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Automated reasoning

A gentle introduction to automated reasoning

Meet Amazon Science’s newest research area.

By Byron Cook
December 1, 2021
18 min read
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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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About the Author

                    

                        
                            
    

        

            
                

                    Byron Cook
                

            
            
            
                

                    Amazon vice president and distinguished scientist Byron Cook is a leader in the field of formal verification, known for his contributions to SAT, SMT, and symbolic model checking, with applications to biological systems, computer operating systems, programming languages, and security. Byron’s work on automated reasoning at Amazon has led to higher levels of assurance in the cloud as well as new customer features.
                

            
        

    



                        
                    

                
            

        


        
    

    
    



    

        
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                Applied Scientist, Prime Video Commerce Insights 
            

        

        

            
                
                    

                
            

            

            

            

            
                
GB, London

            
        


        
    
Come build the future of entertainment with us. Are you interested in shaping the future of movies and television? Do you want to define the next generation of how and what Amazon customers are watching? Prime Video is a premium streaming service that offers customers a vast collection of TV shows and movies — all with the ease of finding what they love to watch in one place. We offer customers thousands of popular movies and TV shows from Originals and Exclusive content to exciting live sports events. We also offer our members the opportunity to subscribe to add-on channels which they can cancel at anytime and to rent or buy new release movies and TV box sets on the Prime Video Store. Prime Video is a fast-paced, growth business — available in over 240 countries and territories worldwide. The team works in a dynamic environment where innovating on behalf of our customers is at the heart of everything we do. If this sounds exciting to you, please read on. Prime Video Commerce's mission is to present the right offer to the right customer at the right time — across subscriptions, channels, and transactional video, in every market and on every device. Our science team replaces static business rules with ML-driven decisions that personalise the entire commerce journey, from discovery through checkout and beyond. We operate at scale across hundreds of millions of customers, and we are expanding into new frontiers — combining the latest advances in agentic and generative AI, behavioural simulation, and causal inference to understand the impact of our decisions before they reach customers. We are looking for an Applied Scientist to join the Prime Video Commerce Insights team in London. You will develop and deploy customer-facing models, understand customer behaviour at scale, and explore emerging techniques that help us make better decisions faster. This is a delivery focused role within a high-visibility multidisciplinary group of engineers and scientists, focused on improving the customer experience for Prime Video. Key job responsibilities - Research, design, and implement machine learning approaches (e.g. reinforcement learning and recommendation systems) that personalise across different customer touch points. - Collaborate with engineers to deploy and integrate successful experiment results into large-scale, complex Amazon production systems with low latency. - Design and execute rigorous experiments to demonstrate the technical efficacy and business value of your methods. - Act as a subject-matter expert and help define the science roadmap and research agenda in line with organisational priorities and production constraints. - Provide machine learning thought leadership to technical and business leaders, thinking strategically about business, product, and technical challenges. - Work with technical product managers to work backwards from what matters to customers and deliver ML-backed solutions. - Share results with the team and wider scientific community through documents that are both statistically rigorous and compellingly relevant. A day in the life You will be a research leader and innovator within the Commerce Insights organisation. You will collaborate with talented engineers and senior leaders to solve problems that are uniquely challenging at Amazon's scale: personalising commerce decisions across multiple business lines, balancing competing objectives, and positively impacting hundreds of millions of customers worldwide. The problems here are technically deep — combining large-scale ML, causal reasoning, and behavioural modelling in a domain where every decision carries real revenue and customer-experience consequences. Your research will ship to production and move metrics that matter. About the team You will join a team of engineers and applied scientists with a proven track record of solving highly complex, ambiguous problems — work that has produced patents and publications at top-tier conferences. The team has direct visibility to senior Prime Video leadership and collaborates broadly across Commerce, Content, and Platform teams to shape how customers discover, subscribe to, and engage with video content. This is a team that operates at the intersection of rigorous research and real-world impact, where your ideas move from whiteboard to production for hundreds of millions of customers.




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Sr. Applied Scientist, SSG Science
            

        

        

            
                
                    

                
            

            

            

            

            
                
IN, KA, Bengaluru

            
        


        
    
Amazon Devices is an inventive research and development company that designs and engineer high-profile devices like the Kindle family of products, Fire Tablets, Fire TV, Health Wellness, Amazon Echo & Astro products. This is an exciting opportunity to join Amazon in developing state-of-the-art techniques that bring Gen AI on edge for our consumer products. We are looking for exceptional scientists to join our Applied Science team and help develop the next generation of edge models, and optimize them while doing co-designed with custom ML HW based on a revolutionary architecture. Work hard. Have Fun. Make History. Key job responsibilities What will you do? - Quantize, prune, distill, finetune Gen AI models to optimize for edge platforms - Fundamentally understand Amazon’s underlying Neural Edge Engine to invent optimization techniques - Analyze deep learning workloads and provide guidance to map them to Amazon’s Neural Edge Engine - Use first principles of Information Theory, Scientific Computing, Deep Learning Theory, Non Equilibrium Thermodynamics - Train custom Gen AI models that beat SOTA and paves path for developing production models - Collaborate closely with compiler engineers, fellow Applied Scientists, Hardware Architects and product teams to build the best ML-centric solutions for our devices - Publish in open source and present on Amazon's behalf at key ML conferences - NeurIPS, ICLR, MLSys.




