A gentle introduction to automated reasoning

Meet Amazon Science’s newest research area.

This week, Amazon Science added automated reasoning to its list of research areas. We made this change because of the impact that automated reasoning is having here at Amazon. For example, Amazon Web Services’ customers now have direct access to automated-reasoning-based features such as IAM Access Analyzer, S3 Block Public Access, or VPC Reachability Analyzer. We also see Amazon development teams integrating automated-reasoning tools into their development processes, raising the bar on the security, durability, availability, and quality of our products.

The goal of this article is to provide a gentle introduction to automated reasoning for the industry professional who knows nothing about the area but is curious to learn more. All you will need to make sense of this article is to be able to read a few small C and Python code fragments. I will refer to a few specialist concepts along the way, but only with the goal of introducing them in an informal manner. I close with links to some of our favorite publicly available tools, videos, books, and articles for those looking to go more in-depth.

Let’s start with a simple example. Consider the following C function:

bool f(unsigned int x, unsigned int y) {
   return (x+y == y+x);
}

Take a few moments to answer the question “Could f ever return false?” This is not a trick question: I’ve purposefully used a simple example to make a point.

To check the answer with exhaustive testing, we could try executing the following doubly nested test loop, which calls f on all possible pairs of values of the type unsigned int:

#include<stdio.h>
#include<stdbool.h>
#include<limits.h>

bool f(unsigned int x, unsigned int y) {
   return (x+y == y+x);
}

void main() {
   for (unsigned int x=0;1;x++) {
      for (unsigned int y=0;1;y++) {
         if (!f(x,y)) printf("Error!\n");
         if (y==UINT_MAX) break;
      }
      if (x==UINT_MAX) break;
   }
}

Unfortunately, even on modern hardware, this doubly nested loop will run for a very long time. I compiled it and ran it on a 2.6 GHz Intel processor for over 48 hours before giving up.

Why does testing take so long? Because UINT_MAX is typically 4,294,967,295, there are 18,446,744,065,119,617,025 separate f calls to consider. On my 2.6 GHz machine, the compiled test loop called f approximately 430 million times a second. But to test all 18 quintillion cases at this performance, we would need over 1,360 years.

When we show the above code to industry professionals, they almost immediately work out that f can't return false as long as the underlying compiler/interpreter and hardware are correct. How do they do that? They reason about it. They remember from their school days that x + y can be rewritten as y + x and conclude that f always returns true.

Re:Invent 2021 keynote address by Peter DeSantis, senior vice president for utility computing at Amazon Web Services
Skip to 15:49 for a discussion of Amazon Web Services' work on automated reasoning.

An automated reasoning tool does this work for us: it attempts to answer questions about a program (or a logic formula) by using known techniques from mathematics. In this case, the tool would use algebra to deduce that x + y == y + x can be replaced with the simple expression true.

Automated-reasoning tools can be incredibly fast, even when the domains are infinite (e.g., unbounded mathematical integers rather than finite C ints). Unfortunately, the tools may answer “Don’t know” in some instances. We'll see a famous example of that below.

The science of automated reasoning is essentially focused on driving the frequency of these “Don’t know” answers down as far as possible: the less often the tools report "Don't know" (or time out while trying), the more useful they are.

Today’s tools are able to give answers for programs and queries where yesterday’s tools could not. Tomorrow’s tools will be even more powerful. We are seeing rapid progress in this field, which is why at Amazon, we are increasingly getting so much value from it. In fact, we see automated reasoning forming its own Amazon-style virtuous cycle, where more input problems to our tools drive improvements to the tools, which encourages more use of the tools.

A slightly more complex example. Now that we know the rough outlines of what automated reasoning is, the next small example gives a slightly more realistic taste of the sort of complexity that the tools are managing for us.

void g(int x, int y) {
   if (y > 0)
      while (x > y)
         x = x - y;
}

Or, alternatively, consider a similar Python program over unbounded integers:

def g(x, y):
   assert isinstance(x, int) and isinstance(y, int)
   if y > 0:
      while x > y:
         x = x - y

Try to answer this question: “Does g always eventually return control back to its caller?”

When we show this program to industry professionals, they usually figure out the right answer quickly. A few, especially those who are aware of results in theoretical computer science, sometimes mistakenly think that we can't answer this question, with the rationale “This is an example of the halting problem, which has been proved insoluble”. In fact, we can reason about the halting behavior for specific programs, including this one. We’ll talk more about that later.

