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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, Sponsored Products Autonomous Campaigns
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, CA, Palo Alto

            
        


        
    
We are looking for a passionate Applied Scientist to help pioneer the next generation of agentic AI applications for Amazon advertisers. In this role, you will design agentic architectures, develop tools and datasets, and contribute to building systems that can reason, plan, and act autonomously across complex advertiser workflows. You will work at the forefront of applied AI, developing methods for fine-tuning, reinforcement learning, and preference optimization, while helping create evaluation frameworks that ensure safety, reliability, and trust at scale. You will work backwards from the needs of advertisers—delivering customer-facing products that directly help them create, optimize, and grow their campaigns. Beyond building models, you will advance the agent ecosystem by experimenting with and applying core primitives such as tool orchestration, multi-step reasoning, and adaptive preference-driven behavior. This role requires working independently on ambiguous technical problems, collaborating closely with scientists, engineers, and product managers to bring innovative solutions into production. Key job responsibilities - Design and build agents for our autonomous campaigns experience. - Design and implement advanced model and agent optimization techniques, including supervised fine-tuning, instruction tuning and preference optimization (e.g., DPO/IPO). - Curate datasets and tools for MCP. - Build evaluation pipelines for agent workflows, including automated benchmarks, multi-step reasoning tests, and safety guardrails. - Develop agentic architectures (e.g., CoT, ToT, ReAct) that integrate planning, tool use, and long-horizon reasoning. - Prototype and iterate on multi-agent orchestration frameworks and workflows. - Collaborate with peers across engineering and product to bring scientific innovations into production. - Stay current with the latest research in LLMs, RL, and agent-based AI, and translate findings into practical applications. About the team The Sponsored Products and Brands team at Amazon Ads is re-imagining the advertising landscape through the latest generative AI technologies, revolutionizing how millions of customers discover products and engage with brands across Amazon.com and beyond. We are at the forefront of re-inventing advertising experiences, bridging human creativity with artificial intelligence to transform every aspect of the advertising lifecycle from ad creation and optimization to performance analysis and customer insights. We are a passionate group of innovators dedicated to developing responsible and intelligent AI technologies that balance the needs of advertisers, enhance the shopping experience, and strengthen the marketplace. If you're energized by solving complex challenges and pushing the boundaries of what's possible with AI, join us in shaping the future of advertising. The Autonomous Campaigns team within Sponsored Products and Brands is focused on guiding and supporting 1.6MM advertisers to meet their advertising needs of creating and managing ad campaigns. At this scale, the complexity of diverse advertiser goals, campaign types, and market dynamics creates both a massive technical challenge and a transformative opportunity: even small improvements in guidance systems can have outsized impact on advertiser success and Amazon’s retail ecosystem. Our vision is to build a highly personalized, context-aware campaign creation and management system that leverages LLMs together with tools such as auction simulations, ML models, and optimization algorithms. This agentic framework, will operate across both chat and non-chat experiences in the ad console, scaling to natural language queries as well as proactively delivering guidance based on deep understanding of the advertiser. To execute this vision, we collaborate closely with stakeholders across Ad Console, Sales, and Marketing to identify opportunities—from high-level product guidance down to granular keyword recommendations—and deliver them through a tailored, personalized experience. Our work is grounded in state-of-the-art agent architectures, tool integration, reasoning frameworks, and model customization approaches (including tuning, MCP, and preference optimization), ensuring our systems are both scalable and adaptive.




