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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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                Sr. Communication Systems Engineer 
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, CA, San Diego

            
        


        
    
Amazon Leo is an initiative to launch a constellation of Low Earth Orbit satellites that will provide low-latency, high-speed broadband connectivity to unserved and underserved communities around the world. Come work at Amazon! The Role: Be part of the team defining the overall communication system and architecture of Leo’s broadband wireless network. This is a unique opportunity to innovate and define groundbreaking wireless technology with few legacy constraints. The team develops and designs the communication system of Leo and analyzes its overall system level performance such as for overall throughput, latency, system availability, packet loss etc. This role in particular will be responsible for leading the effort in integration, verification and testing of the systems especially focused on MAC and higher layer testing. This role will also be responsible developing and testing advanced L1/L2/L3 concept to improve the performance and reliability of the LEO network. This role will also be part of a team and develop simulation tools with particular emphasis on modeling the physical layer aspects such as advanced receiver modeling and abstraction, interference cancellation techniques, FEC abstraction models etc. In this role you will: - Work within a project team and take the responsibility for the Leo’s communication system design, system integration and verification. - Work as a part of the team in building a suite of system and network simulation services in Matlab / C++ / Python - Develop requirements from system level to HW/SW level and define test cases associated with the requirements. - Identify additional HW and SW that are needed for the purposes of verification and guide the HW/SW development team in the development of these test solutions/tools// - Work closely with implementation teams to simulate expected system level performance and provide quick feedback on potential improvements - Write scripts / code for functions / features required for specific simulation, testing and verification of given RF system EXPORT CONTROL REQUIREMENTS 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.




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                PhD Economics Intern – Structural IO, Summer 2027
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, WA, Seattle

            
        


        
    
Amazon Economics is seeking Structural IO Economist (STRUC) Interns who are passionate about applying structural econometric methods to solve real-world business challenges. STRUC economists specialize in the econometric analysis of models that involve the estimation of fundamental preferences and strategic effects. In this full-time internship (40 hours per week, with hourly compensation), you'll work with large-scale datasets to model strategic decision-making and inform business optimization, gaining hands-on experience that's directly applicable to dissertation writing and future career placement. By applying to this role, you are automatically being considered for all our available STRUC internships in 2027. Key job responsibilities As a STRUC Economist Intern, you'll specialize in structural econometric analysis to estimate fundamental preferences and strategic effects in complex business environments. Your responsibilities include: - Analyze large-scale datasets using structural econometric techniques to solve complex business challenges - Applying discrete choice models and methods, including logistic regression family models (such as BLP, nested logit) and models with alternative distributional assumptions - Utilizing advanced structural methods including dynamic models of customer or firm decisions over time, applied game theory (entry and exit of firms), auction models, and labor market models - Building datasets and performing data analysis at scale - Collaborating with economists, scientists, and business leaders to develop data-driven insights and strategic recommendations - Tackling diverse challenges including pricing analysis, competition modeling, strategic behavior estimation, contract design, and marketing strategy optimization - Helping business partners formalize and estimate business objectives to drive optimal decision-making and customer value - Build and refine comprehensive datasets for in-depth structural economic analysis - Present complex analytical findings to business leaders and stakeholders




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Senior Economist, Litigation and Regulatory Economics 
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, VA, Arlington

            
        


        
    
Want to help Amazon tell its customer-centric story around the world and work in a highly cross-functional environment with economists, lawyers, scientists, public policy, public relations, and business teams? If yes, keep reading! You'll join a team of economists, engineers, and lawyers to develop economic analysis and evidence supporting legal and regulatory matters across all our lines of business worldwide—including retail, marketplace services, AWS, consumer experience, shopping and search, and operations. In this role, you will have exposure to complex regulatory issues that are of high strategic importance to the company and will develop significant expertise on the economics of Amazon’s business operations and the industries in which it operates. If you're an economist with a passion for the current legal and policy debate, strong practical judgment and creative problem-solving skills, a love of communicating economic ideas to non-technical audiences, a knack for distilling data and economic models into key insights, and a track record of delivering results fast, we want to talk to you! Key job responsibilities • Provide data-driven guidance on high-stakes legal and regulatory questions facing Amazon worldwide • Collaborate with economists, scientists, engineers, and non-technical partners on high-impact projects with global scope • Partner with global public policy teams to apply economic analyses to current policy debates on competition, AI, and related issues • Engage with external stakeholders to drive deeper understanding of Amazon’s business model and the value it develops for the economy • Support requests for economic analyses and data in ongoing regulatory and litigation matters worldwide • Synthesize business facts and data into compelling economic narratives, translating complex findings into actionable insights • Advise stakeholders across Amazon on a broad spectrum of complex and often novel economic issues • Conduct, direct, and coordinate all phases of research projects—defining key questions, evaluating methodology, executing analysis, and communicating results




