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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, Amazon Music - Search Science, Amazon Music – Search Science
            

        

        

            
                
                    

                
            

            

            

            

            
                
IN, KA, Bengaluru

            
        


        
    
Amazon Music is an immersive audio entertainment service that deepens connections between fans, artists, and creators. From personalized music playlists to exclusive podcasts, concert livestreams to artist merch, Amazon Music is innovating at some of the most exciting intersections of music and culture. The Amazon Music Search Science team is seeking an innovative and driven Applied Scientist to join our engineering and science hub in Bangalore. You will work alongside a world-class team of machine learning experts to break new ground in understanding user intent, classifying complex audio and musical forms, and creating next-generation interactive search experiences that help users find the exact music, podcasts, and audio content they are in the mood for. In this role, you will own the design, development, and deployment of end-to-end machine learning systems. You will balance execution on core search and discovery priorities—such as improving retrieval accuracy, latency, and relevance for millions of daily queries—while laying the foundational modeling capabilities for broader semantic understanding and advanced conversational search experiences across mobile, web, and voice-forward devices (like Alexa and Echo). Key job responsibilities - Core Search & Execution: Collaborate with scientists, software engineers, and product managers to define, frame, and solve complex business and ranking problems as machine learning, information retrieval, or optimization tasks. - Advanced AI & Modeling: Design, build, train, and evaluate production-grade ML models using classical machine learning, deep learning, Large Language Models (LLMs), and Agentic AI techniques to scale music discovery and intent resolution. - End-to-End Production Ownership: Take algorithms from research ideation to production deployment. Build scalable data pipelines, efficient model-serving systems, and robust offline/online evaluation frameworks. - Experimentation & Iteration: Design and analyze large-scale A/B experiments across millions of customers to measure impact on search relevance, engagement, and customer satisfaction, refining models for continuous improvement. - Forward-Looking Innovation: Research and implement novel statistical and machine learning approaches, exploring multi-modal understanding, rich content semantics, and advanced retrieval mechanisms that extend beyond traditional search boundaries. - Technical Communication: Communicate findings, architectural decisions, and technical roadmaps clearly to both technical peers and executive stakeholders, authoring robust design documents and contributing to team standards. Basic Qualifications - PhD, or Master’s degree and 4+ years of relevant experience in Computer Science, Computer Engineering, Machine Learning, Statistics, or a related quantitative field. - 3+ years of hands-on experience building machine learning models or algorithms for business applications and deploying them into production. - Strong programming skills in Python, Java, C++, or related languages, with a solid foundation in data structures, algorithms, and object-oriented design. - Experience in one or more of the following areas: Information Retrieval, Natural Language Processing (NLP), Recommender Systems, Deep Learning, or Numerical Optimization. - Demonstrated ability to work effectively with cross-functional teams in a fast-paced environment. Preferred Qualifications - Experience with large-scale distributed computing frameworks and big data systems (e.g., Spark, Hadoop, AWS infrastructure). - Experience building search ranking, query understanding, or semantic retrieval systems for high-scale consumer applications. - Familiarity with modern foundation models, LLMs, fine-tuning techniques, and efficient inference optimization for production services. - Track record of peer-reviewed publications or patents at top-tier machine learning/AI conferences (e.g., NeurIPS, KDD, ACL, SIGIR, ICML). - Experience in designing, executing, and evaluating rigorous online A/B experiments.




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Senior Applied Scientist, Amazon Music - Search Science, Amazon Music – Search Science
            

        

        

            
                
                    

                
            

            

            

            

            
                
IN, KA, Bengaluru

            
        


        
    