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Manager, Applied Science, International Seller Services
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, WA, Seattle

            
        


        
    
Are you interested in leading growth initiatives for one of Amazon’s most significant and fastest growing businesses? Selling Partners offer hundreds of millions of unique products and are a critical to delivering on our vision of offering the Earth’s largest selection and lowest prices. The Amazon Marketplace enables over 2 million third-party selling partners in eleven marketplaces to list their products for sale to Amazon customers across the world. Within our WW Marketplace business, International Seller Services (ISS) oversees the recruiting and development of Selling Partners for all of our international marketplaces (e.g. UK, Germany, Japan, Middle East etc.). ISS also enables global selling, helping Sellers in one country expand and sell internationally. Are you fascinated by the power of Natural Language Processing (NLP) and Large Language Models (LLM) to transform the way we interact with technology? Are you passionate about applying advanced machine learning techniques to solve complex challenges in the e-commerce space? If so, the Central Science Team of Amazon's International Seller Services has an exciting opportunity for you as an Applied Science Manager. We are seeking an experienced science leader who is adept at a variety of skills; especially in generative AI, computer vision, and large language models that will help international sellers succeed as they sell on Amazon. The right candidate will provide science leadership, establish the right direction and vision, build team mechanisms, foster the spirit of collaboration and innovation within the org, and execute against a roadmap. This leader will provide both technical direction as well as manage a sizable team of scientists. They will need to be adept at recruiting, launching AI models into production, writing vision/direction documents, and building team mechanisms that will foster innovation and execution. Additionally, while the position is based in Seattle, this leader will interact with global leaders and teams in Europe, Japan, China, Australia, and other regions. Key job responsibilities Key job responsibilities Responsibilities include: * Drive end-to-end applied science projects that have a high degree of ambiguity, scale, complexity. * Provide technical / science leadership related to NLP, computer vision and large language models. * Research new and innovative machine learning approaches. * Recruit high performing Applied Scientists to the team and provide mentorship. * Establish team mechanisms, including team building, planning, and document reviews. * Communicate complex technical concepts effectively to both technical and non-technical stakeholders, providing clear explanations and guidance on proposed solutions and their potential impact.




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Senior Economist
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, WA, Seattle

            
        


        
    
Amazon.com strives to be Earth's most customer-centric company where customers can shop in our stores to find and discover anything they want to buy. We hire the world's brightest minds, offering them a fast paced, technologically sophisticated and friendly work environment. Economists in the Forecasting, Macroeconomics & Finance field document, interpret and forecast Amazon business dynamics. This track is well suited for economists adept at combining times-series statistical methods with strong economic analysis and intuition. This track could be a good fit for candidates with research experience in: macroeconometrics and/or empirical macroeconomics; international macroeconomics; time-series econometrics; forecasting; financial econometrics and/or empirical finance; and the use of micro and panel data to improve and validate traditional aggregate models. Economists at Amazon are expected to work directly with our senior management and scientists from other fields on key business problems faced across Amazon, including retail, cloud computing, third party merchants, search, Kindle, streaming video, and operations. The Forecasting, Macroeconomics & Finance field utilizes methods at the frontier of economics to develop formal models to understand the past and the present, predict the future, and identify relevant risks and opportunities. For example, we analyze the internal and external drivers of growth and profitability and how these drivers interact with the customer experience in the short, medium and long-term. We build econometric models of dynamic systems, using our world class data tools, formalizing problems using rigorous science to solve business issues and further delight customers.




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Senior Economist
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, WA, Seattle

            
        


        
    
Amazon.com strives to be Earth's most customer-centric company where customers can shop in our stores to find and discover anything they want to buy. We hire the world's brightest minds, offering them a fast paced, technologically sophisticated and friendly work environment. Economists at Amazon partner closely with senior management, business stakeholders, scientist and engineers, and economist leadership to solve key business problems ranging from Amazon Web Services, Kindle, Prime, inventory planning, international retail, third party merchants, search, pricing, labor and employment planning, effective benefits (health, retirement, etc.) and beyond. Amazon Economists build econometric models using our world class data systems and apply approaches from a variety of skillsets – applied macro/time series, applied micro, econometric theory, empirical IO, empirical health, labor, public economics and related fields are all highly valued skillsets at Amazon. You will work in a fast moving environment to solve business problems as a member of either a cross-functional team embedded within a business unit or a central science and economics organization. You will be expected to develop techniques that apply econometrics to large data sets, address quantitative problems, and contribute to the design of automated systems around the company.




        

        

        

        
    


    



                    

                
            

        

    

    

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