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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 II, Perception
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, MA, N.reading

            
        


        
    
Amazon is on a mission to redefine the future of automation — and we're looking for exceptional talent to help lead the way. We are building the next generation of advanced robotic systems that seamlessly blend cutting-edge AI, sophisticated control systems, and novel mechanical design to create adaptable, intelligent automation solutions capable of operating safely alongside humans in dynamic, real-world environments. At Amazon, we leverage the power of machine learning, artificial intelligence, and advanced robotics to solve some of the most complex operational challenges at a scale unlike anywhere else in the world. Our fleet of robots spans hundreds of facilities globally, working in sophisticated coordination to deliver on our promise of customer excellence — and we're just getting started. As a Applied Scientist in Robot Perception, you will be at the forefront of this transformation. You will develop and deploy state-of-the-art perception algorithms that enable robots to truly understand and interact with the physical world — bridging the gap between theoretical research and real-world impact. Bringing deep expertise in Computer Vision and a nuanced understanding of the capabilities and limitations of modern Vision-Language Models (VLMs), you will innovate boldly and push the boundaries of what's possible. Our vision for the Perception layer is ambitious: to enable seamless, intelligent interaction between the user, the robot, and its environment. This is a rare opportunity to work at the intersection of deep learning, large language models, and robotics — contributing to research that doesn't just advance the field, but reshapes it. You will collaborate with world-class teams pioneering breakthroughs in dexterous manipulation, locomotion, and human-robot interaction, all at an unprecedented scale. Join us in building intelligent robotic systems that will define the future of automation and human-robot collaboration. Key job responsibilities - Design, develop, and deploy perception algorithms for robotics systems, including object detection, segmentation, tracking, depth estimation, and scene understanding - Lead research initiatives in computer vision, sensor fusion and 3D perception - Collaborate with cross-functional teams including robotics engineers, software engineers, and product managers to define and deliver perception capabilities - Drive end-to-end ownership of ML models — from data collection and labeling strategy to training, evaluation, and deployment - Mentor junior scientists and engineers; contribute to a culture of technical excellence - Define and track key metrics to measure perception system performance in real-world environments - Publish research findings in top-tier venues (CVPR, ICCV, ECCV, ICRA, NeurIPS, etc.) and contribute to patents A day in the life - Train ML models for deployment in simulation and real-world robots, identify and document their limitations post-deployment - Drive technical discussions within your team and with key stakeholders to develop innovative solutions to address identified limitations - Actively contribute to brainstorming sessions on adjacent topics, bringing fresh perspectives that help peers grow and succeed — and in doing so, build lasting trust across the team - Mentor team members while maintaining significant hands-on contribution to technical solutions




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Applied Science Manager , C360
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, WA, Seattle

            
        


        
    
We are working on improving shopping on Amazon using the conversational capabilities of large language models and through customer behavioral data to make them more personalized for each customer. We are searching for pioneers who are passionate about technology, innovation, and customer experience, and are ready to make a lasting impact on the industry. In this role, you will be managing a team working on Large Language Model (LLM) and/or Vision-Language Model (VLM) post-training and alignment for new shopping experiences. You’ll be working with talented scientists, engineers, and technical program managers (TPM) to innovate on behalf of our customers. If you’re fired up about being part of a dynamic, driven team, then this is your moment to join us on this exciting journey!




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Sr Data Scientist, Amazon Publisher Services
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, NY, New York

            
        


        
    
