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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 - Agentic AI, Amazon Fulfillment Technology AI Science 
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, WA, Bellevue

            
        


        
    
As an Applied Scientist in Amazon Fullfilment Technology, you will lead the development of agentic systems to assist with operational decision making and orchestration. You will work building full agentic systems leveraging multi-agent orchestration, tool use, memory, and action execution. You will train LLMs using a combination of rejection sampling approaches, SFT, continual post-training, and Reinforcement Learning (RL). These systems are deployed to Amazon buildings, and you will also work on rigorous offline and online evaluations. Your work will leverage the latest LLMs to develop capabilities for agentic reasoning, coding and analytics. You will also lead research projects to tackle unsolved problems, mentor interns, and author academic papers to summarize your findings for external publication. Key job responsibilities - Generating training and preference data for specific use cases (reasoning trajectories, tool traces) - Reward modeling and policy optimization for LLMs: DPO, IPO, RLHF/RLAIF with PPO/GRPO, rejection sampling. - Supervised fine-tuning on step-by-step trajectories and tool-use traces - Verbal Reinforcement Learning and Continual Learning - RL for LLMs, Offline RL and off-policy evaluation - Agentic memory/state management; episodic and semantic memory; vector search; grounding with RAG. - Evaluation: developing decision quality metrics, scaling LLM-based evaluations. About the team Amazon Fulfillment Technologies (AFT) powers Amazon's global fulfillment network. We invent and deliver software, hardware, and data science solutions that orchestrate processes, robots, machines, and people. We harmonize the physical and virtual world so Amazon customers can get what they want, when they want it. Learn more about AFT: https://tinyurl.com/AFTOverview




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Economist, Decision Science - Devices, Decision Science - Devices
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, VA, Arlington

            
        


        
    
Are you excited about making business decisions using science and data? Are you interested in supporting consumer device concepts from idea inception to launch? Do you want to work on a Science Product team focused on scaling statistics and econometrics with custom tools? If so, this may be the role for you! Amazon.com strives to be Earth's most customer-centric company. The Amazon Devices and Services team focuses on delighting customer by enabling seamless functionality in supplying, entertaining, and managing the home -- and beyond. We seek and hire the world's brightest minds, offering them a fast-paced, technologically-sophisticated, and friendly work environment, where economic theory meets real-world industry. The Decision Science team in Devices owns demand estimates and pricing recommendations of concept devices before customers know they exist. We support devices and services ranging from Echo Frames to Kindle Paperwhite to Blink Video Camera …all prior to launch. We are a cross-functional Product team working to scale Econometrics through Amazon and beyond by incorporating Science into internal facing tools and making it easier for others to do so as well. In this role, you will have input in decision meetings with Amazon senior leadership, which include go/no-go decisions for brand new devices and services and build volume decisions for manufacture prior to receiving any customer signal. You will have direct input to pricing decisions. You will leverage Science and Tools produced by the Decision Science team such as conjoint demand models to produce these recommendations. You will work with Scientists, Economists, Product Managers, and Software Developers to provide meaningful feedback about stakeholder problems to inform business solutions and increase the velocity, quality, and scope behind our recommendations. You will also have the opportunity to work on special projects to both guide the business and advance your own knowledge and understanding of specific topics. Key job responsibilities Applies expertise to develop econometric/machine learning models to measure the demand of devices and the business; Reviews models and results for other scientists, mentors junior scientists; Generates economic insights for the Devices and Services business and work with stakeholders to run the business for effectively; Describes strategic importance of vision inside and outside of team; and, Identifies business opportunities, defines the problem and how to solve it; Engages with senior scientists, business leadership outside Devices and Services to understand interplay between different business units.




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Applied Scientist - Agentic AI, Amazon Fulfillment Technology
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, WA, Bellevue

            
        


        
    
As an Applied Scientist in Amazon Fullfilment Technology, you will lead the development of agentic systems to assist with operational decision making and orchestration. You will work building full agentic systems leveraging multi-agent orchestration, tool use, memory, and action execution. You will train LLMs using a combination of rejection sampling approaches, SFT, continual post-training, and Reinforcement Learning (RL). These systems are deployed to Amazon buildings, and you will also work on rigorous offline and online evaluations. Your work will leverage the latest LLMs to develop capabilities for agentic reasoning, coding and analytics. You will also lead research projects to tackle unsolved problems, mentor interns, and author academic papers to summarize your findings for external publication. Key job responsibilities - Generating training and preference data for specific use cases (reasoning trajectories, tool traces) - Reward modeling and policy optimization for LLMs: DPO, IPO, RLHF/RLAIF with PPO/GRPO, rejection sampling. - Supervised fine-tuning on step-by-step trajectories and tool-use traces - Verbal Reinforcement Learning and Continual Learning - RL for LLMs, Offline RL and off-policy evaluation - Agentic memory/state management; episodic and semantic memory; vector search; grounding with RAG. - Evaluation: developing decision quality metrics, scaling LLM-based evaluations. About the team Amazon Fulfillment Technologies (AFT) powers Amazon's global fulfillment network. We invent and deliver software, hardware, and data science solutions that orchestrate processes, robots, machines, and people. We harmonize the physical and virtual world so Amazon customers can get what they want, when they want it. Learn more about AFT: https://tinyurl.com/AFTOverview




