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 bysolved” 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:

  • logic is a formal and mechanical system for defining what is true and untrue. Examples: propositional logic or first-order logic.
  • theorem is a true statement in logic. Example: the four-color theorem.
  • proof is a valid argument in logic of a theorem. Example: Gonthier's proof of the four-color theorem
  • 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. 
  • 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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robotics systems that will transform automation at Amazon's scale. We're building revolutionary robotic systems that combine cutting-edge AI, sophisticated control systems, and advanced mechanical design to create adaptable automation solutions capable of working safely alongside humans in dynamic environments. This is a unique opportunity to shape the future of robotics and automation at unprecedented scale, working with world-class teams pushing the boundaries of what's possible in robotic manipulation, locomotion, and human-robot interaction. This role presents an opportunity to shape the future of robotics through innovative applications of deep learning and large language models. At Amazon Industrial Robotics we leverage advanced robotics, machine learning, and artificial intelligence to solve complex operational challenges at unprecedented scale. Our fleet of robots operates across hundreds of facilities worldwide, working in sophisticated coordination to fulfill our mission of customer excellence. We are pioneering the development of robotics foundation models that: Enable unprecedented generalization across diverse tasks Enable unprecedented robustness and reliability, industry-ready Integrate multi-modal learning capabilities (visual, tactile, linguistic) Accelerate skill acquisition through demonstration learning Enhance robotic perception and environmental understanding Streamline development processes through reusable capabilities The ideal candidate will contribute to research that bridges the gap between theoretical advancement and practical implementation in robotics. You will be part of a team that's revolutionizing how robots learn, adapt, and interact with their environment. Join us in building the next generation of intelligent robotics systems that will transform the future of automation and human-robot collaboration. As an Applied Science Manager in the Foundation Model team, you will build and lead a team that develops and improves machine learning systems that help robots perceive, reason, and act in real-world environments. You will set the technical direction for leveraging state-of-the-art models (open source and internal research), evaluating them on representative tasks, and adapting/optimizing them to meet robustness, safety, and performance needs. You will drive the capability roadmap and the evaluation strategy that defines “what the robot brain can do,” and you will sponsor targeted innovation when gaps remain. You’ll collaborate closely with research, controls, hardware, and product teams, and ensure the team’s outputs can be further customized and deployed by downstream teams on specific robot embodiments.
US, WA, Seattle
Are you ready to revolutionize the future of aerial delivery? Join our pioneering team working on autonomous drone technology that will transform how Amazon delivers to customers. As a Manager III, Applied Science for our Perception team, you'll lead scientists and engineers building world-class machine learning models and computer vision and new frontier model based systems that enable drones to safely navigate complex environments. In this high-impact role, you'll guide the development of perception systems combining radar, machine learning, and distributed computing to solve unprecedented challenges in autonomous flight. Your work will directly shape the future of Amazon's drone delivery service, bringing innovative solutions to customers while pushing the boundaries of what's possible in autonomous systems technology. Key job responsibilities Lead and mentor a team of scientists and software engineers developing advanced perception systems that integrate computer vision, machine learning algorithms, and detect-and-avoid capabilities for autonomous drones Define and drive the technical vision and roadmap for perception technology, making strategic architectural decisions that balance current delivery needs with long-term innovation goals Collaborate with cross-functional teams including hardware engineers, autonomy specialists, and product stakeholders to deliver scalable, distributed systems that meet rigorous safety and performance requirements Establish audit mechanisms and metrics to track system performance, model accuracy, and team progress against goals, using data-driven insights to continuously improve customer experience Build and grow an inclusive, high-performing team through effective hiring, mentoring, and career development while fostering a culture of innovation and technical excellence About the team The Perception team is at the heart of Amazon's autonomous drone delivery initiative, developing the advanced computer vision and machine learning systems that enable safe, reliable flight operations. Our team tackles fascinating challenges at the intersection of radar technology, distributed computing, and real-time detection algorithms, creating solutions that have never been built before. We work in an innovative, collaborative environment where every scientists and engineer's contribution directly impacts the future of delivery technology. As we scale Amazon's drone delivery to new locations, you'll help shape both the technical direction and team culture that will define this transformative service for years to come.
US, MA, North Reading
How should a robot dig a single item out of a cluttered bin, feel when it has made contact, and adjust in real time without crushing what it touches? Single- and dual-arm contact-rich manipulation at production scale is still being solved. We are looking for a Principal Applied Scientist to define the control architectures that our next-generation grasping systems will be built on. The Manipulation Robotics team develops robotic workcells to pick, grasp, and move millions of items and packages across Amazon's fulfillment network. Reaching into clutter and grasping deformable and diverse items depends on solving contact: force-aware control, compliant behaviors, and the layering of classical controllers and learned policies. You will set the technical direction for how the team approaches contact-rich manipulation, working with leaders across the organization to establish this foundational capability. Key job responsibilities - Own the technical direction and contact-control strategy for contact-rich manipulation, including the boundary between low-level control and learned policy, balancing tradeoffs among speed, performance, quality, cost, complexity, and adaptability. - Identify the open scientific problems that gate contact-rich manipulation at production scale, and invent the methods that solve them. - Architect contact control built on force and tactile sensing, covering compliant-contact behaviors, hybrid position and force control, and the safe interaction envelope. Stay hands-on and personally write critical-path code. - Partner with scientists, hardware designers, and systems engineers to treat the platform, arm, and end-effector as one coherent hierarchical system, including co-designing compliant end-effectors as part of the control strategy. - Create mechanisms to learn from fielded production, reasoning rigorously about failure modes and improving recovery from unexpected contact. - Drive how simulation, analytical models, demonstrations, and real-robot data are used together to turn lab results into scaled warehouse performance. - Establish the reference implementations, evaluation standards, and design review practices that let others build on the architecture and scale your impact without your continuous involvement. - Develop other scientists and engineers through mentorship, technical review, and a leading role in hiring. A day in the life Amazon offers a full range of benefits that support you and eligible family members, including domestic partners. Benefits can vary by location, the number of regularly scheduled hours you work, length of employment, and job status such as seasonal or temporary employment. The benefits that generally apply to regular, full-time employees include: 1. Medical, Dental, and Vision Coverage 2. Maternity and Parental Leave Options 3. Paid Time Off (PTO) 4. 401(k) Plan 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 We are a small, high-ownership team building robotic workcells, from early-stage R&D through products fielded 24/7 in Amazon fulfillment centers. We work at the hard edge of grasping: contact-rich tasks, deformable and variable items, and mechanisms that must survive millions of cycles at fleet scale. Our people own problems end to end, and scientists work shoulder to shoulder with mechanical, electrical, software, and controls engineers in a fast-moving, highly empowered environment where research ideas turn into shipped systems.