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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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                Research Scientist, Mobile Manipulation Robotics (I/O)
            

        

        

            
                
                    

                
            

            

            

            

            
                
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!




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Sr. Applied Scientist, DemandTech - AdDelivery, Experience and Measurement 
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, NY, New York

            
        


        
    
Want to work on building a Amazon Ads billion dollar business, innovate on a new product, and have a positive impact on millions of views while working with industry-leading technologies? We're growing a team to support the Sponsored Ads business that powers the advertising experience for millions of viewers and advertisers daily. Amazon is investing heavily in building a world-class advertising business and developing a collection of self-service performance advertising products that drive discovery and sales. We deliver billions of ad impressions and millions of clicks daily and are constantly challenging ourselves to create world-class products and an unparalleled shopping experience for our hundreds of millions of customers worldwide. Key job responsibilities We are building the next-gen smart ads campaign. At its core is an Intelligence Flywheel — an architecture where every component's output is designed to train models that improve every other component. The Model Layer that is meant to power every capability currently has no dedicated science ownership. As the Senior Applied Scientist on this team, you own the science that makes the flywheel turn. You will turn static, threshold-based logic into self-improving, closed-loop intelligence, and define the decision policies that let the system act autonomously with advertiser trust. Concretely, you will: * Build predictive issue-detection models that identify under-delivery, over-delivery, and performance degradation from campaign signals before they materially impact advertisers. * Design the intervention-selection policy — which autonomous action to take — framed as a contextual bandit: choose, observe, update. Engineers implement action execution; you define the policy that selects actions. * Establish causal attribution for autonomous optimization, separating the effect of our interventions from organic performance change, so improvements can be attributed and advertiser trust in autonomy can be earned. * Integrate and adapt cross-model signals. Combine creative-quality, product-relevance, and budget signals into a unified advertiser-intelligence picture, and adapt general-purpose partner models to our product reality via fine-tuning, re-ranking, or thin adaptation layers. * Close recommendation and grading feedback loops — define reward schemas for accept/reject and performance-vs-baseline signals, correlate creative-quality scores with real campaign outcomes, and feed empirical findings back to both our product and partner science teams. You will work backwards from ambiguous business problems, set the science roadmap for the Model Layer, and partner closely with the team's software engineers — who own the services, pipelines, and execution infrastructure — so that model artifacts you produce are deployed and served in production. This is a high-leverage, high-autonomy role: your outputs are consumed by multiple engineering workstreams at once, and you set the abstractions the team builds on. About the team We are focused on goal-oriented, AI powered workflows that help advertisers achieve their marketing objectives. We collect campaign goals, surface relevant data at key decision points, and provide reporting that validates decision-making. Our product suite guides advertisers in building campaigns with optimal targeting, creative formats, inventory, and bid models that are highly likely to hit their goals — reducing the need for manual intervention.




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Applied Scientist, Catalog System Services Science
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, WA, Seattle

            
        


        
    
At Amazon Selection and Catalog Systems (ASCS), our mission is to power the online buying experience for customers worldwide so they can find, discover, and buy any product they want. We innovate on behalf of our customers to ensure uniqueness and consistency of product identity and to infer relationships between products in Amazon Catalog to drive the selection gateway for the search and browse experiences on the website. We're solving a fundamental AI challenge: establishing product relevant information at unprecedented scale with Frontier Models and Agents. The scale is staggering: billions of products, petabytes of multimodal data, millions of sellers, dozens of languages, and infinite product diversity ranging from electronics to groceries to digital content. The research challenges are immense. GenAI and VLMs hold transformative promise for catalog understanding, but we operate where traditional methods fail: ambiguous problem spaces, incomplete and noisy data, inherent uncertainty, reasoning across both images and textual data, and explaining decisions at scale. Enriching product information requires sophisticated models that reason across text, images, and structured data, all while maintaining accuracy and trust for high-stakes business decisions affecting millions of customers daily. Amazon's Catalog System Services Science team is looking for an innovative and customer-focused applied scientist to help us make the world's best product catalog even better. In this role, you will partner with technology and business leaders to build new state-of-the-art algorithms, models, and services. You will pioneer advanced GenAI solutions that power next-generation agentic shopping experiences, working in a collaborative environment where you can experiment with massive data from the world's largest product catalog, tackle problems at the frontier of AI research, rapidly implement and deploy your algorithmic ideas at scale, across millions of customers. Key job responsibilities - Formulate novel research problems at the intersection of GenAI, multimodal learning, and large-scale information retrieval. In essence, translating ambiguous business challenges into tractable scientific frameworks - Design and implement leading models leveraging frontier models, and agentic architectures to enrich catalog information at billion-product scale - Pioneer explainable AI methodologies that balance model performance with scalability requirements for production systems impacting millions of daily customer decisions - Own end-to-end ML pipelines from research ideation to production deployment, processing petabytes of multimodal data with rigorous evaluation frameworks - Represent the team in the broader science community - publishing findings, delivering tech talks, and staying at the forefront of GenAI, VLM, and agentic system research




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Applied Scientist I, Amazon Shipping
            

        

        

            
                
                    

                
            

            

            

            

            
                
IN, HR, Gurugram

            
        


        
    
Building large-scale forecasting and optimization systems that power Amazon’s global transportation network and directly impact customer experience and cost. Key job responsibilities 1. Guide model and system design across a range of techniques, including tree-based models, deep learning (LSTMs, transformers), LLMs, and reinforcement learning. 2. Ensure models are production-ready, scalable, and robust through close partnership with stakeholders. 3. Partner with Product, Operations, and Engineering leaders to enable proactive decision-making and corrective actions. 4 Own end-to-end business metrics, directly influencing customer experience, cost optimization, and network reliability. 5. Help contribute to the broader ML community through publications, conference submissions, and internal knowledge sharing.




        

        

        

        
    


    



                    

                
            

        

    

    

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