Amazon Music is an immersive audio entertainment service that deepens connections between fans, artists, and creators. From personalized music playlists to exclusive podcasts, concert livestreams to artist merch, Amazon Music is innovating at some of the most exciting intersections of music and culture. The Amazon Music Search Science team is looking for an execution-focused Senior Applied Scientist to spearhead core scientific initiatives within our Bangalore hub. In this leadership-by-example role, you will define and execute the applied science roadmap for search and content discovery systems, directly impacting millions of customers worldwide. You will operate at the exciting intersection of large-scale search infrastructure, applied machine learning, and foundation models. You will drive immediate, high-impact business deliverables in music search relevance, personalization, and retrieval performance, while simultaneously architecting the long-term technological vision that expands our search ecosystem toward deeper semantic intelligence, agentic workflows, and cross-domain audio understanding. Key job responsibilities - Strategic Roadmap & Architecture: Define and execute the technical and scientific roadmap for music search systems, making critical architectural decisions that balance short-term feature delivery with long-term scalability, maintainability, and extensibility. - Pioneering Applied Science: Lead the design and implementation of state-of-the-art machine learning solutions leveraging deep learning, LLMs, and agentic workflows to solve complex search, ranking, and intent-matching challenges. - Cross-Functional Leadership: Partner closely with product management, engineering leaders, and peer teams to harmonize technical direction and deliver synchronized customer experiences. - Technical Excellence & Mentorship: Drive engineering and scientific excellence across the team by conducting rigorous design reviews, establishing modeling best practices, setting high bars for artifact delivery, and mentoring junior/mid-level scientists. - Experimentation & Scaling: Establish robust scientific processes for large-scale data analysis, offline model validation, and online A/B experimentation, ensuring high statistical rigor and measurable business impact across millions of active listeners. - Stakeholder Influence & Writing: Author strategic whitepapers, and executive-level documentation. Communicate complex technical options and trade-offs to senior leadership to drive informed decision-making. Basic Qualifications - PhD, or Master’s degree and 6+ years of applied research and industrial machine learning experience in Computer Science, Machine Learning, or a related field. - 3+ years of specialized experience designing, building, and scaling machine learning models for core production business applications (e.g., Search, Recommendation Systems, or Large-Scale Information Retrieval). - Expert programming proficiency in Python, Java, C++, or related languages, combined with deep familiarity with neural network architectures and deep learning frameworks. - Proven track record of owning end-to-end technical deliverables from problem formulation and model architecture to production deployment and performance tuning. - Demonstrated leadership in mentoring technical talent and driving engineering/scientific best practices. Preferred Qualifications - Deep expertise in search retrieval, query understanding, ranking algorithms, and large-scale vector search/embedding systems. - Experience building applied science solutions on top of foundation models, large language models (LLMs), or multi-modal architectures. - Experience with large-scale distributed training and inference optimization on cloud infrastructure (AWS). - A strong publication record or patent portfolio in top-tier peer-reviewed venues (e.g., NeurIPS, SIGIR, KDD, ACL, ICML). - Experience designing and interpreting complex online experimentation frameworks for consumer-facing recommendation or search products.




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Principal Applied Scientist, Sponsored Products and Brands - Demand Utilization
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, CA, Palo Alto

            
        


        
    
The Demand Utilization team within Amazon Advertising is responsible for determining which ads to serve when hundreds of millions of shoppers search for products on Amazon. We sit at the intersection of customer intent understanding and advertiser value, solving one of the most complex matching problems in the industry, identifying the right ad, for the right shopper, at the right moment, across one of the world's largest product catalogs. Our systems deliver billions of ad impressions and millions of clicks daily under strict relevance and latency constraints. We are looking for a Principal Applied Scientist to set the technical vision and drive the science strategy for our ad retrieval and ranking systems. This is a high-impact leadership role where you will tackle unsolved problems at the frontier of large-scale information retrieval, natural language understanding, and multi-objective optimization, all operating in real time at Amazon scale. You will work on challenges such as: - Modeling shopper intent from sparse, ambiguous, and multi-modal signals - Designing retrieval architectures that balance relevance, advertiser and shopper experience across billions of candidate ads - Advancing personalization and cold-start strategies for new advertisers and emerging product categories This is a role for a scientist who wants to shape the future of performance advertising through rigorous research applied to real-world systems that directly impact Amazon's customers, sellers, and business. Key job responsibilities Key Responsibilities: - Own the science roadmap for ad retrieval and ranking within Demand Utilization, defining multi-year research priorities aligned with business goals - Lead the design and development of novel machine learning models and algorithms for relevance, intent understanding, and ad selection at scale - Drive end-to-end execution from problem formulation and experimentation through production deployment, measuring impact on shopper and advertiser outcomes - Mentor and elevate a team of applied scientists and research engineers, raising the technical bar and fostering a culture of scientific rigor - Collaborate cross-functionally with product, engineering, and business leaders to translate science capabilities into product strategy - Represent Amazon externally through publications at top-tier venues, patents, and participation in the broader ML/IR research community




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Applied Scientist II, Reinforcement Learning
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, MA, N.reading

            
        


        
    
Amazon is seeking exceptional talent to help develop the next generation of advanced robotics systems that will transform automation at Amazon's scale. We're building revolutionary robotic systems that combine cutting-edge AI, sophisticated control systems, and advanced mechanical design to create adaptable automation solutions capable of working safely alongside humans in dynamic environments. This is a unique opportunity to shape the future of robotics and automation at an unprecedented scale, working with world-class teams pushing the boundaries of what's possible in robotic dexterous manipulation, locomotion, and human-robot interaction. This role presents an opportunity to shape the future of robotics through innovative applications of deep learning and large language models. At Amazon we leverage advanced robotics, machine learning, and artificial intelligence to solve complex operational challenges at an unprecedented scale. Our fleet of robots operates across hundreds of facilities worldwide, working in sophisticated coordination to fulfill our mission of customer excellence. The ideal candidate will contribute to research that bridges the gap between theoretical advancement and practical implementation in robotics. You will be part of a team that's revolutionizing how robots learn, adapt, and interact with their environment. Join us in building the next generation of intelligent robotics systems that will transform the future of automation and human-robot collaboration. Key job responsibilities - Design and implement whole body control methods for balance, locomotion, and dexterous manipulation - Utilize state-of-the-art in methods in learned and model-based control - Create robust and safe behaviors for different terrains and tasks - Implement real-time controllers with stability guarantees - Collaborate effectively with multi-disciplinary teams to co-design hardware and algorithms for loco-manipulation - Mentor junior engineer and scientists




        

        

        

        
    


    



                    

                
            

        

    

    

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