Kristopher Micinski

The Thing We All Obviously Want

Fri, 26 Jun 2026 00:00:00 -0400

Generated by AI—notice the perspective.

Over the past year, we have seen the rapid development of AI-assisted programming to an astounding degree. Even five years ago, fully-automated program synthesis of large-scale, production systems would have seemed unthinkable. Today, this is not an ambition, it is a reality, at least by some measure. To some computer scientists, natural-language-driven program synthesis was the endgame. On the other hand, the software I use day-to-day doesn’t seem to be getting appreciably better overall. Systems are still broken, apps unresponsive (even on well-resourced hardware), crashes are still common, and interfaces are generally as clunky as before. Personally, I believe we will eventually see many systems adapted by AI-assisted refactoring tools; but I also recognize there are human barriers to deploying those things at full thrust in the short (even medium) term.

In any case, my position is that AI-assisted programming, giving us real-time, on-demand generation of any app, is the thing that we all obviously want. There are a few tensions with this reality: (a) it seriously changes the value proposition of what “code” is in a meaningful way, (b) there are externalities: wasted computation, energy replicating junk, and (c) it challenges the role of humans in the knowledge-generation process.

Note: In the rest of this essay I will use the term “LLM(s).” In general, when I say this, I mean a state-of-the-art integration of a frontier model alongside relatively simple tools (e.g., Claude Code, Codex, etc.). There is some nuance in building these tools, but given that the innovation is the model, I will casually refer to the whole agentic process as the “LLM.”

Program Synthesis: Did it Fail?

Traditional program synthesis (by which I generally mean, SMT-based, search-based, or similar) leveraged a rigorous and formal enumeration / proof to produce a synthesized program–potentially with a certificate of its correctness–driven via a rigorous specification. Like many academic fields, the goal of program synthesis was not only to effect fully-automated programming tools: it was to advance the frontier of understanding in semantics, verification, specification, etc. These were the challenging problems, especially given that traditional search (on the CPU) was so slow.

LLMs allow rigorous concepts to gracefully degrade by using text. The underlying model has such a deep understanding of language that fuzzy, hazily-posed descriptions often still give some sensible interpretation. The obvious issue is hallucination: when you push the embedding space into some inconsistency, won’t it just generate junk? And of course, this is absolutely an issue–but when the error rate is low enough that it’s practically useful, many people will not care.

My position is that LLM-guided software engineering was so wildly successful not just because it nailed the generation part, but also because LLMs ended up practically solving the problem of specification. Humans are simply used to the failure modes of underspecification: even from a young age we’re trained to expect disappointment if we miscommunicate our expectations, and so having the LLM fail doesn’t sting as badly as you might expect.

Granularly-Evolving Formal Specs

One potential issue I foresee with current-generation AI is that they focus the process on a textual-only workflow. In practice, smart humans do want to read something that looks like code most of the time–the issue is that they want to be able to focus their limited mental attention rather than sifting through thousands of lines of code. Most anybody who ever worked on a large codebase (that they did not write entirely themselves) never had more than an LLM-level understanding of parts of the codebase anyway. Instead, we embarked upon code understanding efforts whenever we faced tricky bugs, needed to add new features, etc. We codified this in our own mental model (memory, notes, etc.), but also (sometimes) documentation, bug reports, etc. Hilariously, this is now the kind of thing that the LLM loves to ingest.

As we build software, we want to be able to start with a hazy specification (probably in English, but maybe in a big document) and be able to begin building an application. At key decision-making points we want to be able to solicit input and, finally, be dropped into an exploratory state where we may make our thoughts more granularly precise. For many reasons, I still believe that this should be formal, executable code, not English prose.

The issue is that no single language is perfect. English is great for laypeople: any arbitrarily complex topic can be compressed into an arbitrarily-simple soundbite. Unfortunately, English is imprecise and even lies to you via the embedding. On the other end of the spectrum, we might have Lean in a loop with the LLM. The LLM is speaking Lean and there is some amount of grammar- and semantics-constrained generation occurring. The issue here is one of information overload: Lean is fully-rigorous, but complex propositions and definitions take serious mental bandwidth to unpack as a nonexpert. Sure, you can ask the LLM to explain, but it is not easy if you do not have a background with advanced topics. We need something in between.

There is obviously a market for programming environments which enable granularly-precise constraint-guided software development. Many people will attempt this first with English integration layers on top of preexisting languages (e.g., Lean, C++, etc.). I wonder, however, if there is room for families of languages which are focused on abstracting around granularly-more-correct specifications of behavior, with the understanding that (e.g.,) the LLM (or maybe a human) is filling in the gaps.

Searchable, Visualized, Reactive Knowledge Bases

Many people are now using LLMs to build knowledge bases. The baseline is simple: loose collections of textual notes are spat out by the LLM and digested by a human, who iteratively interacts with a tool (Claude Code) to build the knowledge base. Text is very useful, but it is not always the best way to visualize information–I expect we will see a growing corpus of applications which leverage LLMs to integrate into some higher-level, application-specific knowledge base. As an example, consider CAD components: we will often want some semi-programmatic control over the development of a scene (e.g., laying out an office), but we probably want to reuse previously-designed components (e.g., cabinets with a countertop) with slight tweaks–we want to ensure that crucial constraints are met (e.g., always ensuring a uniform toe kick) while enabling the AI to adapt a previously-built model in our knowledge base. At some point it might be helpful to break apart an item, say “base cabinet” versus “floor cabinet.” I expect there will be a demand for gradually evolving things from unstructured, generative objects into symbolic, computed representations.

Self-Evolving Software

I believe self-evolving software is already possible, perhaps it is already here. Self-evolving software is software which optimizes and evolves itself over time to suit a user’s needs. Right now, we prompt the LLM with directions to build the app we desire. However, we can easily imagine a future where software is optimized and evolved on demand. For example, we might imagine being able to give gentle nudges: “I don’t quite like the theme of the app,” which is then taken as an instruction to evolve the application in the background–whenever a change is tested, we can respond with a thumbs up / down (or some more nuanced feedback signal). Since humans are already doing this in production, it seems reasonable to assume that very quickly this process will be abstracted, and humans will begin to codify the ways in which software is allowed to evolve.

This is partially an issue of modularity–we want to be able to modularly decompose our concerns so that we can make rigorous precisely what concerns us while leaving the rest up to the LLM. Still, we seem to lack a good way to rigorously express the constraints guiding a software system while leaving other parts opaque. The issue seems to be that implementation details often form leaky abstractions in practice, and right now English is the common denominator: once you go beyond English formally, its ambiguity starts leaking in.

Generation, Replication, and Libraries

For a time, humans pressed tab and enter. They would look at the app, observe the feature, give input–it was beautiful. Through rose-colored glasses, one might say that the skilled engineer could spend 2-5 hours reading the code’s structure and architecting it correctly, while the agent did the boring, tedious work. Through a cynic’s perspective, there’s absolutely no sense in generating code that you aren’t at least reading. We reconcile the two perspectives by recognizing that the boundary constitutes the “library.”

Thought experiment: imagine a world where the right library always exists. In this world, you would only ever have to write 10kloc apps; everything beyond 10kloc is pushed into a library. Some systems are not really like this, and they do just take a ton of code. But practically, nearly everything you could learn about in a university class could be written in 10kloc or less, relatively efficiently, with the right libraries.