        

        

        

        
    


    



                    

                
                    

                        

    

    

        
            

                Data Scientist II, AWS Managed Operations Intelligence (MOI) - Data Science
            

        

        

            
                
                    

                
            

            

            

            

            
                
US, VA, Arlington

            
        


        
    
Amazon Web Services (AWS) is the world leader in providing a highly reliable, scalable, low-cost infrastructure platform in the cloud that powers hundreds of thousands of businesses in 190 countries around the world! Passionate about building, owning and operating massively scalable systems? Want to make a billion-dollar impact? If so, we have an exciting opportunity for you. The AWS Managed Operations (MO) organization was founded in April 2023, with the objective to reduce operational load and toil through long-term engineering projects. MO is building the best-in-class engineering and operations team that will own the day-to-day operations for AWS Regions, improving the availability, reliability, latency, performance and efficiency to operate AWS regions. The AWS Managed Operations Intelligence (MOI) Team is looking for a Data Scientist to lead the research and thought leadership to drive our data and insight strategy for AWS. You will be expected to serve as a Full Stack Data Scientist. You will be responsible for driving data-driven transformation across the organization. In this role, you will be responsible for the end-to-end data science lifecycle, from data exploration, ETL, model development and data visualization. You will leverage a broad set of tools and technologies, including general analytical frameworks (Spark, Airflow, etc.), AI frameworks (Hugging Face, etc.) and various machine learning frameworks, to tackle complex business problems. Your analytics research will provide direction on the technology strategy of the Managed Operations organization. Your Decision Science artifacts will provide insights that inform AWS' Operations and Site Reliability Engineering teams. You will work on ambiguous and complex business and research science problems at scale, and you are comfortable working with cross-functional teams and systems. By working together on behalf of our customers, we are building the future one innovative product, service, and idea at a time. Are you ready to embrace the challenge? Come build the future with us. Key job responsibilities - Work with large and complex data sets to solve a wide array of challenging problems using different analytical approaches - Develop ML/AI models. Partner with software teams to productionalize these models. - Data Pipeline and Infrastructure: design and implementation of data pipelines - Metric Development and Monitoring: Define and develop advanced, customized metrics and key performance indicators (KPIs) that capture the nuances of the organization's strategic objectives and operational complexities. Continuously monitor and evaluate the performance of metrics A day in the life Why AWS? Our team is dedicated to supporting new members. We have a broad mix of experience levels and tenures, and we're building an environment that celebrates knowledge-sharing and mentorship. Our senior members enjoy one-on-one mentoring and thorough, but kind, code reviews. We care about your career growth and strive to assign projects that help our team members develop your engineering expertise so you feel empowered to take on more complex tasks in the future. Diverse Experiences AWS values diverse experiences. Even if you do not meet all of the preferred qualifications and skills listed in the job description, we encourage candidates to apply. If your career is just starting, hasn't followed a traditional path, or includes alternative experiences, don't let it stop you from applying. About AWS Amazon Web Services (AWS) is the world's most comprehensive and broadly adopted cloud platform. We pioneered cloud computing and never stopped innovating — that's why customers from the most successful startups to Global 500 companies trust our robust suite of products and services to power their businesses. AWS Infrastructure Services (AIS) AWS Infrastructure Services owns the design, planning, delivery, and operation of all AWS global infrastructure. In other words, we're the people who keep the cloud running. We support all AWS data centers and all of the servers, storage, networking, power, and cooling equipment that ensure our customers have continual access to the innovation they rely on. We work on the most challenging problems, with thousands of variables impacting the supply chain — and we're looking for talented people who want to help. Inclusive Team Culture AWS values curiosity and connection. Our employee-led and company-sponsored affinity groups promote inclusion and empower our people to take pride in what makes us unique. Our inclusion events create stronger, more collaborative teams. Our continual innovation is fueled by the bold ideas, fresh perspectives, and passionate voices our teams bring to everything we do. Mentorship & Career Growth We're continuously raising our performance bar as we strive to become Earth's Best Employer. That's why you'll find endless knowledge-sharing, mentorship and other career-advancing resources here to help you develop into a better-rounded professional. About the team The Managed Operations Intelligence (MOI) Team helps AWS operate its services across the world. We help monitor AWS operations by providing insights and recommendations on AWS operations.




        

        

        

        
    


    



                    

                
            

        

    

    

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