correspondingly `CoUninitialize()` and `AfxDaoTerm()` at completion of the thread.

Monitoring of thread state can be done with
`WaitForSingleObject(myWorkerThreadHandle->m_hThread, 0)` to determine if the thread has completed or `WaitForSingleObject(myWorkerThreadHandle->m_hThread, INFINITE)` for a blocking wait until completion.

Race conditions are always a risk but can be avoided by careful reasoning about execution. Someone once said up to 90% of code errors can be found by desk checking [1]. Race conditions are notoriously hard to debug, partly because they can occur nondeterministically. There are tools for trying to find race condition errors, though I’ve never tried them.

So far I find no rigorous specification of the MFC threading model online that touches on all these concerns. Hopefully this post is useful to someone else working through these issues.

##### References

[1] Dasso, Aristides., Funes, Ana. Verification, Validation and Testing in Software Engineering. United Kingdom: Idea Group Pub., 2007, p. 163.

# How to Organize Technical Research?

64 million scientific papers have been published since 1996 [1].

Assuming you can actually find the information you want in the first place—how can you organize your findings to be able to recall and use them later?

It’s not a trifling question. Discoveries often come from uniting different obscure pieces of information in a new way, possibly from very disparate sources.

Many software tools are used today for notetaking and organizing information, including simple text files and folders, Evernote, GitHub, wikis, Miro, mymind, Synthical and Notion—to name a diverse few.

AI tools can help, though they can’t always recall correctly and get it right, and their ability to find connections between ideas is elementary. But they are getting better [2,3].

One perspective was presented by Jared O’Neal of Argonne National Laboratory, from the standpoint of laboratory notebooks used by teams of experimental scientists [4]. His experience was that as problems become more complex and larger, researchers must invent new tools and processes to cope with the complexity—thus “reinventing the lab notebook.”

While acknowledging the value of paper notebooks, he found electronic methods essential because of distributed teammates. In his view many streams of notes are probably necessary, using tools such as GitLab and Jupyter notebooks. Crucial is the actual discipline and methodology of notetaking, for example a hierarchical organization of notes (separating high-level overview and low-level details) that are carefully written to be understandable to others.

A totally different case is the research methodology of 19th century scientist Michael Faraday. He is not to be taken lightly, being called by some “the best experimentalist in the history of science” (and so, perhaps, even compared to today) [5].

A fascinating paper [6] documents Faraday’s development of “a highly structured set of retrieval strategies as dynamic aids during his scientific research.” He recorded a staggering 30,000 experiments over his lifetime. He used 12 different kinds of record-keeping media, including lab notebooks proper, idea books, loose slips, retrieval sheets and work sheets. Often he would combine ideas from different slips of paper to organize his discoveries. Notably, his process to some degree varied over his lifetime.

Certain motifs emerge from these examples: the value of well-organized notes as memory aids; the need to thoughtfully innovate one’s notetaking methods to find what works best; the freedom to use multiple media, not restricted to a single notetaking tool or format.

Do you have a favorite method for organizing your research? If so, please share in the comments below.

#### References

[1] How Many Journal Articles Have Been Published? https://publishingstate.com/how-many-journal-articles-have-been-published/2023/

[2] “Multimodal prompting with a 44-minute movie | Gemini 1.5 Pro Demo,” https://www.youtube.com/watch?v=wa0MT8OwHuk

[3] Geoffrey Hinton, “CBMM10 Panel: Research on Intelligence in the Age of AI,” https://www.youtube.com/watch?v=Gg-w_n9NJIE&t=4706s

[4] Jared O’Neal, “Lab Notebooks For Computational Mathematics, Sciences, Engineering: One Ex-experimentalist’s Perspective,” Dec. 14, 2022, https://www.exascaleproject.org/event/labnotebooks/

[6] Tweney, R.D. and Ayala, C.D., 2015. Memory and the construction of scientific meaning: Michael Faraday’s use of notebooks and records. Memory Studies8(4), pp.422-439. https://www.researchgate.net/profile/Ryan-Tweney/publication/279216243_Memory_and_the_construction_of_scientific_meaning_Michael_Faraday’s_use_of_notebooks_and_records/links/5783aac708ae3f355b4a1ca5/Memory-and-the-construction-of-scientific-meaning-Michael-Faradays-use-of-notebooks-and-records.pdf

# What’s the Best Code Editor?

Emacs, vi, TextEdit, nano, Sublime, Notepad, Wordpad, Visual Studio, Eclipse, etc., etc.—everyone’s got a favorite.

I used Visual Studio previously and liked the integrated debugger. Recently I started using VS again and found the code editing windows rather cluttered. Thankfully you can tone this down, if you can locate the right options.

Eclipse for Java has instantaneous checking for syntax errors. I have mixed feelings on this. Perhaps you could type a little more code before getting a glaring error message?

Concerning IDEs (integrated development environments) like this—I’ve met people who think that a full GUI-based IDE is the only way to go. Maybe so. However , there’s another view.

You’d think if anyone would know how to write code quickly, accurately and effectively, it would be world-class competitive programmers. They’re the best, right?