We are close (if not already there) to this vision with modern LLMs, but we are not there in an important way: every time you want the library, you regenerate it. This is obviously extremely wasteful: how many to-do list apps do we really need to waste our energy generating? In our thought experiment world where all libraries exist, imagine you had an oracle that told you the right libraries you needed: could you use a much cheaper model?

In a way, this is just semantic caching. There will always be queries made to the LLM where it has seen something “pretty close” before. In an ideal world, this would be cached and we could do only the incremental work necessary to generate the new response, optimally balancing replication, generation, bandwidth, etc. costs.

Security, Trust, OSS, Etc.

What is the point of well-maintained, agreed-upon code? In a world where generating code is cheap, there are both upsides and downsides. The clearest upside is this: there are many components (e.g., container libraries, layout engines, etc.) for which having stable, predictable, repeatable behavior will generally be desirable. While many of us might want our own custom business CRM (hopefully out of documented components), we still want to use Chrome, too. On the other hand, in an age where LLMs have surpassed humans in exploit identification, running the same software as everyone else leaves us open to potential vulnerabilities.

Another concern is the waste of pervasive use of neural models when symbolic models suffice. Due to the current subsidies of frontier AI, we will inevitably see waste when LLMs are used as general-purpose reasoners when more specific reasoners would have sufficed. To some degree, this is just an engineering problem: “LLMs” are complex agentic workflows which build tools on-demand when necessary, and it seems natural to predict that they will do so when valuable. Maybe the question is: what is the use of reusable, OSS components to an LLM?

Human-Human and Human-LLM Teaming

We used to team up to work on software by having a shared artifact: the codebase. When we got hired, we would read the codebase and step through it in a debugger, we would write a test that forced us to read an execution path. But now that LLMs can do it just as well as we can–at least to the functional constraints (the AI uses tests for its reward, so this is predicted)–it is leaving lots of the industry questioning: why read the code?

Unfortunately though, while individual contributors have moved on to mostly using English text to actually write the code, development teams still generally interact by simply pushing the code. Other engineers then use their LLM to help them read and maintain the code. The issue here seems obvious: the substrate is still code tracked in Git; if I’m working with someone who’s also using an LLM, I don’t really just want the code. On the other hand, I personally do like code; I don’t really just want to read a big sequential dump of English either. What I want is to be able to rapidly interact with someone in a way that lets us focus on just as much of the formal specification as we want, leaving the rest underspecified.

AI Acknowledgement

I did not use AI to write this post. However, I did use Claude for two things: (a) I performed spellchecking, and (b) checks for copyediting / consistency / etc.

My Rejected NSF CAREER Proposal

Thu, 09 Apr 2026 00:00:00 -0400

My NSF CAREER proposal was rejected, and I’m making it publicly available: Micinski CAREER Proposal (PDF).

The proposals scores were VG/G, VG/G, and E/VG. While the proposal was rejected as competitive, my position is that the proposal is nicely written and addresses a serious and important challenge in a credible way. I also notice that the only PIs who have had NSF CAREER proposals funded this year have been in AI / quantum–this is seriously frustrating to see, given that NSF is a crucial research vehicle for the US, and I am quite concerned that NSF is becoming a shell of its former self, having been seriously damaged by top-down control. I still strongly support NSF and will gladly review for it in the future. Given that it is my last chance for the NSF CAREER before tenure, I want to upload it for posterity. To all junior faculty applying to NSF CAREER: best wishes, please feel free to reach out and I am absolutely happy to send the full (unedited) reviews, I elide them here to respect the reviewing process.

While I’m frustrated at the rejection, my feeling is that (a) it was somewhat my fault for not applying earlier (it’s well known that second-attempt applicants often fail) and (b) the proposal was “not AI/quantum enough.” I leave the entire thing here for you to read for yourself, so you can be the judge.

NSF is truly one of the most valuable institutions in the history of the USA and the world generally, long live NSF!

– Kristoper Micinski

Why Study CS? Thoughts on LLM-assisted software engineering

Tue, 06 Jan 2026 00:00:00 -0500

Dear students of Computer Science,

Almost everything everyone is telling you about LLMs is probably wrong. Some tell you it’s glorified autocomplete, others say it will cure cancer or prove the loftiest theorems. As with many mass-market takes, these claims contain a grain of truth–but they miss the point. I encourage you to try a modern AI coding assistant for yourself and form your own opinion; here, I’ll offer mine, as someone who wants to give you an optimistic but unvarnished perspective.

Nine months ago (March ‘25), Anthropic’s CEO claimed that AI assistants would be writing 90% of our code within three to six months. We have recently seen that he was only a little too ambitious: the winter break has seen an explosion of programmers uncovering the power of Claude Code, surely a cause for talk around the Kombucha dispenser at OpenAI’s offices.

Claude Code reveals a sobering reality for many software engineering professionals: the vast bulk of the thing we call productivity has now revealed itself to be nothing more than Claude in a loop with command-line tools. The 2023-era copy-and-pasting into the textbox, manually crafting a prompt, and carefully stitching the result into our codebase feels downright manual by comparison. Now, we can boot up Claude Code and iteratively build up ideas, tests, implementations, and (possibly) a complete, working app. The TUI-based interfaces solve one of the main problems inherent to the web-based chatbots: how to meaningfully iterate, and incrementally build larger software artifacts alongside the LLM. I think many of us were telling ourselves that the real ingenuity was in the judiciousness we exercised when carefully copying the LLM’s output and integrating it back into our work: Claude Code shows that doing that for arbitrarily complex domains is often just dirt simple, as long as the tools can communicate via text.

This halcyon turn of events has left many students wondering: should I still study computer science? As someone who sells computer science for a living, I am obligated to sell the position of “yes,” but we all know that there are plenty of good reasons to argue “no” as well. For the sake of this essay, I will assume you lean towards yes, but perhaps with some skepticism. Even if you agree, it’s an important question, because it reveals an uncomfortable systemic issue: many students operate at a level that Claude Code can easily beat, and in the new world order, it will be increasingly hard to get a job. Just so we’re crystal clear, there’s still some code Claude can’t write, even though it can certainly write a lot of very useful code. But “there will always be code that Claude can’t write” is not a very convincing argument when the amount of code Claude does write will be massively higher by comparison.

CS undergraduates face a real issue: beating Claude Code isn’t very easy, even (and especially) on challenging leetcode problems. This may push some students to conclude that the AI is good enough at reasoning such as to absolve them from developing a deep understanding of the technical details. I argue that this is the wrong perspective to take, and that instead you should switch your efforts away from mindless and repetitive generation, and towards problems which stretch your intellectual capacity by making you feel stuck. To the degree the LLM can help get you unstuck in a way which genuinely builds your mental model (so that you’re in the driver’s seat, not a maybe-true understanding you got from the AI): great, do it.

In some ways, I see LLM-assisted software engineering as the Object Orientation of the ’20s: it offers a vast swath of opportunity to give lesser-trained programmers the ability to write huge amounts of good-enough code, thus enabling us to hire a vast army of folks who never had to deal with the pains of memory management, pointers, etc. Unfortunately, however, this time it’s different; we all know the hiring boom of easy-money days is over. Seeing Claude Code, many of us take a more pessimistic stance: those that can’t add value on top of Claude will be out of a job. Who are these folks that will rise to the top? I argue that it is the people who have expended serious intellectual energy to exercise their capacity for understanding deep ideas.

