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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Amazon Advertising is one of Amazon's fastest growing businesses. Amazon's advertising portfolio helps merchants, retail vendors, and brand owners succeed via native advertising, which grows incremental sales of their products sold through Amazon. The primary goals are to help shoppers discover new products they love, be the most efficient way for advertisers to meet their business objectives, and build a sustainable business that continuously innovates on behalf of customers. Our products and solutions are strategically important to enable our Retail and Marketplace businesses to drive long-term growth. We deliver billions of ad impressions and millions of clicks and break fresh ground in product and technical innovations every day! The Creative X team within Amazon Advertising time aims to democratize access to high-quality creatives (audio, images, videos, text) by building AI-driven solutions for advertisers. To accomplish this, we are investing in understanding how best users can leverage Generative AI methods such as latent-diffusion models, large language models (LLM), generative audio (music and speech synthesis), computer vision (CV), reinforced learning (RL) and related. As an Applied Scientist you will be part of a close-knit team of other applied scientists, engineers and product managers who are highly collaborative and at the top of their respective fields. We are looking for talented Applied Scientists who are adept at a variety of skills, especially at the interface of how users can use state-of-the-art generative music and audio, computer vision, latent diffusion or related foundational models that will accelerate our plans to generate high-quality creatives on behalf of advertisers. Every member of the team is expected to build customer (advertiser) facing features, contribute to the collaborative spirit within the team, publish, patent, and bring innovative research to raise the bar within the team. You will: - Drive the invention and development of novel methods for the use of Generative AI in ads - Work closely and integrate end-to-end proof-of-concept Machine Learning projects that have a high degree of ambiguity, scale and complexity. - Build interface-oriented systems that use Machine Learning models, perform proof-of-concept, experiment, optimize, and deploy your models into production; work closely with software engineers to assist in productionizing your ML models. - Run A/B experiments, gather data, and perform statistical analysis. - Establish scalable, efficient, automated processes for large-scale data analysis, machine-learning model development, model validation and serving. - Mentor and help recruit Applied Scientists to the team. - Present results and explain methods to senior leadership. - Willingness to publish research at internal and external top scientific venues. - Write and pursue IP submissions. Key job responsibilities This role is focused on developing Machine Learning and Generative AI methods to augment generative imagery and videos. You will develop new paradigms that will impact how users can engage and benefit from GenAI and that are at the core of advertising-facing tools that we are launching. You will also work on development of ML and GenAI models. You will conduct literature reviews to stay on the top of the field. You will regularly engage with product managers, who will partner with you to productize your work. About the team
US, NY, New York
We are seeking an Applied Scientist to develop and optimize Visual Inertial Odometry (VIO) and sensor fusion systems for our intelligent robots. In this role, you will design, implement, and deploy state estimation and tracking algorithms that enable robots to understand their position and motion in real time, even in challenging and dynamic environments. You will own the full pipeline from algorithm development through embedded deployment, ensuring that perception systems run efficiently on resource-constrained robotic hardware. You will also leverage modern machine learning approaches to push the boundaries of classical perception methods, combining learned representations with geometric techniques to achieve robust, real-time performance. This is a deeply hands-on role. You will work directly with sensors, hardware, and real-world data, while prototyping, testing, and iterating in physical environments. The ideal candidate has strong foundations in VIO and sensor fusion, practical experience optimizing algorithms for embedded platforms, and familiarity with how modern deep learning is transforming perception. Key job responsibilities - Design and implement Visual Inertial Odometry algorithms for robust real-time state estimation on robotic platforms like Sprout - Develop multi-sensor fusion pipelines integrating cameras, IMUs, and other sensing modalities for accurate pose tracking - Optimize perception and tracking algorithms for deployment on embedded hardware (e.g., ARM, GPU-accelerated edge devices) under strict latency and power constraints - Apply modern ML-based perception techniques (learned features, depth estimation, neural odometry) to complement and improve classical geometric approaches - Build and maintain calibration, evaluation, and benchmarking infrastructure for perception systems - Collaborate with hardware, controls, and navigation teams to integrate perception outputs into the robot’s autonomy stack - Lead technical projects from research prototyping through production deployment
US, NY, New York
The FAIM (Full Funnel Agentic Intelligence and Models) builds the agentic AI systems behind next-generation advertising. The team develops multimodal foundation models and LLM-driven agents that power shopper understanding, audience-aware creative generation, and agentic advertising workflows. As a Principal Applied Scientist for the FAIM team, you will invent the models that help advertisers achieve desired outcomes. Today's state of the art models can setup ad campaigns to a specification, but do not understand advertising strategy well enough to help with abstract goals that real advertisers ask about like "how can I make more money." This is a rare charter to build foundational optimization science where little exists today, spanning campaign recommendation, cross-product budget allocation, incrementality measurement, and long-term-sales modeling. You will set technical direction for a growing team, partner with engineering and product to take models from research to production at Amazon scale, and directly move advertiser ROAS and new-to-brand growth. In this role you will - Own the science vision for training models in our RL Gym from end-to-end - Invent models for joint budget allocation, incrementality, and long-term value - Take innovations from prototype to production serving live advertiser campaigns - Raise the technical bar across scientists and engineers Key job responsibilities - Define the long-term scientific vision for our RL Gym based training process focusing on campaign optimization and other advertising workflows - Translating ambiguous advertiser needs and competing objectives into a concrete science roadmap. - Invent, prototype, and productionize machine learning and optimization solutions for joint budget allocation across sponsored ad products, spanning the shopper journey from awareness to purchase. - Develop rigorous approaches to incrementality measurement and long-term-sales modeling that ground optimization in true advertiser value. - Design and lead large-scale experiments and analyses to validate hypotheses and guide product direction. - Partner closely with engineering and product to define technical contracts, data schemas, and serving systems that carry models into production. - Raise the technical bar across science and engineering through mentorship, design reviews, and hands-on collaboration. - Grow scientific talent and publish impactful research internally and at top-tier venues.