One of the very top people is Gennady Korotkevich. He’s won many international competitions.

What does he use? Far Manager, a text-based user interface tool with a mere two panels and command prompt. It’s based on 1980s pre-GUI file manager methodologies that were implemented under DOS.

It reminds me of a conversation I had with our admin when I was in grad school. I asked, “Why do you use vi instead of MS Word for editing documents?” Answer: “I like vi because it’s faster—your fingers never need to leave the keyboard.”

Admittedly, not all developer workflows would necessarily find this approach optimal. But still it makes you think. Sometimes the conventional answer is not the best one.

Do you have a favorite code editor? Please let us know in the comments.

# Avoiding Multiprocessing Errors in Bash Shell

Suppose you have two Linux processes trying to modify a file at the same time and you don’t want them stepping on each other’s work and making a mess.  A common solution is to use a “lock” mechanism (a.k.a. “mutex”). One process “locks the lock” and by this action has sole ownership of a resource in order to make updates, until it unlocks the lock to allow other processes access.

Writing a custom lock in Linux bash shell is tricky. Here’s an example that DOESN’T work right:

```#/bin/bash
let is_locked=1 # helper variable to denote locked state
mylockvariable=\$(cat mylockfile 2>/dev/null)  # read the lock
while [ "\$mylockvariable" != \$is_locked ]  # loop until unlocked
do
sleep 5 # wait 5 seconds to try again
mylockvariable=\$(cat mylockfile 2>/dev/null)  # read again
done
echo \$is_locked > mylockfile  # lock the lock
# >>> do critical work safely here <<<
# >>> ERROR: NOT SAFE <<<
rm mylockfile  # unlock the lock
```

Here the lock value is stored in a shared resource, the file “mylockfile”. If the file exists and contains the character “1”, the lock is considered locked; otherwise, it is considered unlocked.  The code will loop until the lock is unlocked, then acquire the lock, do the required single-process work, and then release the lock.

However, this code can fail without warning: suppose two processes A and B execute this code concurrently. Initially the lock is in an unlocked state. Process A reads the lockfile. Then suppose immediately after this, Process A is temporarily interrupted, perhaps to give CPU cycles to run Process B. Then, suppose Process B begins, reads the lock, locks the lock and starts doing its critical work. Suppose now Process B is put into wait state and Process A is restarted. Process A, since it previously read the lockfile, wrongly believes the lock is unlocked, thus proceeds to also lock the lock and do the critical work—resulting in a mess.

This is an example of a classic race condition, in which the order of execution of threads or processes can affect the final outcome of execution.

A solution to this conundrum is found in the excellent book, Unix Power Tools [1,2]. This is a hefty tome but very accessibly written, for some people well worth a read-through to pick up a slew of time-saving tips.

The problem with the example code is the need to both read and set the lock in a single, indivisible (atomic) operation. Here’s a trick to do it:

```#/bin/bash
until (umask 222; echo > mylockfile) 2>/dev/null  # check and lock
do  # keep trying if failed
sleep 5 # wait 5 seconds to try again
done
# >>> do critical work safely here <<<
rm -f mylockfile  # unlock the lock
```

Here, the existence of the lockfile itself is the indicator that the lock is set. Setting the umask makes this file creation fail if the file already exists, triggering the loop to activate to keep trying. This works because the existence of a file can either be true or false and nothing else; the existence of a file is guaranteed atomicity by the OS and the filesystem. Thus, assuming the system is working correctly, this code is guaranteed to produce the desired behavior.

Race conditions can be a nuisance to find since their occurrence is nondeterministic and can be rare but devastating. Writing correct code for multiple threads of execution can be confusing to those who haven’t done it before. But with experience it becomes easier to reason about correctness and spot such errors.

## References:

[1] Peek, Jerry D., Shelley Powers, Tim O’Reilly and Mike Loukides. “Unix Power Tools, Third Edition.” (2002), https://learning.oreilly.com/library/view/unix-power-tools/0596003307/

# Is Low Precision Arithmetic Safe?

The popularity of low precision arithmetic for computing has exploded since the 2017 release of the Nvidia Volta GPU. The half precision tensor cores of Volta offered a massive 16X performance gain over double precision for key operations. The “race to the bottom” for lower precision computations continues: some have even solved significant problems using 1-bit precision arithmetic hardware ([1], [2]). And hardware performance is getting even better: the Nvidia H100 tensor core-enabled FP16 is a full 58X faster than standard FP64, and 1-bit precision is yet another 16X faster than this, for total speedup of over 900X for algorithms that can use it [3].

This eye-popping speedup certainly draws attention. However, in scientific computing, low precision arithmetic has typically been seen as unsafe for modeling and simulation codes. Indeed, lower precision can sometimes be used to advantage [4], commonly in a “mixed precision” setting in which only parts of the calculation are done in low precision. However, in general anything less than double precision is considered inadequate to model complex physical phenomena with fidelity (see, e.g., [5]).