I personally believe the purpose of a university class is to stress your intellectual capacity through repeated exposure to increasingly-complex technical ideas which build coherent vision. In doing this, we teach students skills that can be broadly binned into generation (synthesis) and verification (checking). In many contexts (e.g., debugging), we build our mental models via iterating between synthesizing candidate ideas, then carefully checking our work, and reflecting upon what to do next.

As a student, I saw learning as a way of “compiling” things into my mental representation: I would spend days at the library (I do not do this anymore, my productivity has unfortunately declined) carefully rewriting books in my own words, which taught me the difference between what it meant to deeply grok something, and to possess an LLM-level understanding of it. One thing I remember from those times, however, is how mentally exhausted I felt at the end of the day. The reason I was exhausted is because I was playing the verifier. I learned a hard truth: after reading a sentence of a book, things often make perfect sense, but when you try to explain it to someone else, you realize you have no clue what is going on. When we do it to ourselves we just feel stupid, but if it happens in an environment that encourages rapid failure, we can grow: effective instructors incentivize rapid failure (e.g., by allowing students to rapidly test their solutions against an autograder).

However, in the case of an LLM, failures often manifest in ways which can be confusing and subtle: students may find themselves frustrated to realize that the LLM led them into a hallucination. Feeling confused about what to believe can be a very jarring experience–leading many of us to feelings of repulsion when we read some of our old code–but the LLM can further amplify this anxiety; because it is so shockingly capable of solving a problem, we feel utterly paranoid about hearing the music stop, “what will I do if I actually have to figure this part of the code out for myself?”

One crucial skill software engineering students learn is how to employ abstraction when they understand a software system: being able to read just enough of the whole system to understand its architecture, and then employing iterative deepening to learn about subsystems (e.g., the memory manager, networking, etc.) in an on-demand fashion. Right now, many students are telling themselves a comfortable lie: that iteratively prompting the LLM is a substitute for the true iterative understanding we do to build our mental model. Confirmation bias is a big concern: the tools are often so good, they give us the impression that there is essentially no point in checking their correctness most of the time. This sounds great, until we get stuck, and then realize we actually didn’t understand things nearly as well as necessary to tackle the task at hand.

This is not a new problem, in fact it is the problem that has plagued CS undergraduates since the beginning of time. Being stuck is obviously deeply uncomfortable, because the real world has deadlines, value-measuring systems, all of which depend on our work being complete and correct in a timely fashion. No student wants to be stuck, so they work to exit the stuck state as quickly as possible, often latching on to an incorrect hypothesis (“oh wait, this line must be broken…”) to get back to harmony; this is bad. As you become a senior software developer, you learn that being stuck is in fact the default state when you are working on challenging problems. The mental energy comprising your engagement with courses should not be primarily bottlenecked by generation, instead, it should meaningfully harmonize generation and verification in a way that allows you to build an ever-expanding mental model.

I recall years ago I chatted with a teaching professor about a worrying trend in CS. This was around 2016, and they mentioned that students were increasingly ignoring the “nuts and bolts” of applications to focus only on the flashy bits that were easy to get not-wrong. The specific concern was hackathons: do they encourage students to do something hard and deep that exercises real engineering skill? Or do they incentivize students to focus on startup-pitch-style motivation with impressive-looking web interfaces backed by fake data? The trend of students avoiding the genuinely-hard parts of the problem to focus on the flashy parts has been occurring since long before the days of AI coding assistants.

Another professor recently told me that they switched their assessment style to “code comprehension” rather than code synthesis. I think this is an excellent way to see the problem, because it stresses that–when generation is cheap–explanation, reconstructing a reason for generated content, becomes the dominating factor in terms of assessing understanding. You might ask: in a world where generators are perfect, what is the role of the verifiers? This is an apt question.

The first answer is that English is genuinely fuzzy, and in almost every case, the ways in which humans describe specifications lead to contradictory, unrealizable, or intractable results. In a world run by humans, software engineers expend serious effort interfacing with other humans to refine fuzzy conceptual architectures into correct production software. While it seems reasonable to expect that Claude Code could refine an English specification, there is always an intrinsic value to working with another human. The issue with automation is that this may no longer be worth the price: there is still a market for freelance web developers, for example, but it’s been severely diminished by sites like SquareSpace and WordPress.

LLMs take English-based automated code synthesis to the next level, giving us a general-purpose tool to refine hazy specs into mostly-working software. I predict different impacts across sectors. Startups will be the most impacted: it has never been easier to iterate to “something that looks like a passable minimum-viable product.” If the goal is to demonstrate a cool market gap that is tantamount to a JavaScript app, you can now do that very quickly with AI coding assistants. Enterprise apps, on the other hand, will persist: organizational bloat, vendor lock-in, and sales play a huge role in the momentum of those settings.

The truly useful applications of AI will involve deeply leveraging AI in a way that allows it to be built into the kinds of symbolic workflows humans love in things like Keynote and Premiere Pro. For example, Adobe Photoshop’s “generative fill” produces cool raster images right now, but it doesn’t do so in a manner that gives you a layered, element-by-element breakdown: this is the thing all of us obviously actually want, because it lets us refine the results of the AI, and interact with it, just like Claude Code did for our repos.

Thus, to aspiring computer scientists, I urge you to seriously consider how AI will impact your career. But I also urge you to seriously resist the temptation to resort to mindlessly iterating with the AI, toiling in unchecked guesses because you never bothered to go check the notes or read the book. Of course such students existed before LLMs as well: upon facing broken code, we all have the temptation to prove it is just a minor typo, we edit the code superficially: sometimes it works, but sometimes it leads us to a significantly more subtle error.

There is a vast amount of potential in leveraging AI to expand our intellectual capacity, by pushing us to rapidly confront our misunderstandings and push past them. However, we often overestimate the impact of change on the fundamental nature of things. At least in your case, a CS student, your goal is to rise above regurgitation, mimicry, or rearranging: we all know the parable of the junior engineer who spent all their time trying to rearrange bits and pieces of StackOverflow answers. The goal is thus, as it has always been, to take from our time what we may use to most-optimally execute our vision. In your time as a student, your objective is to learn the key principles of computer science (or whatever field you choose) in a way which maximally stresses your intellectual capacity while offering you the ability to grow from repeated failure. Throughout your career, I urge you to develop a taste for a specific set of topics in your courses (or computing as a whole) for which you possess enough genuine intellectual curiosity necessary to grok the true essence of things.

– Kristopher Micinski, Syracuse

P.s., a few caveats to this post: (a) like everyone else, I am being reductive, there are plenty of domains where Claude Code completely fails right now–paren matching in Racket is only one example. (b) I’m specifically discussing Claude Code due to the recent explosion, but the ideas were pioneered in other tools, and (c) there will surely be plenty of reasonable folks who believe that AI is of no (or very limited) use to them, it is impossible for me to prove that all of these people are wrong since utility depends on workload; I personally found AI to be surprisingly helpful in my hobbyist woodworking, however.

