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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, Amazon Leo Satellite Build Systems
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, WA, Bellevue

            
        


        
    
Build the scientific intelligence layer powering Amazon’s satellite manufacturing system. As an Applied Scientist, you will develop machine learning models that transform fragmented manufacturing, test, quality, and operational data into actionable intelligence that improves how satellites are built. You will tackle ambiguous, high-impact problems where data is incomplete, noisy, and distributed, and where model outputs influence real-world manufacturing decisions. Your work will power AI-enabled workflows such as non-conformance disposition, root-cause analysis, and predictive test optimization - reducing defects, accelerating production, and helping create more intelligent, data-driven manufacturing systems. Export Control Requirement: Due to applicable export control laws and regulations, candidates must be a U.S. citizen or national, U.S. permanent resident (i.e., current Green Card holder), or lawfully admitted into the U.S. as a refugee or granted asylum. Key job responsibilities - Translate ambiguous manufacturing and operational problems into well-defined scientific problems, modeling approaches, and evaluation criteria - Design, train, and deploy machine learning models, including LLM-based systems, retrieval models, and task-specific models - Develop and evaluate models using large-scale, noisy, heterogeneous datasets with incomplete, delayed, or imperfect ground truth - Apply state-of-the-art techniques in areas such as anomaly detection, root-cause inference, multimodal learning, information retrieval, and generative AI, adapting or extending them to meet project requirements - Design experiments and evaluation frameworks that capture real-world failure modes, distribution shift, and decision risk - Make principled tradeoffs among model complexity, data quality, accuracy, latency, cost, and maintainability - Build production-quality scientific components with appropriate testing, documentation, monitoring, and operational mechanisms - Work with Manufacturing, Quality, Test, and engineering partners to understand customer needs and translate them into effective scientific solutions - Analyze model and system performance, identify gaps and root causes, and iteratively improve deployed solutions - Clearly document scientific approaches, experimental results, design decisions, and lessons learned so that others can understand and reproduce the work - Contribute to technical discussions, mentor less experienced teammates, and help advance scientific and engineering best practices within the team A day in the life You may start by partnering with Quality and Manufacturing teams to define a training dataset for a root-cause prediction model, including how historical cases should be labeled and evaluated. You then design experiments and train models, comparing approaches across architectures, features, and data slices. Later, you analyze benchmark results to identify failure modes, data-quality issues, and generalization gaps, and refine the evaluation set to better represent real-world cases. You work with engineers to integrate the model into a production workflow, adding testing, monitoring, and feedback mechanisms. Throughout the day, you balance scientific rigor with practical constraints such as data availability, latency, reliability, and operational cost. About the team Leo Satellite Build Systems is the centralized AI team within Leo Production Operations. We build shared capabilities for AI across Production Operations, including governed data assets, machine learning models, retrieval systems, evaluation frameworks, and knowledge services. We work on real-world systems where scientific decisions can influence physical outcomes. We value rigorous experimentation, strong data foundations, clear documentation, and production-ready engineering. Our team is helping enable AI-native manufacturing by turning fragmented operational knowledge and data into reliable intelligence that improves production outcomes.




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Thermal Scientist, Hardware - Product Integrity
            

        

        

            
                
                    

                
            

            

            

            

            
                
CN, 44, Shenzhen

            
        


        
    
You will be working with a unique and gifted team developing exciting products for consumers. The team is a multidisciplinary group of engineers and scientists engaged in a fast paced mission to deliver new products. The team faces a challenging task of balancing cost, schedule, and performance requirements. You should be comfortable collaborating in a fast-paced and often uncertain environment, and contributing to innovative solutions, while demonstrating leadership, technical competence, and meticulousness. Your deliverables will include development of thermal solutions, concept design, feature development, product architecture and system validation through to manufacturing release. You will support creative developments through application of analysis and testing of complex electronic assemblies using advanced simulation and experimentation tools and techniques. Key job responsibilities * Evaluate and optimize thermal solution requirements of consumer electronic products * Use simulation tools like Star-CCM+ or FloTherm XT/EFD for analysis and design of products * Validate design modifications for thermal concerns using simulation and actual prototypes * Establish temperature thresholds for user comfort level and component level considering reliability requirements * Have intimate knowledge of various materials and heat spreaders solutions to resolve thermal issues * Use of programming languages like Python and Matlab for analytical/statistical analyses and automation * Collaborate as part of device team to iterate and optimize design parameters of enclosures and structural parts to establish and deliver project performance objectives * Design and execute of tests using statistical tools to validate analytical models, identify risks and assess design margins * Create and present analytical and experimental results * Develop and apply design guidelines based on project learnings




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Applied Science Manager, Advertising Trust
            

        

        

            
                
                    

                
            

            

            

            

            
                
IN, KA, Bengaluru

            
        


        
    