External job description Job summary Amazon Publisher Services (APS) helps digital publishers around the world build and grow thriving businesses. We provide services and advanced technologies to web, mobile app and advanced TV publishers of all sizes, including many of comScore’s global top 100, to help them monetize their content with demand from multiple programmatic buyers. Our server-side header bidding solutions are fast and reliable across devices, handling billions of queries per day, delivering ads in milliseconds. The result is more profitable advertising for publishers and more relevant ads for customers. As a Data Scientist on this team, you will: • Solve real-world problems by getting and analyzing large amounts of data, diving deep to identify business insights and opportunities, design simulations and experiments, developing statistical and ML models by tailoring to business needs, and collaborating with Scientists, Engineers, BIE's, and Product Managers. • Write code (Python, R, Scala, etc.) to analyze data and build statistical models to solve specific business problems. • Apply statistical and machine learning knowledge to specific business problems and data. • Build decision-making models and propose solution for the business problem you define. • Retrieve, synthesize, and present critical data in a format that is immediately useful to answering specific questions or improving system performance. • Analyze historical data to identify trends and support optimal decision making. • Formalize assumptions about how our systems are expected to work, create statistical definition of the outlier, and develop methods to systematically identify outliers. Work out why such examples are outliers and define if any actions needed. • Given anecdotes about anomalies or generate automatic scripts to define anomalies, deep dive to explain why they happen, and identify fixes. • Conduct written and verbal presentations to share insights to audiences of varying levels of technical sophistication. Why you will love this opportunity: Amazon is investing heavily in building a world-class advertising business. This team defines and delivers a collection of advertising products that drive discovery and sales. Our solutions generate billions in revenue and drive long-term growth for Amazon’s Retail and Marketplace businesses. We deliver billions of ad impressions, millions of clicks daily, and break fresh ground to create world-class products. We are a highly motivated, collaborative, and fun-loving team with an entrepreneurial spirit - with a broad mandate to experiment and innovate. Impact and Career Growth: You will invent new experiences and influence customer-facing shopping experiences to help suppliers grow their retail business and the auction dynamics that leverage native advertising; this is your opportunity to work within the fastest-growing businesses across all of Amazon! Define a long-term science vision for our advertising business, driven from our customers' needs, translating that direction into specific plans for research and applied scientists, as well as engineering and product teams. This role combines science leadership, organizational ability, technical strength, product focus, and business understanding. About the team The Marketplace Services team within Amazon Publisher Services organization primarily focuses on improving monetization for our STV, Web, Mobile and Audio publisher customers. We directly work with 60+ 3p buyers to enable optimal connectivity for publishers to improve their yield. We also own products such as Connections Marketplace (CxM) and Signal IQ that help publishers connect to myriad of 3p and 1p ad tech vendors to boost their bid request quality, while measuring the value of each signal on their bid stream through rigorous A/B testing. Internal job description The candidate would work with Product, Engineering, BIEs and Scientist across Supply and Demand organization to help make APS the best performing supply path for Amazon ads advertiser customers. They would spearhead efforts to conduct experiments alongside demand and measurement teams to identify optimal perfomance path for advertisers while improving APS Share of Wallet. About the team The Marketplace Services team within Amazon Publisher Services organization primarily focuses on improving monetization for our STV, Web, Mobile and Audio publisher customers. We directly work with 60+ 3p buyers to enable optimal connectivity for publishers to improve their yield. We also own products such as Connections Marketplace (CxM) and Signal IQ that help publishers connect to myriad of 3p and 1p ad tech vendors to boost their bid request quality, while measuring the value of each signal on their bid stream through rigorous A/B testing.




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Applied Scientist, Stores Economics & Science (SEAS)
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, WA, Seattle

            
        


        
    
Stores Economics and Science (SEAS) is an interdisciplinary science and engineering team in Amazon's Stores organization with a peak-jumping mission: we apply expertise in science and engineering to move from local to global optima in methods, models, and software. We pursue this mission by leveraging frontier science; collaborating with partner teams; and learning from the tools, experience, and perspective of others. We scale by solving problems, first in the small to prove concepts, and then in the large by building scalable solutions. We also help other teams within Amazon scale by hiring and developing the best and embedding them in other business units. In 2026, we are focused on economics and science in areas related to (1) lowering cost-to-serve, (2) optimizing selection, and (3) emerging machine learning. We also have some ongoing and highly-leveraged collaborations that help partner teams inside Amazon short-circuit months of R&D or otherwise look around corners. We are looking for an Applied Scientist to build and deliver state-of-the-art science and engineering solutions to improve our Stores business. In this role, you will work in a team of scientists and engineers with backgrounds in machine learning, NLP, IR, statistics, and economics to identify bottlenecks in our business, conceive new ideas to overcome those challenges, and deploy scientific solutions in partnership with product teams. Your responsibilities include developing and maintaining the scientific models, benchmarks, and services. Graduate education or hands-on experience in machine learning, optimization, causal inference, Bayesian statistics, deep learning, or other quantitative scientific fields is a big plus. To be successful in this role, you should be a quick learner and comfortable with a high degree of ambiguity. Key job responsibilities The successful candidate will lead large-scale science initiatives from research to production and translate complex business problems into mathematical frameworks. They will design and implement large-scale algorithms for complex supply chain and marketplace problems, and design incentive-compatible mechanisms for marketplace challenges. The ideal candidate will have a strong publication record in top-tier conferences/journals (INFORMS, EC, WINE, ICML, NeurIPS, etc.) and experience coordinating cross-functional projects. Hands-on experience building science solutions to mechanism design problems (e.g., optimal auction design, welfare maximization under constraints, incentive compatible coordination), with expertise in statistical learning and algorithm development. Leadership responsibilities include influencing technical strategy and roadmaps for complex initiatives, influencing senior stakeholders and shaping technical direction, and fostering team growth.




        

        

        

        
    


    



                    

                
            

        

    

    

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