Build a Compiler in Five Projects

Sun, 23 Nov 2025 00:00:00 -0500

Class website here: https://kmicinski.com/cis531-f25

Are you interested in building a compiler? Learning how functional languages are implemented? Gaining a bit of practical experience with x86-64 assembly language? If so, I invite you to try your hand at the projects in my class, CIS531. CIS531 is a masters-level class on compiler design which assumes that (a) you know how to program, (b) you’ve had some exposure to C (know about stack allocation, malloc, etc.), and (c) have seen some assembly code. My class projects are in the Racket programming language, but if you don’t know Racket, it is quite easy to learn: I have a set of YouTube video lectures that teach Racket quickly! If you’ve never heard of Racket before, or you’re skeptical of functional programming, indulge me for a bit: there’s no hardcore FP theory or math in this course, and Racket is genuinely the best language to use for this specific setup.

My class follows Prof. Jeremy Siek’s excellent book, “Essentials of Compilation.” While I highly recommend buying the book and supporting Prof. Siek, I will also note that there are free online preliminary editions floating around; in my class, I followed the free version and suggested that students buy the book if doing so fit their goals. However, along with the book, I also have a set of class slides along with sporadic course videos, both available on the class website.

This class builds up to a compiler with the following features:

Variables and assignment via let
Integer arithmetic via + and -
Reading inputs / printing output
Booleans, conjunctions/disjunctions (and/or)
Branching via if, integer comparisons (<, etc.)
Heap-allocated vectors
Assignment / mutation (set!)
While loops
Fixed-arity functions and function application
Lambdas (closures at runtime)

The unique combination of features lets us tour an interesting cross-section of programming languages, exploring both imperative programming with loops and mutation but also functional programming with lists and recursion.

The Projects

To be specific, I challenge you to complete five projects, each including a comprehensive test suite that will seriously stress the correctness of your implementation. p1 is a warmup project (you should skip if you already know Racket), but p2-5 build a compiler for a set of increasingly-complex languages to x86-64. The languages nest inside of each other, with p2 giving us straight-line arithmetic, p3 giving us decision trees, p4 giving us loops and mutation, and p5 giving us functions, recursion, and lambdas.

p1 – Stack interpreter. This is a warmup project, if you know Racket and have some PL background, feel free to skip.
p2 – Straight-line arithmetic / variables → x86-64 assembly language
p3 – Booleans and branching (if, and, or) → x86-64 assembly language
p4 – Vectors, heap allocation, set!, and loops → x86-64 assembly language
p5 – Functions, lambdas, and closure conversion → x86-64 assembly language

The projects are designed with one key principle in mind: get us to the most expressive/fun language possible, as fast as possible. In doing this, we sacrifice a lot that might be typically covered:

Our languages aren’t type/memory safe, we assume the programmer is correct
No register allocation (possible to add, not too hard)
No garbage collection of any kind: we just use malloc. We could trivially support the Boehm GC (I have done that in the past), but it was another static library to link in and I really wanted to make this self contained.
We support a very limited set of builtins (but it is trivial to add more)

So even after project 5, getting to a “real” compiler would take a bit of effort. The most important (in my opinion) are (a) memory safety (the language needs to be safe, period) via dynamic type tagging and (b) slightly more builtins, and (c) register allocation. That would get us to a respectable compiler. After that, we could add more language features, or optimize the ones we have, e.g., by using abstract interpretation.

An Example Program

Our language will include functions, loops, branching, assignment, and even heap-allocated vectors. As an example of the power, here’s a Sudoku solver written in the language

(program
 ;; =========================
 ;; List primitives
 ;; Empty list is (void)
 ;; =========================
 (define (is_nil x) (eq? x (void)))

 ;; cons cell as 2-element vector: [0] = head, [1] = tail
 (define (cons h t)
   (let ([c (make-vector 2)])
     (let ([_ (vector-set! c 0 h)])
       (let ([_ (vector-set! c 1 t)])
         c))))

 (define (head c) (vector-ref c 0))
 (define (tail c) (vector-ref c 1))

 ;; =========================
 ;; Cell representation
 ;; cell = (row col val) as nested cons
 ;; =========================
 (define (make_cell r c v)
   (cons r (cons c (cons v (void)))))

 (define (cell_row cell)
   (head cell))

 (define (cell_col cell)
   (head (tail cell)))

 (define (cell_val cell)
   (head (tail (tail cell))))

 ;; =========================
 ;; Block indexing (0,1,2) for rows/cols
 ;; =========================
 (define (block_index3 x)
   (if (< x 3)
       0
       (if (< x 6)
           1
           2)))

 (define (same_block? r1 c1 r2 c2)
   (if (eq? (block_index3 r1) (block_index3 r2))
       (eq? (block_index3 c1) (block_index3 c2))
       #f))

 ;; =========================
 ;; Lookup current value at (row, col) in board
 ;; board is a list of cells
 ;; Return 0 if not assigned
 ;; =========================
 (define (lookup board row col)
   (if (is_nil board)
       0
       (let ([cell (head board)])
         (let ([r (cell_row cell)])
           (let ([c (cell_col cell)])
             (if (and (eq? r row) (eq? c col))
                 (cell_val cell)
                 (lookup (tail board) row col)))))))

 ;; =========================
 ;; Conflict check:
 ;; #t if some cell in board has:
 ;;   - same value, and
 ;;   - same row OR same col OR same 3x3 block
 ;; =========================
 (define (conflicts? board row col val)
   (if (is_nil board)
       #f
       (let ([cell (head board)])
         (let ([r (cell_row cell)])
           (let ([c (cell_col cell)])
             (let ([v (cell_val cell)])
               (if (and (eq? v val)
                        (or (eq? r row)
                            (or (eq? c col)
                                (same_block? r c row col))))
                   #t
                   (conflicts? (tail board) row col val))))))))

 ;; =========================
 ;; Recursive backtracking solver over (row, col)
 ;; board: list of assignments
 ;; rows, cols = 0..8
 ;; =========================
 (define (solve_cell row col board)
   (if (eq? row 9)
       ;; All rows done: solved
       board
       (if (eq? col 9)
           ;; End of row: go to next row
           (solve_cell (+ row 1) 0 board)
           ;; Otherwise, try this cell
           (let ([existing (lookup board row col)])
             (if (eq? existing 0)
                 ;; Empty cell: try values 1..9
                 (let ([candidate 1])
                   (let ([solution (void)])
                     (begin
                       (while (and (< candidate 10)
                                   (eq? solution (void)))
                              (begin
				(if (conflicts? board row col candidate)
                                    ;; conflict, skip
                                    (set! solution solution)
                                    ;; no conflict, extend board and recurse
                                    (let ([s (solve_cell row
                                                         (+ col 1)
                                                         (cons (make_cell row col candidate)
                                                               board))])
                                      (if (eq? s (void))
                                          (set! solution solution)
                                          (set! solution s))))
				(set! candidate (+ candidate 1))))
                       solution)))
                 ;; Pre-filled cell: just move on
                 (solve_cell row (+ col 1) board))))))

 ;; =========================
 ;; Read initial board from input:
 ;; 81 integers, row-major, 0 = empty, 1..9 = given
 ;; Returns list of cells
 ;; =========================
 (define (read_board)
   (let ([board (void)])
     (let ([i 0])
       (begin
         (while (< i 9)
		(begin
                  (let ([j 0])
                    (while (< j 9)
			   (begin
			     (let ([v (read)])
                               (if (eq? v 0)
				   (set! board board)
				   (set! board (cons (make_cell i j v) board))))
			     (set! j (+ j 1)))))
                  (set! i (+ i 1))))
         board))))

 ;; =========================
 ;; Entry: read board, solve from (0,0), return solution
 ;; Solution is a list of (row col val) cells
 ;; =========================
 (let* ([board (read_board)]
        [solution (solve_cell 0 0 board)])
   (lookup solution 8 8)))

The Full Language

The final language you’ll implement will be this one. In comments, I’ve also highlighted the sublanguages: for example, project 2 includes only numbers, input (read), binary plus, unary minus, variable references and let binding. It grows to all of R5.

