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 bysolved” 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:

  • logic is a formal and mechanical system for defining what is true and untrue. Examples: propositional logic or first-order logic.
  • theorem is a true statement in logic. Example: the four-color theorem.
  • proof is a valid argument in logic of a theorem. Example: Gonthier's proof of the four-color theorem
  • 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. 
  • 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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Interested in solving challenging problems using latest developments in Large Language Models and Artificial Intelligence to protect the global AWS platform from fraud ? The AWS Fraud Prevention organisation is looking for a talented Applied Scientist with a solid background in the design and development of scalable AI and ML systems and services, deep passion for building ML-powered products, a proven track record of executing complex projects, and delivering high business and customer impact. You will help us develop fraud prevention strategies leveraging AI technologies and add to our expansive portfolio of ML based mitigations. As a member of our team, you'll work on projects that directly impact millions of customers, organizations , and employees every single day. This role will provide exposure to state-of-the-art innovations in AI/ML systems (including GenAI). Technologies you will have exposure to, and/or will work with, include AWS Bedrock, Amazon Q, SageMaker, and Foundational Models such as Titan, Anthropic’s Claude, Mistral, among others. You’ll be part of a team of Applied Scientists, Research Scientists, Data Scientists, Investigations Analysts, and Technical & non-Technical Program Managers and Software Engineers. As an Applied Scientist, you will work directly with these folks to monitor the flavor/ trend of exploits AWS worldwide and design appropriate solutions to respond in a collaborative environment. There are no walls, and success is based on by your ability to dive deep, and understand the subtle demands new and complex services will place upon systems and teams. As an Applied Scientist your responsibilities will include: - Develop and apply state-of-the-art Machine Learning methods to large amounts of data from different sources to build and productionalize fraud prevention, detection and mitigation solutions - Actively contribute to the development of our ML Ops infrastructure - Actively contribute to the creation and delivery of our science roadmap which includes Gen AI efforts - Work closely with software engineering teams to deploy your innovations. - Mentor and coach junior scientists - Publish your work at internal and external conferences/journals - Providing on-call product support approximately once every 3 months On Call Responsibility. This position involves on-call responsibilities to support emergent customer impacting events. Our on-call fraud escalation support occurs approximately once every 12 weeks. This Applied Scientist is also expected to play a key role in raising the AI/ML skillset within the organization, so a passion for teaching/mentoring is important. This involves bringing deep AI/ML expertise to collaborations with other scientists, leading lunch & learn session, conducting tech talks, setting up regular office hours, evangelizing lessons learned & industry best practices, and more. The ideal candidate is one who possesses notable pedigree in ML-related academic work, along with deep, real-world, hands-on experience in executing/shipping AI-based products in a fast-paced setting. You will be looked up to as an ML expert that our organization’s leadership can rely on for guidance on AI/ML topics. Mentorship & Career Growth: Our team is dedicated to supporting new members. We have a broad mix of experience levels, functions and tenures, and we’re building an environment that celebrates knowledge sharing and mentorship. We care about your career growth and strive to assign projects based on what will help each team member develop into a better-rounded professional and enable them to take on more complex tasks in the future. Learn and Be Curious: We have a formal mentor search application that lets you find a mentor that works best for you based on location, job family, job level etc. Your manager can also help you find a mentor or two, because two is better than one. In addition to formal mentors, we work and train together so that we are always learning from one another, and we celebrate and support the career progression of our team members. Inclusion and Diversity. Our team is diverse! We drive towards an inclusive culture and work environment. We are intentional about attracting, developing, and retaining amazing talent from diverse backgrounds. Team members are active in Amazon’s 10+ affinity groups, sometimes known as employee resource groups, which bring employees together across businesses and locations around the world. These range from groups such as the Black Employee Network, Latinos at Amazon, Indigenous at Amazon, Families at Amazon, LGBTQ+, Warriors at Amazon (Military), Amazon People With Disabilities, and more. Learn more about Amazon on our Day 1 Blog: https://blog.aboutamazon.com About AWS 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. Why 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. Inclusive Team Culture Here at AWS, it’s in our nature to learn and be curious. Our employee-led affinity groups foster a culture of inclusion that empower us to be proud of our differences. Ongoing events and learning experiences, including our Conversations on Race and Ethnicity (CORE) and AmazeCon conferences, inspire us to never stop embracing our uniqueness. 