A gentle introduction to automated reasoning

Meet Amazon Science’s newest research area.

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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MULTIPLE POSITIONS AVAILABLE Employer: AMAZON.COM SERVICES LLC Offered Position: Applied Scientist III Job Location: Seattle, Washington Job Number: AMZ10037670 Position Responsibilities: Participate in the design, development, evaluation, deployment and updating of data-driven models and analytical solutions for machine learning (ML) and/or natural language (NL) applications. Develop and/or apply statistical modeling techniques (e.g. Bayesian models and deep neural networks), optimization methods, and other ML techniques to different applications in business and engineering. Routinely build and deploy ML models on available data, and run and analyze experiments in a production environment. Identify new opportunities for research in order to meet business goals. Research and implement novel ML and statistical approaches to add value to the business. Mentor junior engineers and scientists. Position Requirements: Master’s degree or foreign equivalent degree in Computer Science, Machine Learning, Engineering, or a related field and two years of research or work experience in the job offered, or as a Research Scientist, Research Assistant, Software Engineer, or a related occupation. Employer will accept a Bachelor’s degree or foreign equivalent degree in Computer Science, Machine Learning, Engineering, or a related field and five years of progressive post-baccalaureate research or work experience in the job offered or a related occupation as equivalent to the Master’s degree and two years of research or work experience. Must have one year of research or work experience in the following skill(s): (1) programming in Java, C++, Python, or equivalent programming language; and (2) conducting the analysis and development of various supervised and unsupervised machine learning models for moderately complex projects in business, science, or engineering. Amazon.com is an Equal Opportunity-Affirmative Action Employer – Minority / Female / Disability / Veteran / Gender Identity / Sexual Orientation. 40 hours / week, 8:00am-5:00pm, Salary Range $167,100/year to $226,100/year. Amazon is a total compensation company. Dependent on the position offered, equity, sign-on payments, and other forms of compensation may be provided as part of a total compensation package, in addition to a full range of medical, financial, and/or other benefits. For more information, visit: https://www.aboutamazon.com/workplace/employee-benefits. #0000
US, WA, Seattle
Amazon Web Services (AWS) provides a highly reliable, scalable, low-cost infrastructure platform in the cloud that powers hundreds of thousands of businesses in 190 countries around the world. We are hiring an Applied Scientist in our AWS Sales Science & Economics team. The team uses machine learning, econometrics, and data science to optimize AWS’s service catalogue, driving customer engagement and generating insights to guide AWS sales strategy. We use detailed customer behavioral and usage data to predict and understand what customers want from AWS. Key job responsibilities As an Applied Scientist, you will apply the latest research, state of the art machine learning and GenAI to build highly scalable models in support of predictive research and products used across AWS Sales and leadership. You will be responsible for designing and executing science experiments in short and agile cycles. You will use the latest AWS technologies in Machine Learning, Data Science, and AI. You will work closely with business leaders to identify opportunities to drive business growth by applying machine learning and building out predictive models to support analytics and insights. The ideal candidate will be curious and able to learn new technologies quickly. You will have hands-on experience building data science and machine learning pipelines in production working with large scale data sets. You are passionate about sharing your knowledge to enable others. You empathize with the needs of our customers and align your work accordingly, and provide thought leadership and guidance across the organization. A day in the life You will start your day reviewing experimental results and refining model architectures based on new insights. You might spend the morning prototyping a novel approach to a customer-facing problem, then shift to a design review with engineering partners in the afternoon to align on how your solution integrates into production systems. Along the way, you will mentor junior scientists, contribute to peer reviews, and stay connected to the latest research through reading groups and internal science discussions. Your contributions will move from research notebooks to live systems that customers interact with every day. About the team Our team is focused on applying advanced science and machine learning to build intelligent systems that improve the Amazon experience. We value collaboration, intellectual rigor, and a willingness to explore new ideas. You will work alongside scientists and engineers who are committed to turning ambitious research into practical, high-impact solutions. We are investing in the next generation of machine learning capabilities and are looking for team members who are excited to help define that future.