Here’s the reasoning that most industry professionals use when looking at this problem:

  1. In the case where y is not positive, execution jumps to the end of the function g. That’s the easy case.
  2. If, in every iteration of the loop, the value of the variable x decreases, then eventually, the loop condition x > y will fail, and the end of g will be reached.
  3. The value of x always decreases only if y is always positive, because only then does the update to x (i.e., x = x - y) decrease x. But y’s positivity is established by the conditional expression, so x always decreases.

The experienced programmer will usually worry about underflow in the x = x - y command of the C program but will then notice that x > y before the update to x and thus cannot underflow.

If you carried out the three steps above yourself, you now have a very intuitive view of the type of thinking an automated-reasoning tool is performing on our behalf when reasoning about a computer program. There are many nitty-gritty details that the tools have to face (e.g., heaps, stacks, strings, pointer arithmetic, recursion, concurrency, callbacks, etc.), but there’s also decades of research papers on techniques for handling these and other topics, along with various practical tools that put these ideas to work.

Policy-code.gif
Automated reasoning can be applied to both policies (top) and code (bottom). In both cases, an essential step is reasoning about what's always true.

The main takeaway is that automated-reasoning tools are usually working through the three steps above on our behalf: Item 1 is reasoning about the program’s control structure. Item 2 is reasoning about what is eventually true within the program. Item 3 is reasoning about what is always true in the program.

Note that configuration artifacts such as AWS resource policies, VPC network descriptions, or even makefiles can be thought of as code. This viewpoint allows us to use the same techniques we use to reason about C or Python code to answer questions about the interpretation of configurations. It’s this insight that gives us tools like IAM Access Analyzer or VPC Reachability Analyzer.

An end to testing?

As we saw above when looking at f and g, automated reasoning can be dramatically faster than exhaustive testing. With tools available today, we can show properties of f or g in milliseconds, rather than waiting lifetimes with exhaustive testing.

Can we throw away our testing tools now and just move to automated reasoning? Not quite. Yes, we can dramatically reduce our dependency on testing, but we will not be completely eliminating it any time soon, if ever. Consider our first example:

bool f(unsigned int x, unsigned int y) {
   return (x + y == y + x);
}

Recall the worry that a buggy compiler or microprocessor could in fact cause an executable program constructed from this source code to return false. We might also need to worry about the language runtime. For example, the C math library or the Python garbage collector might have bugs that cause a program to misbehave.

What’s interesting about testing, and something we often forget, is that it’s doing much more than just telling us about the C or Python source code. It’s also testing the compiler, the runtime, the interpreter, the microprocessor, etc. A test failure could be rooted in any of those tools in the stack.

Automated reasoning, in contrast, is usually applied to just one layer of that stack — the source code itself, or sometimes the compiler or the microprocessor. What we find so valuable about reasoning is it allows us to clearly define both what we do know and what we do not know about the layer under inspection.

Furthermore, the models of the surrounding environment (e.g., the compiler or the procedure calling our procedure) used by the automated-reasoning tool make our assumptions very precise. Separating the layers of the computational stack helps make better use of our time, energy, and money and the capabilities of the tools today and tomorrow.

Unfortunately, we will almost always need to make assumptions about something when using automated reasoning — for example, the principles of physics that govern our silicon chips. Thus, testing will never be fully replaced. We will want to perform end-to-end testing to try and validate our assumptions as best we can.

An impossible program

I previously mentioned that automated-reasoning tools sometimes return “Don’t know” rather than “yes” or “no”. They also sometimes run forever (or time out), thus never returning an answer. Let’s look at the famous "halting problem" program, in which we know tools cannot return “yes” or “no”.

Imagine that we have an automated-reasoning API, called terminates, that returns “yes” if a C function always terminates or “no” when the function could execute forever. As an example, we could build such an API using the tool described here (shameless self-promotion of author’s previous work). To get the idea of what a termination tool can do for us, consider two basic C functions, g (from above),

void g(int x, int y) {
   if (y > 0)
      while (x > y)
         x = x - y;
}

and g2:

void g2(int x, int y) {
   while (x > y)
      x = x - y;
}

For the reasons we have already discussed, the function g always returns control back to its caller, so terminates(g) should return true. Meanwhile, terminates(g2) should return false because, for example, g2(5, 0) will never terminate.

Now comes the difficult function. Consider h:

void h() {
   if terminates(h) while(1){}
}

Notice that it's recursive. What’s the right answer for terminates(h)? The answer cannot be "yes". It also cannot be "no". Why?

Imagine that terminates(h) were to return "yes". If you read the code of h, you’ll see that in this case, the function does not terminate because of the conditional statement in the code of h that will execute the infinite loop while(1){}. Thus, in this case, the terminates(h) answer would be wrong, because h is defined recursively, calling terminates on itself.