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. Work/Life Balance We value work-life harmony. Achieving success at work should never come at the expense of sacrifices at home, which is why we strive for flexibility as part of our working culture. When we feel supported in the workplace and at home, there’s nothing we can’t achieve in the cloud. About the team Utility Computing (UC) AWS Utility Computing (UC) provides product innovations — from foundational services such as Amazon’s Simple Storage Service (S3) and Amazon Elastic Compute Cloud (EC2), to consistently released new product innovations that continue to set AWS’s services and features apart in the industry. As a member of the UC organization, you’ll support the development and management of Compute, Database, Storage, Internet of Things (Iot), Platform, and Productivity Apps services in AWS, including support for customers who require specialized security solutions for their cloud services.
IL, Tel Aviv
Come build the future of entertainment with us. Are you interested in helping shape the future of movies and television? Do you want to help define the next generation of how and what Amazon customers are watching? Prime Video is a premium streaming service that offers customers a vast collection of TV shows and movies - all with the ease of finding what they love to watch in one place. We offer customers thousands of popular movies and TV shows from Originals and Exclusive content to exciting live sports events. We also offer our members the opportunity to subscribe to add-on channels which they can cancel at any time and to rent or buy new release movies and TV box sets on the Prime Video Store. Prime Video is a fast-paced, growth business - available in over 240 countries and territories worldwide. The team works in a dynamic environment where innovating on behalf of our customers is at the heart of everything we do. If this sounds exciting to you, please read on We are seeking an exceptional Applied Scientist to join our Prime Video Sports tech team in Israel. Our team is dedicated to developing state-of-the-art science to allow for personalizing the customers’ experience and customers to seamlessly find any live event in our selection. You will have the opportunity to work on innovative, large-scale projects that push the boundaries of what's possible in sports content delivery and engagement. Your expertise will be crucial in tackling complex challenges such as temporal information retrieval, leveraging Generative AI and Large Language Models (LLMs), and building state-of-the-art recommender systems. Key job responsibilities We are looking for an Applied Scientist with domain expertise in Personalization, Information Retrieval, and Recommender Systems, or general ML to lead the development of new algorithms and end-to-end solutions. As part of our team of applied scientists and software development engineers, you will be responsible for researching, designing, developing, and deploying algorithms into production pipelines. Your role will involve working with cutting-edge technologies such as Gen AI/LLMs to enhance content discovery and search capabilities. You'll also tackle unique challenges like temporal information retrieval to improve real-time sports content recommendations. As a technologist, you will drive the publication of original work in top-tier conferences in Machine Learning and Information Retrieval. We expect you to thrive in ambiguous situations, demonstrating outstanding analytical abilities and comfort in collaborating with cross-functional teams and systems. The ideal candidate is a self-starter with the ability to learn and adapt quickly in our fast-paced environment. About the team We are the Prime Video Sports team. In September 2018 Prime Video launched its first full-scale live streaming experience to world-wide Prime customers with NFL Thursday Night Football. That was just the start. Now Amazon has exclusive broadcasting rights to major leagues like NFL Thursday Night Football, Tennis major like Roland-Garros and English Premium League to list few and are broadcasting live events across 30+ sports world-wide. Prime Video is expanding not just the breadth of live content that it offers, but the depth of the experience. This is a transformative opportunity, the chance to be at the vanguard of a program that will revolutionize Prime Video, and the live streaming experience of customers everywhere.