(define (R5-exp? e)
  (match e
    ;; Project 2
    [(? fixnum?) #t]
    ['(read) #t]
    [`(+ ,(? R5-exp? e0) ,(? R5-exp? e1)) #t]
    [`(- ,(? R5-exp? e)) #t]
    [(? symbol?) #t]
    [`(let ([,(? symbol? x) ,(? R5-exp? e)]) ,(? R5-exp? eb)) #t]
	;; Project 3
    [#t #t]
    [#f #t]
    ['(void) #t]
    [`(- ,(? R5-exp? e0) ,(? R5-exp? e1)) #t]
    [`(and ,(? R5-exp? e0) ,(? R5-exp? e1)) #t]
    [`(or  ,(? R5-exp? e0) ,(? R5-exp? e1)) #t]
    [`(not ,(? R5-exp? e1)) #t]
    [`(,(? cmp? c) ,(? R5-exp? e0) ,(? R5-exp? e1)) #t]
    [`(if ,(? R5-exp? e-g) ,(? R5-exp? e-t) ,(? R5-exp? e-f)) #t]
    ;; Project 4
    [`(let* ([,(? symbol? xs) ,(? R5-exp? es)] ...) ,(? R5-exp? eb)) #t]
    [`(begin ,(? R5-exp?) ... ,(? R5-exp? ret)) #t]
    [`(while ,(? R5-exp? e-g) ,(? R5-exp? es) ...) #t]
    [`(make-vector ,(? R5-exp? len)) #t]
    [`(vector-ref ,(? R5-exp? v) ,(? fixnum? i)) #t]
    [`(vector-set! ,(? R5-exp? v) ,(? fixnum? i) ,(? R5-exp? e-v)) #t]
    [`(set! ,(? symbol? x) ,(? R5-exp? e)) #t]
    ;; Project 5
    [`(,(? R5-exp? e-f) ,(? R5-exp? a-args) ...) #t]
    [`(lambda (,(? symbol? xs) ...) ,(? R5-exp? e-body)) #t]
	[_ #f]))

(define (R5-defn? defn)
  (match defn
    ;; Project 5 adds multiple function definitions
    [`(define (,(? symbol? f) ,(? symbol? formals) ...)  ,(? R5-exp? e-b)) #t]
    [_ #f]))

(define (R5? p)
  (match p
    [`(program ,(? R5-defn? defns) ... ,(? R5-exp?)) #t]
    [_ #f]))

The Compiler’s Structure

To get you booted up fast as possible, every single project is designed the same way:

compile.rkt – Your pass implementations. You will edit functions provided here. -> This is the only file you will edit! The rest are read-only
irs.rkt – IR definitions and predicates like anf-program?, c1-program?, etc. (see also typed/shrunk variants)
interpreters.rkt – Reference interpreters for several IRs (used by tests and for your own debugging).
system.rkt – System/ABI configuration, pass names, runtime filenames, output paths, etc.
main.rkt – Driver that runs all passes, can build a binary, and can launch a debug server.
test.rkt – Test harness. Runs isolation tests or end-to-end native tests depending on -m mode.
runtime.c – Minimal runtime (read_int64, print_int64, etc.).
test-programs/ – Example programs (.scm).
input-files/ – Input streams for programs (lines of integers).
goldens/ – Instructor goldens (IR snapshots, interpreter outputs, and stdout baselines).

You write your code in compile.rkt, which consists of a set of passes. Each pass transforms an input language into an output language, and these intermediate languages (IRs) are codified via predicates in irs.rkt. To define the meaning of each IR, we give an interpreter for each in interpreters.rkt. For the compiler to be correct, it needs to be the case that–for all input streams–the compiler produces the same output stream across all intermediate IRs. There is some system-specific stuff in system.rkt, which takes care of things like Linux vs. Mac ABI issues, specifying register names, etc. The main.rkt file acts as a main compiler entrypoint, and it carefully runs each pass of the compiler, checking predicates before/after each pass and interpreting each IR, checking to ensure consistency. This is a huge win for debugging, in my opinion: you always want to localize errors to the proximate pass which causes misinterpretation, and main.rkt seriously aids debugging in my experience. There is also more comprehensive test infrastructure in test.rkt; this test script is invoked by the Python-based test scripts in test/. These tests check the behavior of the compiler on the programs in the test-programs/ directory, using the files from input-files as inputs and comparing to the outputs in goldens/.

Why Is This Course Unique and Cool?

You build a real compiler, all the way to actual x86-64 assembly.
Each IR has a corresponding interpreter, which is easy to find/read and written in a familiar style, giving semantic clarity and testable correctness.
The project is language scalable, meaning that you can use it as a base for building your own language. Of course, this is thanks to Dr. Siek’s great “incremental” design.
It is fully testable across multiple passes, which helps anticipate the thing we all fear most about writing a compiler: seeing a problem that is the ramification of far-away code from higher up in the compilation pipeline.
It is written in a simple, pure recursive style. Just plain old pattern matching and recursion here, no need for any complex abstractions.

How Do I Get Started?

Familiarize yourself with the course webpage: https://kmicinski.com/cis531-f25
If you don’t know Racket, start with project 1: https://kmicinski.com/cis531-f25/projects/1
Otherwise, start with project 2: https://kmicinski.com/cis531-f25/projects/2
When you finish each project, move on to the next!
When you’re done, start building your own language. Consider adding type (checking/inference), classes, more builtins, pattern matching, continuations, exceptions, algebraic effects. The options are myriad, but once you’ve finished projects 2-5, you’ve built a whole compiler for a surprisingly expressive language.

Thank you to the National Science Foundation and Others

If you like this work and live in the United States, please feel commensurately less bad about paying your taxes. I made the whole class free, at least as free as I could given practical constraints. This class work on compilation is partially supported by our NSF PPoSS large, which has already produced many cool major results. In subsequent explorations, I am hoping that I can use this class compiler as a baseline for highly-scalable engines that reason about programs. Given the simple, self-contained nature–and the presence of per-pass interpreters and consistency testing–I see this as an awesome potential baseline for cool extensions.

My course is of course heavily inspired by Prof. Siek’s book and course, along with inspiration from Thomas Gilray at Washington State. Eight years ago, Tom and I took a spontaneous trip to see the eclipse halfway across the country (skipping out on the ICSE ‘17 deadline basically); we discussed compiler design over a steamed seafood buffet in Myrtle Beach after napping in a cheap motel, having been awake for over 24 hours and feeling the eclipse had made it worth it. We sketched out his whole compiler on that roadtrip, and ever since that night eating steamed crabs, I wanted to build my own course compiler. Now that I have, I am not sure it compares to waking up for just four hours of twilight, only to consume copious amounts of butter and shellfish as the brisk ocean air wisps over your face, the closures and continuations softly washing rhythmically through the conversation as you walk along the beach back to your $50 motel room.