US, WA, Seattle
Amazon Advertising is one of Amazon's fastest growing and most profitable businesses. As a core product offering within our advertising portfolio, Sponsored Products (SP) helps merchants, retail vendors, and brand owners succeed via native advertising, which grows incremental sales of their products sold through Amazon. The SP team's primary goals are to help shoppers discover new products they love, be the most efficient way for advertisers to meet their business objectives, and build a sustainable business that continuously innovates on behalf of customers. Our products and solutions are strategically important to enable our Retail and Marketplace businesses to drive long-term growth. We deliver billions of ad impressions and millions of clicks and break fresh ground in product and technical innovations every day! Within Sponsored Products, the Bidding team is responsible for defining and delivering a collection of advertising products around bid controls (dynamic bidding, bid recommendations, etc.) that drive discovery and sales. Our solutions generate billions in revenue and drive long-term growth for Amazon’s Retail and Marketplace businesses. We deliver billions of ad impressions, millions of clicks daily, and break fresh ground to create world-class products. We are highly motivated, collaborative, and fun-loving team with an entrepreneurial spirit - with a broad mandate to experiment and innovate. You will invent new experiences and influence customer-facing shopping experiences to help suppliers grow their retail business and the auction dynamics that leverage native advertising; this is your opportunity to work within the fastest-growing businesses across all of Amazon! Define a long-term science vision for our advertising business, driven fundamentally from our customers' needs, translating that direction into specific plans for research and applied scientists, as well as engineering and product teams. This role combines science leadership, organizational ability, technical strength, product focus, and business understanding. Key job responsibilities • Lead a new initiative across Sponsored Products Bidding focused on AI/ML based features. • Be the technical leader in AI, Machine Learning; lead efforts within this team and across other teams. • Perform hands-on analysis and modeling of enormous data sets to develop insights that increase traffic monetization and merchandise sales, without compromising the shopper experience. • Drive end-to-end AI/Machine Learning projects that have a high degree of ambiguity, scale, complexity. • Build models, perform proof-of-concept, experiment, optimize, and deploy your models into production; work closely with software engineers to assist in productionizing your AI/ML models. • Run A/B experiments, gather data, and perform statistical analysis. • Establish scalable, efficient, automated processes for large-scale data analysis, machine-learning model development, model validation and serving. • Research new and innovative AI/ machine learning approaches. A day in the life Why you will love this opportunity: Amazon is investing heavily in building a world-class advertising business. This team defines and delivers a collection of advertising products that drive discovery and sales. Our solutions generate billions in revenue and drive long-term growth for Amazon's Retail and Marketplace businesses. We deliver billions of ad impressions, millions of clicks daily, and break fresh ground to create world-class products. We are a highly motivated, collaborative, and fun-loving team with an entrepreneurial spirit - with a broad mandate to experiment and innovate. Impact and Career Growth: You will invent new experiences and influence customer-facing shopping experiences to help suppliers grow their retail business and the auction dynamics that leverage native advertising; this is your opportunity to work within the fastest-growing businesses across all of Amazon! Define a long-term science vision for our advertising business, driven from our customers' needs, translating that direction into specific plans for research and applied scientists, as well as engineering and product teams. This role combines science leadership, organizational ability, technical strength, product focus, and business understanding. About the team The Sponsored Products and Brands team at Amazon Ads is re-imagining the advertising landscape through the latest generative AI technologies, revolutionizing how millions of customers discover products and engage with brands across Amazon.com and beyond. We are at the forefront of re-inventing advertising experiences, bridging human creativity with artificial intelligence to transform every aspect of the advertising lifecycle from ad creation and optimization to performance analysis and customer insights. We are a passionate group of innovators dedicated to developing responsible and intelligent AI technologies that balance the needs of advertisers, enhance the shopping experience, and strengthen the marketplace. If you're energized by solving complex challenges and pushing the boundaries of what's possible with AI, join us in shaping the future of advertising. The SPB Bidding team within Sponsored Products and Brands is focused on guiding and supporting Millions of advertisers to meet their advertising needs of creating and managing ad campaigns. At this scale, the complexity of diverse advertiser goals, campaign types, and market dynamics creates both a massive technical challenge and a transformative opportunity: even small improvements in bidding systems can have outsized impact on advertiser success and Amazon’s retail ecosystem. Our vision is to build a highly personalized, context-aware bidding system that leverages auction simulations, ML models, and optimization algorithms. This framework, will operate across SPB bidding system and proactively delivering value based on deep understanding of the advertiser. To execute this vision, we collaborate closely with stakeholders across Ad Console, Sales, and Marketing to identify opportunities—from high-level product guidance down to granular recommendations—and deliver them through a tailored, personalized experience. Our work is grounded in state-of-the-art bidding agent architectures, tool integration, reasoning frameworks, and model customization approaches (including tuning and preference optimization), ensuring our systems are both scalable and adaptive.