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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, Special Projects
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, WA, Seattle

            
        


        
    
Innovators wanted! Are you an entrepreneur? A builder? A dreamer? This role is part of an Amazon Special Projects team that takes the company’s Think Big leadership principle to the limits. If you’re interested in innovating at scale to address big challenges in the world, this is the team for you. As an Applied Scientist on our team, you will focus on building state-of-the-art ML models for healthcare. Our team rewards curiosity while maintaining a laser-focus in bringing products to market. Competitive candidates are responsive, flexible, and able to succeed within an open, collaborative, entrepreneurial, startup-like environment. At the forefront of both academic and applied research in this product area, you have the opportunity to work together with a diverse and talented team of scientists, engineers, and product managers and collaborate with other teams. This role offers a unique opportunity to work on projects that could fundamentally transform healthcare outcomes. Key job responsibilities In this role, you will: • Design and implement novel AI/ML solutions for complex healthcare challenges • Drive advancements in machine learning and data science • Balance theoretical knowledge with practical implementation • Work closely with customers and partners to understand their requirements • Navigate ambiguity and create clarity in early-stage product development • Collaborate with cross-functional teams while fostering innovation in a collaborative work environment to deliver impactful solutions • Establish best practices for ML experimentation, evaluation, development and deployment • Partner with leadership to define roadmap and strategic initiatives You’ll need a strong background in AI/ML, proven leadership skills, and the ability to translate complex concepts into actionable plans. You’ll also need to effectively translate research findings into practical solutions. A day in the life You will solve real-world problems by getting and analyzing large amounts of data, generate insights and opportunities, design simulations and experiments, and develop statistical and ML models. The team is driven by business needs, which requires collaboration with other Scientists, Engineers, and Product Managers across the Special Projects organization. You will prepare written and verbal presentations to share insights to audiences of varying levels of technical sophistication. About the team We represent Amazon's ambitious vision to solve the world's most pressing challenges. We are exploring new approaches to enhance research practices in the healthcare space, leveraging Amazon's scale and technological expertise. We operate with the agility of a startup while backed by Amazon's resources and operational excellence. We're looking for builders who are excited about working on ambitious, undefined problems and are comfortable with ambiguity.




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Sr. Comm Systems Engineer, Wireless Systems 
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, TX, Austin

            
        


        
    
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. As a Systems Engineer, this role is primarily responsible for the design, development and integration of communication payload and customer terminal systems. The Role: Be part of the team defining the overall communication system and architecture of Amazon Leo’s broadband wireless network. This is a unique opportunity to innovate and define groundbreaking wireless technology at global scale. The team develops and designs the communication system for project 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 designing and developing advanced technology and solutions for communication system. This role will also be responsible developing advanced physical layer + protocol stacks systems as proof of concept and reference implementation to improve the performance and reliability of the LEO network. In particular this role will be responsible for using concepts from digital signal processing, information theory, wireless communications to develop novel solutions for achieving ultra-high performance 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. This role will also play a critical role in the integration and verification of various HW and SW sub-systems as a part of system integration and link bring-up and verification. 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.




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Senior Applied Scientist, Real-Time Conversational AI , AGI
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, CA, Sunnyvale

            
        


        
    
