A gentle introduction to automated reasoning

Meet Amazon Science’s newest research area.

The new automated-reasoning icon

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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Job summaryDo you want to be part of a new team at Amazon that is making history? Do you want to build technology and new science that millions of people will use? Are you excited about working on large scale Natural Language Processing (NLP), Machine Learning (ML), and Deep Learning (DL)?We are embarking on a multi-year journey to improve the shopping experience for customers globally. Amazon Search team creates customer-focused search solutions and technologies that makes shopping delightful and effortless for our customers. Our goal is to understand what customers are looking for in whatever language happens to be their choice at the moment and help them find what they need in Amazon's vast catalog of billions of products. As Amazon expands to new geographies, we are faced with the unique challenge of maintaining the bar on Search Quality due to the diversity in user preferences, multilingual search and data scarcity in new locales.We are looking for an applied researcher to work on improving search on Amazon using NLP, ML, and DL technology. As an Applied Scientist, you will lead our efforts in query understanding, semantic matching (e.g. is a drone the same as quadcopter?), relevance ranking (what is a "funny halloween costume"?), language identification (did the customer just switch to their mother tongue?), machine translation (猫の餌を注文する). This is a highly visible role with a huge impact on Amazon customers and business. As part of this role, you will develop high precision, high recall, and low latency solutions for search. Your solutions should work for all languages that Amazon supports and will be used in all Amazon locales world-wide. You will develop scalable science and engineering solutions that work successfully in production. You will work with leaders to develop a strategic vision and long term plans to improve search globally. We are growing our collaborative group of engineers and applied scientists by expanding into new areas. This is a position on Global Search Quality team in Bangalore (India). We are moving fast to change the way Amazon search works. Together with a multi-disciplinary team you will work on building solutions with NLP/ML/DL at its core. Along the way, you’ll learn a ton, have fun and make a positive impact on millions of people. Come and join us as we invent new ways to delight Amazon customers.Do not hesitate to reach out if you have some of the following: ability to apply state of the art in large scale Machine Learning (e.g. semi-weakly-un-supervised deep learning, natural language understanding), curiosity to learn through controlled experimentation or experience with low latency production systems. Apply now or ping Shailendra Agarwal (https://www.linkedin.com/in/shailendra-agarwal-893a3b11/) to learn more about the different ways you can have huge impact with us.We are an inclusive employer and value diversity at Amazon. We do not discriminate on the basis of race, religion, color, national origin, gender, sexual orientation, age, marital status, veteran status, or disability status.
US, VA, Virtual Location - Virginia
Job summaryJob summaryMachine learning (ML) has been strategic to Amazon from the early years. We are pioneers in areas such as recommendation engines, product search, eCommerce fraud detection, and large-scale optimization of fulfillment center operations.The Amazon ML Solutions Lab team helps AWS customers accelerate the use of machine learning to solve business and operational challenges and promote innovation in their organization. We are looking for a passionate, talented, and inventive Applied Scientist with a strong machine learning background to help develop solutions by pushing the envelope in Time Series, Automatic Speech Recognition (ASR), Natural Language Understanding (NLU), Machine Learning (ML), Computer Vision (CV) and More.As a ML Solutions Lab Applied Scientist, you are proficient in designing and developing advanced ML models to solve diverse challenges and opportunities. You will be working with terabytes of text, images, and other types of data and develop novel models to solve real-world problems. You'll design and run experiments, research new algorithms, and find new ways of optimizing risk, profitability, and customer experience. You will apply classical ML algorithms and cutting-edge deep learning (DL) and reinforcement learning approaches to areas such as drug discovery, customer segmentation, fraud prevention, capacity planning, predictive maintenance, pricing optimization, call center analytics, player pose estimation, event detection, and virtual assistant among others.Key job responsibilitiesThe primary responsibilities of this role are to:Design, develop, and evaluate innovative ML/DL models to solve diverse challenges and opportunities across industriesInteract with customer directly to understand their business problems, and help them with defining and implementing scalable ML/DL solutions to solve themWork closely with account teams, research scientist teams, and product engineering teams to drive model implementations and new algorithmsThis position requires travel of up to 25%. Role can be preferably based in DC, Maryland, Virginia / New York, New Jersey area; However candidates in Boston or Atlanta area are encouraged to apply.
FR
Job summaryThe ER Solutions team uses economics, behavioral science, statistics, and machine learning to proactively identify mechanisms and process improvements which solve for defects in the Employee Experience. We are an interdisciplinary team which combines the talents of science, psychology and industry specialists to develop and deliver solutions that measurably achieve this goal. We are looking for an Economist who is able to provide structure around complex business problems, hone those complex problems into specific, scientific questions, and test those questions to generate insights. The ideal candidate will work with various science, PXT, ER, operations and tech teams to estimate models and algorithms on large scale data, design pilots and measure their impact, and transform successful prototypes into improved policies and programs at scale. Ideal candidates will work closely with business partners to develop science that solves the most important business challenges. They will need to work well in a team setting with individuals from diverse disciplines and backgrounds. Serve as an ambassador for science and a scientific resource for business teams, so that scientific processes permeate throughout the PXT organization to the benefit of Amazonians and Amazon. Ideal candidates will own the development of scientific models and manage the data analysis, modeling, and experimentation that is necessary for estimating and validating model. They will be customer-centric – clearly communicating scientific approaches and findings to business leaders, listening to and incorporate their feedback, and delivering successful scientific solutions.
US, WA, Seattle
Job summaryLooking for a challenging and fun role in a new and growing part of Amazon? At Alexa Shopping, we strive to enable shopping in everyday life. We make it easy for customers to shop by interacting with Alexa on Amazon devices such as Show, Dot, Fire TV or on Amazon Shopping app on mobile or on any 3rd party device with an Alexa endpoint. Our Services allow you to shop, no matter where you are or what you are doing, you can go from 'I want that' to 'that's on the way' in a matter of seconds. We are seeking the industry's best to help us create new ways to interact, search and shop. Join us, and you'll be taking part in changing the future of everyday life. 