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Applied Scientist, SSG Science
            

        

        

            
                
                    

                
            

            

            

            

            
                
IN, KA, Bengaluru

            
        


        
    
Amazon Devices is an inventive research and development company that designs and engineer high-profile devices like the Kindle family of products, Fire Tablets, Fire TV, Health Wellness, Amazon Echo & Astro products. This is an exciting opportunity to join Amazon in developing state-of-the-art techniques that bring Gen AI on edge for our consumer products. We are looking for exceptional scientists to join our Applied Science team and help develop the next generation of edge models, and optimize them while doing co-designed with custom ML HW based on a revolutionary architecture. Work hard. Have Fun. Make History. Key job responsibilities What will you do? - Quantize, prune, distill, finetune Gen AI models to optimize for edge platforms - Fundamentally understand Amazon’s underlying Neural Edge Engine to invent optimization techniques - Analyze deep learning workloads and provide guidance to map them to Amazon’s Neural Edge Engine - Use first principles of Information Theory, Scientific Computing, Deep Learning Theory, Non Equilibrium Thermodynamics - Train custom Gen AI models that beat SOTA and paves path for developing production models - Collaborate closely with compiler engineers, fellow Applied Scientists, Hardware Architects and product teams to build the best ML-centric solutions for our devices - Publish in open source and present on Amazon's behalf at key ML conferences - NeurIPS, ICLR, MLSys.




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Member of Technical Staff, Frontier AI & Robotics (FAR)
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, CA, San Francisco

            
        


        
    
Join the next revolution in robotics at Amazon's Frontier AI & Robotics team, where you'll work alongside world-renowned AI pioneers to push the boundaries of what's possible in robotic intelligence. As a Member of Technical Staff, you'll be at the forefront of developing breakthrough foundation models that enable robots to perceive, understand, and interact with the world in unprecedented ways. You'll drive independent research initiatives in areas such as perception, manipulation, science understanding, locomotion, manipulation, sim2real transfer, multi-modal foundation models and multi-task robot learning, designing novel frameworks that bridge the gap between state-of-the-art research and real-world deployment at Amazon scale. In this role, you'll balance innovative technical exploration with practical implementation, collaborating with platform teams to ensure your models and algorithms perform robustly in dynamic real-world environments. You'll have access to Amazon's vast computational resources, enabling you to tackle ambitious problems in areas like very large multi-modal robotic foundation models and efficient, promptable model architectures that can scale across diverse robotic applications. Key job responsibilities - Drive independent research initiatives across the robotics stack, including robotics foundation models, focusing on breakthrough approaches in perception, and manipulation, for example open-vocabulary panoptic scene understanding, scaling up multi-modal LLMs, sim2real/real2sim techniques, end-to-end vision-language-action models, efficient model inference, video tokenization - Design and implement novel deep learning architectures that push the boundaries of what robots can understand and accomplish - Lead full-stack robotics projects from conceptualization through deployment, taking a system-level approach that integrates hardware considerations with algorithmic development, ensuring robust performance in production environments - Collaborate with platform and hardware teams to ensure seamless integration across the entire robotics stack, optimizing and scaling models for real-world applications - Contribute to the team's technical strategy and help shape our approach to next-generation robotics challenges A day in the life - Design and implement novel foundation model architectures and innovative systems and algorithms, leveraging our extensive infrastructure to prototype and evaluate at scale - Collaborate with our world-class research team to solve complex technical challenges - Lead technical initiatives from conception to deployment, working closely with robotics engineers to integrate your solutions into production systems - Participate in technical discussions and brainstorming sessions with team leaders and fellow scientists - Leverage our massive compute cluster and extensive robotics infrastructure to rapidly prototype and validate new ideas - Transform theoretical insights into practical solutions that can handle the complexities of real-world robotics applications About the team At Frontier AI & Robotics, we're not just advancing robotics – we're reimagining it from the ground up. Our team is building the future of intelligent robotics through innovative foundation models and end-to-end learned systems. We tackle some of the most challenging problems in AI and robotics, from developing sophisticated perception systems to creating adaptive manipulation strategies that work in complex, real-world scenarios. What sets us apart is our unique combination of ambitious research vision and practical impact. We leverage Amazon's massive computational infrastructure and rich real-world datasets to train and deploy state-of-the-art foundation models. Our work spans the full spectrum of robotics intelligence – from multimodal perception using images, videos, and sensor data, to sophisticated manipulation strategies that can handle diverse real-world scenarios. We're building systems that don't just work in the lab, but scale to meet the demands of Amazon's global operations. Join us if you're excited about pushing the boundaries of what's possible in robotics, working with world-class researchers, and seeing your innovations deployed at unprecedented scale.




        

        

        

        
    


    



                    

                
            

        

    

    

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