We are looking for a Senior Applied Scientist to help drive the research and development of real-time multimodal conversational AI. You will contribute across two focus areas: advancing foundation models for speech and audio, and building the post-training systems (reward modeling, reinforcement learning) that shape natural, human-like conversational behavior. You will own a significant research area and contribute across the full model lifecycle — from pre-training and architecture design through post-training alignment and real-time deployment. You will work at the frontier of what’s possible in conversational AI, with the compute, data, and runway to pursue problems that few teams in the world have the resources to tackle. As a Senior Scientist, you will drive the technical execution of your research area, contribute to the team’s roadmap, and work closely with inference engineers to ensure your models are designed for real-time production deployment. Key job responsibilities What You’ll Do Foundation Model Scaling - Help build and train large-scale multimodal foundation models for real-time speech and audio generation, from architecture design through production-scale training - Advance the scaling and efficiency of conversational models, including the relationship between data, model size, and real-time performance - Design model architectures informed by hardware constraints and inference requirements, working with inference engineers to ensure models are servable from inception - Develop training methodologies for multimodal models that jointly process and generate speech, language, and audio in real-time streaming contexts - Contribute to the state of the art on efficient architectures and training methods for conversational AI at scale Post-Training & Reinforcement Learning - Design and build reward models and reward functions for speech systems — capturing naturalness, fluency, conversational quality, and real-time responsiveness - Develop and apply reinforcement learning methods to shape conversational behavior — teaching models natural timing, responsiveness, and fluid interaction - Build parts of the post-training pipeline from SFT through RL alignment, optimized for real-time multimodal outputs rather than text-only generation - Design evaluation frameworks that capture the quality dimensions unique to real-time conversation (latency sensitivity, audio quality, prosody, interaction naturalness) Real-Time Perception & Generation - Advance the team’s capabilities in real-time perception — the ability of the model to process incoming audio/speech while simultaneously generating responses - Develop techniques for natural interactive systems where the model handles concurrent input and output with human-like timing - Work at the intersection of model architecture and production constraints to ensure multimodal capabilities function within hard real-time latency budgets




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Senior Applied Scientist, Fauna Robotics
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, WA, Bellevue

            
        


        
    
Robots are about to step out of the movies and into our homes. If you're excited about this robotics revolution, come join Amazon's Fauna Robotics. As an Applied Scientist in this org, you'll push the state-of-the-art techniques in Robotics. You'll drive technical excellence in areas such as perception, manipulation, sim2real transfer, reinforcement learning and multi-task learning, designing novel algorithms that bridge the gap between research and real-world deployment. In this role, you will integrate hands-on technical expertise with scientific leadership, ensuring your team delivers robust solutions for dynamic real-world environments. While staying updated with the latest advancements in robotics research, you will leverage your experience and expertise to guide the team towards the most promising direction. Collaborating with a talented group of software engineers and scientists, you will bring your ideas to fruition. Deploying robots in home environments requires you to anticipate challenges not encountered in structured settings like warehouses. Additionally, consumer-grade sensors and actuators necessitate working around their limitations, requiring out-of-the-box innovative ideas. As such, you are someone who truly enjoys thinking about problems using first principles at a holistic and systems level to address the aforementioned challenges. Key job responsibilities - Lead technical initiatives in areas such as robotics foundation models, reinforcement learning and manipulation - - Design experiments to identify the limitations of current state-of-the-art models and developing new models or techniques that can surpass these models - Design and implement novel deep learning architectures that push the boundaries of what robots can understand and accomplish - Mentor fellow scientists while maintaining strong individual technical contributions - Collaborate with engineering teams to optimize and scale models for real-world applications - Influence technical decisions and implementation strategies within your area of focus - Experienced in communicating complex technical work to a non-technical audience - The ability to work with minimal guidance, be proactive and to handle ambiguity and the challenge of quickly evolving goals A day in the life - Train ML models for deployment in simulation and real-world robots, identify and document their limitations post-deployment - Drive technical discussions within your team and with key stakeholders to develop innovative solutions to address identified limitations - Guide fellow scientists in solving complex technical challenges, from sim2real transfer to training RL policies - Mentor team members while maintaining significant hands-on contribution to technical solutions About the team Fauna Robotics, an Amazon company, is building capable, safe, and genuinely delightful robots for everyday life. Our goal is simple: make robots people actually want to live and interact with in everyday human spaces. We believe that future won’t arrive until building for robotics becomes far more accessible. Today, too much effort is spent reinventing the fundamentals. We’re changing that by developing tightly integrated hardware and software systems that make it faster, safer, and more intuitive to create real-world robotic products. Our work spans the full stack: mechanical design, control systems, dynamic modeling, and intelligent software. The focus is not just functionality, but experience. We’re building robots that feel responsive, expressive, and genuinely useful. At Fauna, you’ll work at the frontier of this space, helping define how robots move, manipulate, and interact with people in natural environments. It’s an opportunity to solve hard problems across hardware and software with a team focused on making robotics accessible and joyful to build. If you care about making robotics real for everyone and building systems that are as delightful as they are capable, we’re interested in hearing from you.




        

        

        

        
    


    



                    

                
            

        

    

    

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