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                PhD Economics Intern – Reduced Form Causal Inference, Summer 2027
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, WA, Seattle

            
        


        
    
Amazon Economics is seeking Reduced Form Causal Analysis (RFCA) Economist Interns who are passionate about applying econometric methods to solve real-world business challenges. RFCA represents the largest group of economists at Amazon, and these core econometric methods are fundamental to economic analysis across the company. In this a full-time internship (40 hours per week, with hourly compensation). You'll work with large-scale datasets to analyze causal relationships and inform strategic business decisions, gaining hands-on experience that's directly applicable to dissertation writing and future career placement. By applying to this role, you are automatically being considered for all our available RFCA internships in 2027. Key job responsibilities As an RFCA Economist Intern, you'll specialize in econometric analysis to determine causal relationships in complex business environments. Your responsibilities include: - Analyze large-scale datasets using advanced econometric techniques to solve complex business challenges - Applying econometric techniques such as regression analysis, binary variable models, cross-section and panel data analysis, instrumental variables, and treatment effects estimation - Utilizing advanced methods including differences-in-differences, propensity score matching, synthetic controls, and experimental design - Building datasets and performing data analysis at scale - Collaborating with economists, scientists, and business leaders to develop data-driven insights and strategic recommendations - Tackling diverse challenges including program evaluation, elasticity estimation, customer behavior analysis, and predictive modeling that accounts for seasonality and time trends - Build and refine comprehensive datasets for in-depth economic analysis - Present complex analytical findings to business leaders and stakeholders




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Senior Applied Scientist, FBA AI Science & Analytics
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, WA, Bellevue

            
        


        
    
FBA AI Science and Analytics accelerates the AI-native transformation of Fulfillment by Amazon by building, integrating, and scaling AI-powered data & science products and seller-facing experiences that drive operational efficiency and growth across Fulfillment by Amazon globally. We learn seller behaviors, design the policies and incentives that shape their experience, and ship science products that help third-party sellers grow topline and cut operating costs at Amazon scale. Our work sits at the intersection of machine learning, statistics, economics, operations research, and GenAI/LLMs. We're looking for a Senior Applied Scientist who wants to put GenAI to work on a hard, high-visibility problem: building next-generation multi-agent systems that interact with millions of sellers and guide them through their toughest challenges at scale. You'll own solutions spanning supervised and unsupervised learning, recommendation systems, statistical learning, LLMs, harness engineering, and reinforcement learning. The ambition is to make AI a native layer in every seller decision rather than a separate tool sellers must adopt, delivering actionable insight in minutes, not days. You'll shape end-to-end experiences across the highest-frequency seller workflows, including inventory optimization, inbound efficiency, defect improvements, reimbursements, and capacity planning. The role carries direct visibility with senior Amazon business leaders and works together with fellow scientists, engineers, and product teams to launch production-grade agentic capabilities. Key job responsibilities - Design, build and deploy FBA’s GenAI architectures end to end. - Apply state-of-the-art ML and GenAI solve diverse business problems across seller supply chain systems. - Define the team’s long-term science vision and roadmap, driven fundamentally from our customers' needs, translating those directions into specific plans for scientists, engineers, and product partners. - Partner closely with scientists and software engineers to drive real-time model implementations and deliver high-impact features. - Establish scalable, efficient, automated processes for large scale data analyses, model benchmarking, model evaluation and model implementation. - Advocate the right ML solutions to business stakeholders, engineering teams, as well as executive level decision makers




        

        

        

        
    


    



                    

                
            

        

    

    

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