In closing, thanks for checking this out, this compiler was a ton of fun to build. Even as someone who has some amount of expertise in compiler design, building it and getting it 100% right (I hope!) was such a rewarding experience. My real sincere hope is that it offers students (and you!) a fun journey. If you end up doing anything this, please get in touch: kkmicins@syr.edu. I’d love to see what you come up with. Best wishes,

Kristopher Micinski – Syracuse, November, 2025

Why Tail-Recursive Functions are Loops

Fri, 01 Aug 2025 00:00:00 -0400

One story every computing enthusiast should hear is the lesson of how loops and tail-recursion are equivalent. We like recursive functions because they’re amenable to induction, and we can derive them in a way that is in direct correspondence with the definition of the datatype over which they recur. We like loops because they’re fast and make intuitive sense as long as variables don’t change in too tricky a way.

In general, recursive functions are slower than loops because they push stack frames: the performance of most programs today is dominated by memory reads/writes. The data we touch the most lives in the cache–we do not want to evict a ton of stuff from the cache, under any circumstance. In a direct-style implementation of a recursive function, the recursive call has to push a stack frame to remember what to do once the function returns:

;; Racket
(define (sum l)
  (if (empty? l)
      0
      (+ (first l) (sum (rest l)))))

// C 
int sum(int *l, int length) {
    if (length == 0)
        return 0;
    else
        return l[0] + sum(l + 1, length - 1);
}

When we get to (+ (first l) (sum (rest l))), we first call (first l) (which returns the first element). While we’re making that call, we have to remember to come back and do (sum (rest l))–to be fully precise, we remember that we need to do (rest l), then take its result x and call (sum x), remembering to come back and finally take that result and add it to the result of (first l). The reason we have to do this is because we need to remember those partial results (in this case the result of (first l)): we have to store them somewhere after all, and each time we make the recursive call, we need to remember the result of (first l) from this call–we need O(n) stack space for a list of size n.

Of course, if we use iteration this all goes away:

;; Racket
(define (sum l)
  (define x 0)
  (for ([elt l])
    (set! x (+ x elt)))
  x)

// C
int sum(const int *l, int length) {
    int x = 0;
    for (int i = 0; i < length; i++) {
        x += l[i];
    }
    return x;
}

We all have an intuitive sense of what the loop is doing: once we hit the end of the loop, we do not make a recursive call (we never issue a call instruction in assembly), we simply jump up to the beginning of the loop. The key is that x is being used as an accumulator, growing a partial result in a bottom-up fashion as the computation proceeds, eventually yielding the final value at the end. Instead of keeping partial results on the stack, the loop takes a constant amount of space but linear time.

In a tail-recursive implementation, the rule is that every recursive call must be a tail call. Intuitively, a tail call is a call which is “immediately returned.” More formally, a subexpression of an expression is in tail position if the return value from that expression is the return value from the whole expression. For example, in (if guard e-t e-f), both e-t and e-f are in tail position, but the guard is not: after we decide which branch to take, we’re committed:

(define (foo ...)
  ...
  (if guard
    (f x y ...)
    (g z ...)))

Once we finish executing guard, it would be useless (but correct) to (a) push a stack frame, (b) wait on the result of the subordinate call, and (c) merely return that same result, because all we’d be doing is copying the return value from the callee and propagating it back as the return value of the caller. Being a tail call is a syntactic property of a callsite: we (and the compiler) can easily look at a piece of code and cheaply decide when a call is a tail call versus not.

This reasoning above generalizes to any call expression in tail position: because a tail call will necessarily evaluate to its result, administratively copying it up/down the stack is extensionally a no-op. Now, the tail-recursive version uses a simple trick I teach to all of my students: (a) identify an accumulator variable, (b) instead of computing with the result of the recursive call, compute with the current accumulator, (c) return the accumulator in the base case:

;; Racket
(define (sum l acc) ;; note: acc got added
  (if (empty? l)
      acc ;; this is the *true* return!
      (sum (rest l) (+ acc (first l)))))

// C -- we pass in length manually because we're using arrays
int sum(const int* l, int length, int acc) {
    if (length == 0) return acc;
    return sum(l + 1, length - 1, acc + l[0]);
}

Both of these functions are tail recursive: because the only recursive call to sum is also the return value from sum (or, more directly: because both calls to sum are in tail position). Since the compiler knows that these are tail calls, a compiler with tail-call optimization will ensure that both of these tail calls compile into jmp statements–with zero implication on stack usage–rather than the more burdensome (on the cache, stack, etc.) direct-style calls. Something that should concern you is this: if the function is using constant stack space, how are the variables being updated / represented!? The answer is that the arguments get stomped over, and mutably updated, yielding the exact same performance profile as a loop!.

Now time for an exercise, what about this program, can you convert it to using tail-recursion?

;; return a pair (cons cell) of the number of even numbers,
;; and the number of odd numbers.
;; HINT: use multiple accumulators. 
(define (even-odd l)
  (if (empty? l)
      (cons 0 0)
      (let ([v (even-odd (rest l))])
            (if (first l)
                (cons (add1 (car v)) (cdr v))
                (cons (car v) (add1 (cdr v))))))))

What about this program?

;; flattens a tree? into a list
;; It's hard because there are *two* calls to linearize--can you do anything?
(define (linearize t)
  (match t
    ['empty '()]
    [`(node ,v ,t0 ,t1) (append (list v) (linearize t0) (linearize t1))]))

One surprising fact is that we can systematically compile any program so that every call is a tail-call, by completely transforming the program into continuation-passing-style (CPS), this essentially eliminates the stack. Indeed, some compilers for functional languages work precisely this way: those languages cannot fall prey to a stack overflow, because they have essentially traded a monolithic (efficient, array-like) stack for a deeply-nested stack, strewn throughout the heap—because the continuations will be heap-allocated and nested. There are various exciting trade-offs here, but for now I will leave this as is–we will continue next week.

Modern Deduction Post 1: Chain-Forward Computation

Sat, 11 May 2024 00:00:00 -0400

Our setting is logic programming, a field which attempts to design programming languages whose semantics have a close relationship to formal logic. The reason we might want to do this is that it suits our application domain more precisely than an implementation in a traditional programming language. Thus, using a logic programming language allows us to write more obviously-correct code, and perhaps even code that can be extracted cleanly from a certified implementation. Alternatively, if we did it ourselves, we’d have to do what our compiler (interpreter, …) would do anyway, so there’s no sense in doing it manually. Unfortunately, when we see a powerful tool, we are tempted to use it for everything: if our application is not ultimately-suited to the operationalization strategy of the logic programming engine we’re using, we simply obfuscate the issue in a veneer of formalism and end up with leaky abstractions. This is, I speculate, why logic programming languages have never caught on broadly for general-purpose programming. In this blog, I will detail the various trade-offs and implementation paradigms for modern logic programming engines, starting from Datalog and with a focus on program analysis.