Similarly, if terminates(h) were to return "no", then h would in fact terminate and return control to its caller, because the if case of the conditional statement is not met, and there is no else branch. Again, the answer would be wrong. This is why the “Don’t know” answer is actually unavoidable in this case.

The program h is a variation of examples given in Turing’s famous 1936 paper on decidability and Gödel’s incompleteness theorems from 1931. These papers tell us that problems like the halting problem cannot be “solved”, if by“solved” we mean that the solution procedure itself always terminates and answers either “yes” or “no” but never “Don’t know”. But that is not the definition of “solved” that many of us have in mind. For many of us, a tool that sometimes times out or occasionally returns “Don’t know” but, when it gives an answer, always gives the right answer is good enough.

This problem is analogous to airline travel: we know it’s not 100% safe, because crashes have happened in the past, and we are sure that they will happen in the future. But when you land safely, you know it worked that time. The goal of the airline industry is to reduce failure as much as possible, even though it’s in principle unavoidable.

To put that in the context of automated reasoning: for some programs, like h, we can never improve the tool enough to replace the "Don't know" answer. But there are many other cases where today's tools answer "Don't know", but future tools may be able to answer "yes" or "no". The modern scientific challenge for automated-reasoning subject-matter experts is to get the practical tools to return “yes” or “no” as often as possible. As an example of current work, check out CMU professor and Amazon Scholar Marijn Heule and his quest to solve the Collatz termination problem.

Another thing to keep in mind is that automated-reasoning tools are regularly trying to solve “intractable” problems, e.g., problems in the NP complexity class. Here, the same thinking applies that we saw in the case of the halting problem: automated-reasoning tools have powerful heuristics that often work around the intractability problem for specific cases, but those heuristics can (and sometimes do) fail, resulting in “Don’t know” answers or impractically long execution time. The science is to improve the heuristics to minimize that problem.

Nomenclature

A host of names are used in the scientific literature to describe interrelated topics, of which automated reasoning is just one. Here’s a quick glossary:

  • A logic is a formal and mechanical system for defining what is true and untrue. Examples: propositional logic or first-order logic.
  • A theorem is a true statement in logic. Example: the four-color theorem.
  • A proof is a valid argument in logic of a theorem. Example: Gonthier's proof of the four-color theorem. 
  • A mechanical theorem prover is a semi-automated-reasoning tool that checks a machine-readable expression of a proof often written down by a human. These tools often require human guidance. Example: HOL-light, from Amazon researcher John Harrison. 
  • Formal verification is the use of theorem proving when applied to models of computer systems to prove desired properties of the systems. Example: the CompCert verified C compiler. 
  • Formal methods is the broadest term, meaning simply the use of logic to reason formally about models of systems. 
  • Automated reasoning focuses on the automation of formal methods. 
  • A semi-automated-reasoning tool is one that requires hints from the user but still finds valid proofs in logic. 

As you can see, we have a choice of monikers when working in this space. At Amazon, we’ve chosen to use automated reasoning, as we think it best captures our ambition for automation and scale. In practice, some of our internal teams use both automated and semi-automated reasoning tools, because the scientists we've hired can often get semi-automated reasoning tools to succeed where the heuristics in fully automated reasoning might fail. For our externally facing customer features, we currently use only fully automated approaches.

Next steps

In this essay, I’ve introduced the idea of automated reasoning, with the smallest of toy programs. I haven’t described how to handle realistic programs, with heap or concurrency. In fact, there are a wide variety of automated-reasoning tools and techniques, solving problems in all kinds of different domains, some of them quite narrow. To describe them all and the many branches and sub-disciplines of the field (e.g. “SMT solving”, “higher-order logic theorem proving”, “separation logic”) would take thousands of blogs posts and books.

Automated reasoning goes back to the early inventors of computers. And logic itself (which automated reasoning attempts to solve) is thousands of years old. In order to keep this post brief, I’ll stop here and suggest further reading. Note that it’s very easy to get lost in the weeds reading depth-first into this area, and you could emerge more confused than when you started. I encourage you to use a bounded depth-first search approach, looking sequentially at a wide variety of tools and techniques in only some detail and then moving on, rather than learning only one aspect deeply.

Suggested books:

International conferences/workshops:

Tool competitions:

Some tools:

Interviews of Amazon staff about their use of automated reasoning:

AWS Lectures aimed at customers and industry:

AWS talks aimed at the automated-reasoning science community:

AWS blog posts and informational videos:

Some course notes by Amazon Scholars who are also university professors:

A fun deep track:

Some algorithms found in the automated theorem provers we use today date as far back as 1959, when Hao Wang used automated reasoning to prove the theorems from Principia Mathematica.