What you will do: You will lead a team of talented and experienced scientists and engineers that implement solutions to improve automated speech recognition (ASR) for Alexa Shopping customers. 
You will build a world class team, demonstrate thought-leadership and guide the team to make the right scientific or technical trade-offs to meet long term/short-term business needs. You will lead research and drive greenfield invention leading a team of data scientists and software engineers. You should be comfortable with ambiguity and enjoy working in a fast-paced and dynamic environment. 
You will own: 
(1) Creating a road map of the most challenging business questions from product owners, program managers and senior leadership, and use data to find solutions.
(2) Partnering with scientists across Shopping and Alexa to design and run experiments, research new algorithms, and find new ways to improve Shopping ASR. 
(3) Working with stakeholders and internal customer teams to solve business and technology problems to delight both internal stakeholder teams and Alexa end customers. 
(4) Creating systems and data pipelines to deliver solutions at Amazon scale.And finally, you will have the satisfaction of being able to look back and say you were a key contributor to something special from its earliest stages. You will be working closely with executive leadership, multiple product managers and leaders from partner teams in Amazon Retail, Alexa, and Speech Recognition teams. 

What we are looking for: We are looking for a talented Data Science Manager with a strong technical background and solid people management skills to build, manage and develop a highly-talented and experienced data science team. We are seeking leaders that can guide technical and product innovation in the areas of voice experiences, machine learning models and the distributed systems to bring our vision together. Strong judgment and communication skills, long term technical vision, and continuous focus on engineering and operational excellence are essential for the success in this role.Key job responsibilitiesYou will own: 
(1) Creating a road map of the most challenging business questions from product owners, program managers and senior leadership, and use data to find solutions.
(2) Partnering with scientists across Shopping and Alexa to design and run experiments, research new algorithms, and find new ways to improve Shopping ASR. 
(3) Working with stakeholders and internal customer teams to solve business and technology problems to delight both internal stakeholder teams and Alexa end customers. 
(4) Creating systems and data pipelines to deliver solutions at Amazon scale.(5) Growing and developing the careers of scientists and SDEs in the team.A day in the lifeYou'll review KPIs for the business and the team and look for ways each day to improve them or to create new metrics and insights. Partner with Alexa ASR, Retail Search and Shopping CX owners to align goals, tech strategy and roadmaps. Guide your team to build new data science solutions, ML models and data pipelines.About the teamThe Shopping ASR team is a mix of data scientists and SDEs that own the charter of improving speech accuracy for all shopping customers across devices (Amazon App, Echo, Show) and experiences such as Purchasing, Your Account, Shopping Lists and Product Q&A. We work closely with Alexa ASR, Shopping NLU and Automated Metrics as well as teams in retail such as M5.