The history of logic is rich, and I will not attempt to recount it all. Here I will focus on more restricted, application-specific languages, especially Datalog and its derivatives. The specific features of these languages, and the particulars of their implementation, often dovetail with a “right place, right time” effect. For example, Datalog backed by BDDs was a significant step forward in terms of production program analyses. More modern implementations eschew BDDs for more explicit representations, but it remains the case that engineers and computer scientists are on the lookout for logic-programming-based approaches to hard problems, especially those which deal intrinsically in the enumeration of large state spaces.

Perhaps one of logic programming’s most exciting motivations is program analysis. Program analysis systems automatically prove properties about, find bugs in, or simply help us understand our programs. These can come in a variety of forms from on-demand in-editor type-checking to whole-program (runs in microseconds), context-sensitive points-to analysis (days). Program analyses are notoriously hard to specify, and are especially hard to implement in a way that provides a close relationship to the formal specification. Additionally, program analyses often grapple with large state spaces in practice to solve interesting problems, and require some amount of thought regarding high-performance implementation.

Chain-Forward Computation and Datalog

The central evaluation mechanism in Datalog, and its derivatives, is to saturate a set of rules (e.g., Horn clauses) to a fixed point, to obtain a knowledge database in some domain. We call the computation “chain forward” because the evaluation of such languages is guided by an ordering on knowledge, typically set inclusion (in the case of traditional Datalogs). In these settings, we define an “immediate consequence” operator, which tells us everything which must be known, as a consequence of what we currently know; crucially, this operator is typically monotonic: we do not lose knowledge over time, though we will discuss some interesting departures from this assumption. Applying this immediate consequence operator repeatedly yields a stream of knowledge databases over time. Assuming the immediate consequence operator is monotone, this stream of knowledge databases over time forms an ascending chain according to the ordering on knowledge.

Datalog’s syntax consisting of “facts” and “rules.” Facts are “known statements” and always have the following form: R(c0, ...), where R is a relation name (identifier) and the arguments are constants. For example >(3,2) might be a fact, along with reaches("n0","n1"). But facts may not include variables, for example: reaches("n0",x) is disallowable as a fact, because x is not bounded in any way.

(define relation? symbol?)
(define atom? symbol?) ;; atomic lits are symbols

;; variables must be explicitly tagged, not 'x, but '(var x)
(define (var? x) (match x [`(var ,(? symbol? x)) #t] [_ #f]))
(define (var-or-atom? x) (or (atom? x) (var? x)))

;; tuples are untagged
(define (tuple? f)
  (match f
    [`((,(? atom?) ...)) #t]
    [_ #f]))

;; facts add a relation name (otherwise how would we know it)
(define (fact? f)
  (match f
    [`((,(? relation? r) ,(? atom?) ...)) #t]
    [_ #f]))

;; notice that ∧ is implicit in the body
(define (rule? r)
  (match r
    [`((,(? relation? head-rel) ,(? var-or-atom?) ...) <-- 
       (,(? relation? body-rels) ,(? var-or-atom?) ...) ...)
     #t]
    [_ #f]))

Let’s look at simplest interesting example: transitive closure. The Datalog program (technically in Soufflé here) to implement transitive closure example (tc.dl) is here (I elide some declarations):

.output path
.decl edge(x:number,y:number)
.decl path(x:number,y:number)
edge(1,2). edge(2,3). edge(3,5). edge(5,4). edge(4,1). edge(4,8).
path(x, y) :- edge(x, y).
path(x, y) :- path(x, z), edge(z, y).

Running this program in Soufflé (souffle tc.dl) yields an output database, the transitively-closed graph, in the file path.csv.

Datalog cannot search (disjunction), saturated conjunctions only

Disjunctions in the head of a rule is disallowed: this is not semantically within reach of Datalog. The following is invalid:

Q(x,...) ∨ R(x,...) ← P(...) ∧ R(...)

Once we have more than one positive literal in a clause, we need a SAT solver. SAT solvers combine search (“guess new things”) and deduction (“derive consequences”); Datalog solvers only employ deduction. Both SAT and Datalog engines share some overlapping ideas; for example, both use indexing, to accelerate joins (Datalog) and for efficient unit propagation (SAT). But (>2)-SAT is strictly harder than Datalog: 2-SAT can be written as Horn clauses (Q ← R is ¬R ∨ Q), but 3-SAT and beyond are out of reach. By contrast, ∧ in the head of a rule presents no serious semantic issue: Q(x,...) ∧ R(x,...) ← ... can easily be desugared into two rules: Q(x,...) ← ... and R(x,...) ← ....

SAT solvers have an importance difference from Datalog solvers: they forget information. This is important because, while you can use Datalog to do SAT solving just fine, you will pay a massive performance due to the concomitant over-materialization induced by enumerating all possibilities: you would be literally materializing O(2ⁿ). Of course, there are languages that embrace such cartoonishly-huge state spaces by design (answer-set programming, for example), and their designs are subsequently guided around these issues; indeed, grounding on-the-fly is an active research topic within ASP.

As an example, think about how you could write 5-clique:

clique3(A, B, C) :-
    edge(A, B), edge(A, C), edge(B, C),
    A != B, A != C, B != C.
clique4(A, B, C, D) :-
    clique3(A, B, C),
    edge(A, D), edge(B, D), edge(C, D),
    A != D, B != D, C != D.
clique5(A, B, C, D, E) :-
    clique4(A, B, C, D),
    edge(A, E), edge(B, E), edge(C, E), edge(D, E),
    A != E, B != E, C != E, D != E.

Semi-Naïve Evaluation

One issue with the repeated application of the rules is that if we use an explicit set-based representation of tuples, each iteration we’ll “rediscover” all knowledge from every previous iteration—this translates to additional data load, without commensurate knowledge throughput. In Datalog, the solution is to employ a compilation into an incrementalized IR, ala semi-naïve evaluation. For example, the recursive rule in transitive closure becomes:

path(x,z) ← path(x,y) ∧ edge(y,z)
          | 
   becomes| No Δ versions for edge as it is static
          |
Δpath(x,z) ← (Δpath(x,y) ∧ edge(y,z)) - path
i.e., 
Δpath(x,z) ∪= (Δpath(x,y) ⋈ edge(y,z)) - path

The rule expands into a single rule, because the relation edge never changes–thus, tracking a delta version would be irrelevant. Additionally, we assume that at the end of each iteration, Δpath is merged into path. In the more general case such as:

g(y,x) ∧ p(x,z) ← p(x,y) ∧ g(y,z)

We would need to split the rule into several versions: one to join Δp with g, one to join p with Δg, and one to join Δp with Δg. Think about what would happen if we have only Δp ⋈ Δg: if we have facts (x,y) in Δp and (y,z) in Δg, everything works fine. But what happens if (x,y) skips an iteration? It would be hard to ensure that doesn’t happen (though some work certainly explores this to a degree), and so we diversify our rules to enable us to catch things in Δp and Δg.

Previously, I mentioned the resulting rule would look something like:

Δpath(x,z) ∪= (Δpath(x,y) ⋈ edge(y,z)) - path

In fact, all rules within a fixedpoint (more specifically, an SCC of rules) will have the structure:

ΔR(x...) ∪= (ΔR(x,...) ⋈ Q(y,...)) - R

There is a crucial tacit point to be explained here: deduplication, i.e., - R is dirt cheap when implemented thoughtfully, and it is possible to parallelize nicely. There is no explicit scan of R in implementing subtraction, rather every rule generates a set of possibly-new tuples, which are deduplicated in some efficient manner to add to ΔR at the end of each iteration, before emptying ΔR in preparation for the next iteration.