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Fauna Robotics is seeking a Principal Applied Scientist to lead the development of evaluation frameworks and data collection protocols that measure, stress-test, and improve robot behavior across real-world tasks. Operating at the intersection of robotics, machine learning, and human-in-the-loop systems, this role owns the infrastructure and methodologies that connect teleoperation, evaluation, and learning — shaping how policies are validated and how high-quality datasets are generated to accelerate system performance. Key job responsibilities What you'll do: - Build and own evaluation frameworks that measure and stress-test robot behavior across real-world tasks - Design data collection protocols that generate high-quality datasets to accelerate system performance - Connect teleoperation, evaluation, and learning systems into cohesive, measurable infrastructure - Translate ambiguous capability goals into actionable, measurable evaluation systems What we're looking for: - A highly experimental, systems-oriented scientist comfortable working across software, robotics, and data pipelines - Deep expertise in machine learning and robotics, with a track record of building evaluation methodologies About the team Fauna Robotics, an Amazon company, is building capable, safe, and genuinely delightful robots for everyday life. Our goal is simple: make robots people actually want to live and interact with in everyday human spaces. We believe that future won’t arrive until building for robotics becomes far more accessible. Today, too much effort is spent reinventing the fundamentals. We’re changing that by developing tightly integrated hardware and software systems that make it faster, safer, and more intuitive to create real-world robotic products. Our work spans the full stack: mechanical design, control systems, dynamic modeling, and intelligent software. The focus is not just functionality, but experience. We’re building robots that feel responsive, expressive, and genuinely useful. At Fauna, you’ll work at the frontier of this space, helping define how robots move, manipulate, and interact with people in natural environments. It’s an opportunity to solve hard problems across hardware and software with a team focused on making robotics accessible and joyful to build. If you care about making robotics real for everyone and building systems that are as delightful as they are capable, we’re interested in hearing from you. an opportunity to solve hard problems across hardware and software with a team focused on making robotics accessible and joyful to build. If you care about making robotics real for everyone and building systems that are as delightful as they are capable, we’re interested in hearing from you.
US, WA, Seattle
Amazon's eCommerce Foundation (eCF) organization provides the core technologies that drive and power Amazon's Stores, Digital, and Other (SDO) businesses. Millions of customer page views and orders per day are enabled by the systems eCF builds from the ground up. CloudTune, within eCF, empowers growth and business agility needs by automatically and efficiently managing AWS capacity and business processes needed to safely meet Amazon's customer demand. CloudTune serves its primary customers, internal software teams, through forecast-driven automation of cost controllership, capacity management, and scaling. We predict expected load, and drive procurement and allocation of AWS capacity for new product launches and high-velocity events like Prime Day and Cyber Monday. CloudTune is looking for a Data Scientist to join our forecasting and optimization team. The team develops sophisticated algorithms that combine machine learning-based demand forecasting with mathematical optimization to solve large-scale capacity planning problems under uncertainty. We work with massive datasets — spanning thousands of availability zone and instance family combinations across global regions — to determine optimal resource allocation strategies that balance infrastructure holding costs against service availability requirements. These models directly inform multi-million dollar capacity investment decisions and drive automated procurement, retention, and release policies across Amazon's compute infrastructure. As a Data Scientist in CloudTune, you will work with other scientists, software engineers, data engineers, and product managers on a variety of important research problems in the areas of stochastic optimization, time series modeling, and operations research. You will work on statistical problems with a high level of ambiguity. You will analyze and process large amounts of data, develop new algorithms and improve existing approaches based on statistical models, machine learning algorithms and big data solutions to automatically scale Amazon’s compute infrastructure, optimizing the balance between availability risk and cost efficiency for all of Amazon businesses. Key job responsibilities - Process and analyze large data sets, mining additional data sources as needed - Build statistical models and drive scalable solutions for multi-year capacity demand forecasting horizons - Create, enhance, and maintain technical documentation, and present to other scientists and business leaders - Process and analyze large-scale operational data to identify patterns and inform model development - Create, enhance, and maintain technical documentation, and present research findings to scientists, engineering teams, and senior leadership
US, WA, Seattle