Amazon Ads delivers advertising experiences across Amazon's owned-and-operated properties and third-party networks, reaching hundreds of millions of customers worldwide. Within Amazon Ads, Advertising Trust is the science-first organization responsible for ensuring every ad shown to customers meets Amazon's content policies — at massive scale, across all ad formats and global marketplaces. The Ads Trust Science team builds the ML systems that automate content moderation decisions: multimodal classification, retrieval-based labeling, LLM reasoning, and agentic self-improvement architectures. This requires inventing new approaches at the intersection of computer vision, NLP, information retrieval, and generative AI. We are seeking an Applied Science Manager to lead a team of applied scientists building next-generation content moderation intelligence. You will own the science roadmap for one of the highest-impact automation programs in Amazon Advertising, defining how multimodal content understanding, retrieval-first classification, and LLM-based reasoning combine into a production system that serves global advertising at scale. Key job responsibilities * Lead a team of applied scientists working across multimodal ML (vision-language models, video understanding), large-scale retrieval systems (embedding-based similarity and deduplication), and generative AI (LLM-based policy reasoning, knowledge distillation, agentic architectures, reinforcement learning). * Define the science strategy for ads trust. * Own end-to-end delivery of ML solutions: problem formulation, offline experimentation, online A/B testing, and production deployment. Your models directly move automation and defect metrics reported to senior leadership. * Build and grow scientists — hire, mentor, and develop team members. Raise the science bar through structured review processes and a publication culture within Amazon. * Partner with engineering, product, and operations teams to translate science investments into measurable automation improvements. Influence roadmaps across dependent teams. * Communicate science strategy and results to senior leadership through narratives, technical deep-dives, and roadmap documents.




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Applied Scientist II, Amazon Search
            

        

        

            
                
                    

                
            

            

            

            

            
                
IN, KA, Bengaluru

            
        


        
    
We are embarking on a multi-year journey to improve the shopping experience for customers globally. Amazon Search team creates customer-focused search solutions and technologies that make shopping delightful and effortless for our customers. Our goal is to understand what customers are looking for in whatever language happens to be their choice at the moment and help them find what they need in Amazon's vast catalog of billions of products — starting from the very first keystroke. As Amazon expands to new interfaces, we are faced with the unique challenge of maintaining the bar on Search Results Quality and Search Autocomplete. We are looking for a Applied Scientist II to work on improving search on Amazon using NLP, ML, and DL technology. As an Applied Scientist, you will lead our efforts in query understanding, semantic matching, and ranking. You will build systems that anticipate search query intent and surface the right results. As part of this role, you will develop high precision, high recall, and low latency solutions for search. Your solutions should work for all languages that Amazon supports and will be used in all Amazon locales world-wide. You will develop scalable science and engineering solutions that work successfully in production. Key job responsibilities As an Applied Scientist on the team, you will lead science innovation to improve the customer search experience through higher-quality search results. You will: - Develop and deploy ML models to produce relevant search results. - Design and train semantic matching models (bi-encoders, cross-encoders, and distillation from large foundation models) for ranking and relevance. - Develop reinforcement learning and reward-modeling approaches to continuously improve search results quality. - Train multi-objective ranking and scoring systems that balance suggestion diversity, specificity, and relevance. - Design and implement scalable model architectures optimized for strict latency constraints, including knowledge distillation, quantization, and efficient inference strategies for production deployment. - Lead end-to-end science projects from problem formulation through production launch, collaborating closely with engineers and scientists within and outside the team to deliver customer-facing impact.




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Applied Scientist II, Amazon Search
            

        

        

            
                
                    

                
            

            

            

            

            
                
IN, KA, Bengaluru

            
        


        
    
We are embarking on a multi-year journey to improve the shopping experience for customers globally. Amazon Search team creates customer-focused search solutions and technologies that make shopping delightful and effortless for our customers. Our goal is to understand what customers are looking for in whatever language happens to be their choice at the moment and help them find what they need in Amazon's vast catalog of billions of products — starting from the very first keystroke. As Amazon expands to new interfaces, we are faced with the unique challenge of maintaining the bar on Search Results Quality and Search Autocomplete. We are looking for a Applied Scientist II to work on improving search on Amazon using NLP, ML, and DL technology. As an Applied Scientist, you will lead our efforts in query understanding, semantic matching, and ranking. You will build systems that anticipate search query intent and surface the right results. As part of this role, you will develop high precision, high recall, and low latency solutions for search. Your solutions should work for all languages that Amazon supports and will be used in all Amazon locales world-wide. You will develop scalable science and engineering solutions that work successfully in production. Key job responsibilities As an Applied Scientist on the team, you will lead science innovation to improve the customer search experience through higher-quality search results. You will: - Develop and deploy ML models to produce relevant search results. - Design and train semantic matching models (bi-encoders, cross-encoders, and distillation from large foundation models) for ranking and relevance. - Develop reinforcement learning and reward-modeling approaches to continuously improve search results quality. - Train multi-objective ranking and scoring systems that balance suggestion diversity, specificity, and relevance. - Design and implement scalable model architectures optimized for strict latency constraints, including knowledge distillation, quantization, and efficient inference strategies for production deployment. - Lead end-to-end science projects from problem formulation through production launch, collaborating closely with engineers and scientists within and outside the team to deliver customer-facing impact.




        

        

        

        
    


    



                    

                
            

        

    

    

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