US, CA, San Francisco
The Central Science Team within Amazon’s People Experience and Technology org (PXTCS) uses economics, behavioral science, statistics, and machine learning to proactively identify mechanisms and process improvements which simultaneously improve Amazon and the lives, well-being, and the value of work to Amazonians. We are an interdisciplinary team, which combines the talents of science and engineering to develop and deliver solutions that measurably achieve this goal. We are looking for an Applied Scientist to build production machine learning solutions that improve employee experiences and inform workforce decisions. You will work on problems spanning predictive modeling, natural language processing, recommendation systems, personalization, generative AI, and causal evaluation. These solutions will predict workforce outcomes, identify individual and organizational needs, recommend targeted actions, and measure the causal impact of programs, products, and policies across diverse employee populations and business contexts. As an Applied Scientist, you will develop models and algorithms from prototype through production. You will collaborate with other scientists and economists to combine machine learning with causal inference and experimentation, allowing teams to understand both what is likely to happen and which actions will improve outcomes. You will also work with product managers, engineers, and business partners to translate business needs into technical solutions and communicate methods and results to technical and nontechnical audiences. Key job responsibilities • Design, develop, and deploy machine learning solutions in production environments. • Develop solutions using techniques that include predictive modeling, deep learning, natural language processing, recommendation systems, personalization, and generative AI. • Build models that predict workforce outcomes, identify employee and organizational needs, and recommend targeted actions. • Partner with economists and other scientists to design causal evaluations and experimental frameworks that measure the impact of programs, products, and policies. • Combine predictive models with causal estimates to help business partners identify effective actions and make evidence-based decisions. • Collaborate with product managers, engineers, and business partners to translate business requirements into modeling approaches and contribute to technical design and roadmap decisions. • Evaluate model accuracy, performance, calibration, and limitations, and identify opportunities for improvement. • Communicate technical concepts, methods, results, and recommendations to technical peers and business partners. A day in the life • Analyze data to investigate model performance or understand an unexpected result from a causal evaluation. • Brainstorm modeling and evaluation approaches with applied scientists and economists on the team. • Develop and test model features, recommendation methods, or generative AI applications. • Design or analyze experiments and observational studies to estimate the causal impact of programs, products, or policies. • Run simulations, offline evaluations, or backtests to assess model performance. • Meet with product, engineering, and business partners to review requirements, data, designs, and implementation decisions. • Review code changes or technical design documents from scientists and engineers. • Write and present documents covering methods, results, limitations, and recommendations. About the team The People eXperience and Technology Central Science Team (PXTCS) uses economics, behavioral science, statistics, and machine learning to proactively identify mechanisms and process improvements which simultaneously improve Amazon and the lives, wellbeing, and the value of work to Amazonians. We are an interdisciplinary team that combines the talents of science and engineering to develop and deliver solutions that measurably achieve this goal.
US, WA, Seattle
Are you excited to play a strategic role in defining the customer experience on Tablets? Come join a dynamic, customer-obsessed, world-class science team to completely reshape the customer experience on our tablets. Key job responsibilities Using Amazon’s large-scale computing resources, you will design and deploy state-of-the-art recommendation models. You will ask research questions about customer behavior, design state-of-the-art models that help customers discover content, and deploy these models to production alongside other engineers. You will participate in the Amazon ML community and mentor other engineers with a strong interest in and knowledge of ML and generative AI. Your work will directly benefit customers! A day in the life We are looking for a passionate, hard-working, and talented Applied Scientist who has experience developing state-of-the-art models and deploying them to production. You will have an opportunity to make an enormous impact on the design, architecture, and implementation of products used everyday by real customers! About the team The Alexa Tablet team is a world-class team that delivers high-quality, innovative products that are loved by our customers. Our team focuses on spinning many of Amazon’s flywheels with bar-raising mobile experiences on our tablets. Over the last 12 years, we have built 4.0+ star rated products, and we continue to look for ways to scale our business. Join us as we continue our ambitious journey which will give you the opportunity to deliver disruptive products with global impact.
IN, KA, Bengaluru
Amazon.com’s Product Detail Page team is looking for talented, motivated and passionate applied scientist to be part of the design and development of a highly scalable multi-tiered shopping application to provide the best possible online shopping experience for Amazon customers world-wide. Our team is comprised of talented applied scientists, developers, testers, program managers, designers and product managers tasked with the singular goal to create THE world's best buying experience. Scientists on this team develop the next-generation technologies and experiences that change how millions interact and shop online. To provide the best possible online shopping at the scale of the web requires ideas from every area of computer science, including distributed computing, large-scale system design, machine learning, natural language processing, data compression and user interface design; the list goes on and is growing every day. We need our scientists to be versatile and always eager to tackle new problems as we continue to push technology forward. Our team leverages sophisticated econometric, machine learning, and big data technologies to help customers to discover the right products at the right prices from millions of trusted sellers billions of times a day. If you are looking for a career-defining opportunity on one of the most customer centric and business impacting teams within Amazon, we’d love to hear from you. We are looking for an Applied Scientist to help build the next generation of Detail Page optimization algorithms. These new set of algorithms will incorporate the continually changing preferences of our customers and continue to scale with numerous new programs that Amazon is introducing for our customers. You will work with multiple Amazon businesses and programs to identify big business opportunities and propose new business features and technical systems to improve customer experience on Amazon Detail Page, Search Page and many other widgets throughout the website. You will be responsible for the quality of algorithm design and will get the opportunity to present your ideas and share results of your deliverables with Amazon executives on a frequent basis. You will get an opportunity to work with senior scientists to define and enforce broad, company-wide technical standards in optimization techniques, statistical modeling and simulation techniques, and/or data analytics.