In response, developers have created tools to measure the safety of reduced precision arithmetic in application codes [6]. Some tools can even identify which variables or arrays can be safely demoted to lower precision without loss of accuracy in the final result. However, use of these tools in a blind fashion, not backed by some kind of reasoning process, can be hazardous.

An example will illustrate this. The conjugate gradient method for linear system solving and optimization [7] and the closely related Lanczos method for eigenvalue problem solving [8] showed great promise following their invention in the early 1950s. However, they were considered unsafe due to catastrophic roundoff errors under floating point arithmetic—even more pronounced as floating point precision is reduced. Nonetheless, Chris Paige showed in his pioneering work in the 1970s [9] that the roundoff error, though substantial, did not preclude the usefulness of the methods when properly used. The conjugate gradient method has gone on to become a mainstay in scientific computing.

Notice that no tool could possibly arrive at this finding, without a careful mathematical analysis of the methods. A tool would detect inaccuracy in the calculation but could not certify that these errors could cause no harm to the final result.

Some might propose instead a purely data-driven approach: just try low precision on some test cases, if it works then use low precision in production. This approach is fraught with peril, however: the test cases may not capture all situations that could be encountered in production.

For example, one might test an aerodynamics code only on smooth flow regimes, but production runs may encounter complex flows with steep gradients—that low precision arithmetic cannot correctly model. Academic papers that test low precision methods and tools must rigorously evaluate in challenging real-world scenarios like this.

Sadly, computational science teams frequently don’t have the time to evaluate their codes for potential use of lower precision arithmetic. Tools could certainly help. Also, libraries that encapsulate mixed precision methods can provide benefits to many users. A great success story here is mixed precision dense linear solvers, founded on the solid theoretical work of Nick Highnam and colleagues [10], which has found its way into libraries such as [11].

So the final answer is, “it depends.” Each new case must be looked at carefully, and a determination made based on some combination of analysis and testing.

### References

[1] Zhang, Y., Garg, A., Cao, Y., Lew, Ł., Ghorbani, B., Zhang, Z. and Firat, O., 2023. Binarized Neural Machine Translation. arXiv preprint arXiv:2302.04907, https://openreview.net/forum?id=XAyPlfmWpu

[2] Lagergren, J., Cashman, M., Vergara, V.G.M., Eller, P.R., Gazolla, J.G.F.M., Chhetri, H.B., Streich, J., Climer, S., Thornton, P., Joubert, W. and Jacobson, D., 2023. Climatic clustering and longitudinal analysis with impacts on food, bioenergy, and pandemics. Phytobiomes Journal, 7(1), pp.65-77, https://apsjournals.apsnet.org/doi/10.1094/PBIOMES-02-22-0007-R.

[3] “NVIDIA H100 Tensor Core GPU Datasheet,” https://resources.nvidia.com/en-us-tensor-core/nvidia-tensor-core-gpu-datasheet.

[4] G. Alvarez et al., “New algorithm to enable 400+ TFlop/s sustained performance in simulations of disorder effects in high-Tc superconductors,” SC ’08: Proceedings of the 2008 ACM/IEEE Conference on Supercomputing, Austin, TX, USA, 2008, pp. 1-10, doi: 10.1109/SC.2008.5218119.

[5] Spafford, K., Meredith, J., Vetter, J., Chen, J., Grout, R., Sankaran, R. (2010). Accelerating S3D: A GPGPU Case Study. In: Lin, HX., et al. Euro-Par 2009 – Parallel Processing Workshops. Euro-Par 2009. Lecture Notes in Computer Science, vol 6043. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14122-5_16.

[6] “Mixed precision analysis tools,” https://scholar.google.com/scholar?q=mixed+precision+analysis+tools

[7] Hestenes, M.R. and Stiefel, E., 1952. Methods of conjugate gradients for solving linear systems. Journal of research of the National Bureau of Standards49(6), pp.409-436, https://nvlpubs.nist.gov/nistpubs/jres/049/jresv49n6p409_a1b.pdf.

[8] Cornelius Lanczos, An Iteration Method for the Solution of the Eigenvalue Problem of Linear Differential and Integral Operators, Journal of Research of the National Bureau of Standards Vol. 45, No. 4, October 1950, https://nvlpubs.nist.gov/nistpubs/jres/045/4/V45.N04.A01.pdf.

[9] Paige, Christopher C.. “The computation of eigenvalues and eigenvectors of very large sparse matrices.” (1971), https://www.cs.mcgill.ca/~chris/pubClassic/PaigeThesis.pdf.

[10] Higham, N.J., Pranesh, S. and Zounon, M., 2019. Squeezing a matrix into half precision, with an application to solving linear systems. SIAM Journal on Scientific Computing41(4), pp.A2536-A2551, https://epubs.siam.org/doi/abs/10.1137/18M1229511.

[11] Lu, Hao; Matheson, Michael; Wang, Feiyi; Joubert, Wayne; Ellis, Austin; Oles, Vladyslav. “OpenMxP-Opensource Mixed Precision Computing,” https://www.osti.gov/biblio/1961398.