Pricing is one of the most consequential decisions Amazon makes — and the science behind it needs to be causally rigorous, not just predictive. The P2 Optimization Science (P2OS) team builds the machine learning systems that power Amazon's pricing decisions at scale: demand lift models, customer lifetime value frameworks, and the experimentation infrastructure that validates whether our pricing changes actually work. We're hiring an Applied Scientist to own causal inference at the intersection of ML and pricing experimentation. This role exists because our team has identified a real gap: the methodological bridge between econometric analysis (owned by our economists) and production-scale ML pipelines (owned by our engineers) needs a practitioner who lives in both worlds. You'll build CATE estimation models, design analysis workflows for pricing weblabs, and develop the reusable causal ML infrastructure that the broader team — including non-ML scientists — can rely on. This is not a research role. The bias here is toward shipping production-quality causal pipelines with real downstream business impact. You'll measure success by what changes in LTV estimates, what pricing errors your models help avoid, and whether the economists on your team can actually use what you build. If you're a scientist who wants to work on hard causal identification problems in a high-stakes production environment — and who finds satisfaction in making rigorous methods accessible to a broader team — this role is for you.
US, MA, North Reading
Amazon Robotics is transforming warehouse automation through edge AI and machine learning applied to real-world robotics challenges. We're seeking a Research Scientist to advance our mobile manipulation capabilities by developing novel learning-based approaches that enable robots to navigate and manipulate objects in dynamic fulfillment environments. This role offers the opportunity to conduct original research and translate state-of-the-art findings into production systems operating at Amazon's unprecedented scale. Key job responsibilities Research and Algorithm Development: Formulate novel research problems in robot learning and manipulation, design new model architectures, validate hypotheses through rigorous experimentation, and advance the state of the art in learning-based robotics. Data Strategy and Pipeline Design: Define data requirements for research initiatives, design scalable collection and curation strategies, establish governance and provenance standards, and build reusable pipelines ensuring data quality and reproducibility. Experimentation and Scientific Validation: Design and execute experiments in simulation and real-world embodiments, develop evaluation methodologies and benchmarks, perform ablation studies and statistical analyses, and iterate systematically to advance model performance. Prototyping and Research Infrastructure: Develop clean, well-documented research codebases, build experimentation frameworks and evaluation tooling, contribute to shared training infrastructure, and implement interfaces for broader robotics integration. Scientific Leadership and Publication: Drive an independent research agenda aligned with team objectives, publish at top-tier venues (e.g., RSS, CoRL, ICRA, NeurIPS), identify research gaps through literature reviews, and present findings via technical reports and talks. Cross-Functional Collaboration: Partner with scientists, engineers, and leaders across teams to translate research into deployable solutions, mentor junior researchers, contribute to the team's scientific culture, and support integration with robotics hardware teams. A day in the life If you are not sure that every qualification on the list above describes you exactly, we'd still love to hear from you! At Amazon, we value people with unique backgrounds, experiences, and skillsets. If you’re passionate about this role and want to make an impact on a global scale, please apply! About the team Are you inspired by invention? Is problem solving through teamwork in your DNA? Do you like the idea of seeing how your work impacts the bigger picture? Answer yes to any of these and you’ll fit right in here at Amazon Robotics. We are a smart, collaborative team of enthusiastic doers that work passionately to apply innovative advances in robotics and software to solve real-world challenges that will transform our customers’ experiences in ways we can’t even image yet. We invent new improvements every day. We are Amazon Robotics and we will give you the tools and support you need to invent with us in ways that are rewarding, fulfilling and fun!
US, NY, New York
Amazon Advertising is one of Amazon's fastest growing and most profitable businesses. Amazon's advertising portfolio helps merchants, retail vendors, and brand owners succeed via native advertising, which grows incremental sales of their products sold through Amazon. The primary goals are to help shoppers discover new products they love, be the most efficient way for advertisers to meet their business objectives, and build a sustainable business that continuously innovates on behalf of customers. Our products and solutions are strategically important to enable our Retail and Marketplace businesses to drive long-term growth. We deliver billions of ad impressions and millions of clicks and break fresh ground in product and technical innovations every day! The Creative X team within Amazon Advertising time aims to democratize access to high-quality creatives (text, audio, images, videos) by building AI-driven solutions for advertisers. To accomplish this, we are investing in latent-diffusion models, large language models (LLM), generative audio (music and speech synthesis), computer vision (CV), reinforced learning (RL) and related methods. You will be part of a close-knit team of applied scientists and product managers who are highly collaborative and at the top of their respective fields. We are looking for talented Applied Scientists who have a proven record of developing Generative AI science at the highest standards of innovation, and experience working and delivering product-ready code, are adept at a variety of skills, especially with LLMs, generative audio, computer vision, latent diffusion or related foundational models that will accelerate our plans to generate high-quality creatives on behalf of advertisers. Every member of the team is expected to build customer (advertiser) facing features, contribute to the collaborative spirit within the team, publish, patent, and bring innovation research to raise the bar within the team.