I will not code up semi-naïve evaluation here, but will likely discuss a general approach to forward differentation and incrementalization in subsequent posts.

Relational Algebra

The explicit focus on bound variables has made the above presentation informal, as substitution was never defined. Indeed, substitution is “where computation happens” here in much the same way as in the λ-calculus. However, handling binders is a bit tedious, and it turns out there is a better, more systematic way. In the same way that category theory will avoid mentioning concrete points for products A × B, relational algebra will avoid mentioning explicitly-named points for manipulating relations.

The lack of explicit binders allows us to write an obviously-correct (albeit slow) interpreter for a conjunctive query that produces sets of tuples as its output:

;; q: query?
;; db: hashmap from relation name ↦ ℘(Tuple)
;; returns a set of tuples
(define (interpret-query q db)
  (match q
    [`(literal-tuple ,es ...) (set es)]
    [`(scan ,R) (hash-ref db R)]
    [`(select from ,q+ where column ,n equals ,k)
     (list->set (filter (lambda (x) (equal? (list-ref x n) k))
                        (set->list (interpret-query q+ db))))]
    [`(,q0 ∪ ,q1) (set-union (interpret-query q0 db)
                             (interpret-query q1 db))]
    [`(,q0 ∩ ,q1) (set-intersect (interpret-query q0 db)
                                 (interpret-query q1 db))]
    [`(reorder ,q ,order ...)
     (let ([ts (set->list (interpret-query q db))])
       (set->list
        (map (λ (t) (map (λ (i) (list-ref t i)) order))
             ts)))]
    [`(project ,q to first ,n)
     (list->set (map (λ (t) (take t n)) (set->list (interpret-query q db))))]
    [`(,q0 ⋈ ,q1 on first ,N)
     (list->set
      (foldl
       (λ (t0 tups)
         (foldl (λ (t1 tups)
                  (if (equal? (take t0 N) (take t1 N))
                      (set-add tups (append (take t0 N) (drop t0 N) (drop t1 N)))
                      tups))

The implementation is mostly standard: notice that ⋈ degenerates to the cartesian product in the case that N=0 (since (take t 0) returns the empty list for all t). We include only binary joins, k-ary joins are typically decomposed into chains or trees of binary joins: R ⋈ Q ⋈ S can be construed as either (R ⋈ Q) ⋈ S or R ⋈ (Q ⋈ S) (3 or more joins would allow us to consider trees, too). Unfortunately, the wrong choice could force the enumeration to explode: if a subordinate join produces lots of junk which is later filtered out by the outer join, we may end up doing more work than an alternative join plan. Join planning–-the process of optimally deciding on a join decomposition–is a challenging aspect of the implementation; for some programs, any particular static join plan is be inefficient, as relation sizes dynamically evolve, the optimal join plan may shift as well. While traditional databases put a lot of effort into query planning, Datalog has taken a slightly different approach, with only scant recent work exploring dynamic plans.

With this implementation, we can write query to find us all of the one-hop transitive edges: the query we want is edge(x,y), path(y,z), which operationalizes to a join between edge and path; since we want to line up the ys we reorder edge, then reorder the join result (to gather x and z), then project the first two elements.

(interpret-query
 '(project (reorder ((reorder (scan edge) 1 0) ⋈ (scan path) on first 1) 1 2 0)
           to first 2)
 (hash 'edge (set '(a b) '(b c) '(c d) '(b e))
       'path (set '(a b) '(b c) '(c d) '(b e))))
;; yields (set '(a e) '(b d) '(a c))

Assuming we’re willing to write programs in terms of relational algebra, we can iteratively generate new tuples. An RA-rule? extends a specific relation (recall we’re taking a point-free style) with the result of a query. We can interpret rules by evaluating the query and unioning the result set into the necessary relation. A program is a set of rules, which we can evaluate by iteratively (to a fixed point) applying all rules until we see no changes in the database.

(define (RA-rule? r)
  (match r
    [`(,(? relation? R) ∪= ,(? RA-query? q)) #t]
    [_ #f]))

(define (interpret-rule r db)
  (match-define `(,R ∪= ,q) r)
  (hash-set db R (set-union (hash-ref db R (set)) (interpret-query q db))))

(define (interpret-program rules)
  (define (next-iteration db)
    (foldl (λ (r db) (interpret-rule r db)) db rules))
  (define (do-some-more db)
    (if (equal? db (next-iteration db))
        db
        (do-some-more (next-iteration db))))
  (do-some-more (hash)))

Now we can write small programs:

(interpret-program
 '((edge ∪= (literal-tuple a b))
   (edge ∪= (literal-tuple b c))
   (edge ∪= (literal-tuple c d))
   (path ∪= (scan edge))
   (path ∪= (project (reorder ((reorder (scan edge) 1 0) ⋈ (scan path) on first 1) 1 2 0)
                     to first 2))))

Next time, we’ll plan to talk a bit about how to systematically compile our high-level rules into the RA plans, and discuss the various trade-offs in doing so.

The full code

Is here in this gist

;; Relational algebra
#lang racket

(define relation? symbol?)
(define atom? symbol?) ;; atomic lits are symbols

;; variables must be explicitly tagged, not 'x, but '(var x)
(define (var? x) (match x [`(var ,(? symbol? x)) #t] [_ #f]))
(define (var-or-atom? x) (or (atom? x) (var? x)))

;; tuples are untagged
(define (tuple? f)
  (match f
    [`((,(? atom?) ...)) #t]
    [_ #f]))

;; facts add a relation name (otherwise how would we know it)
(define (fact? f)
  (match f
    [`((,(? relation? r) ,(? atom?) ...)) #t]
    [_ #f]))

;; notice that ∧ is implicit in the body
(define (rule? r)
  (match r
    [`((,(? relation? head-rel) ,(? var-or-atom?) ...) <-- 
       (,(? relation? body-rels) ,(? var-or-atom?) ...) ...)
     #t]
    [_ #f]))

;; should all be true
(rule? '((p) <--))
(rule? '((r) <-- (q)))
(rule? '((path x y) <-- (edge x x)))
(rule? '((path x y) <-- (edge x y) (path y z)))

;; RA queries
(define (RA-query? q)
  (match q
    [`(literal-tuple ,(? atom? es) ...) #t]
    ;; scan a relation
    [`(scan ,(? relation? R)) #t]
    [`(select from ,(? RA-query?)
              where column ,(? nonnegative-integer?)
              equals ,(? atom?))
     #t]
    ;; natural join on the first N columns, values must be equal
    [`(,(? RA-query? q0) ⋈ ,(? RA-query? q1) on first ,(? nonnegative-integer? N)) #t]
    [`(,(? RA-query? q0) ∪ ,(? RA-query? q0)) #t]
    [`(,(? RA-query? q0) ∩ ,(? RA-query? q0)) #t]
    ;; reorder tuples
    [`(reorder ,(? RA-query? q) ,(? nonnegative-integer?) ...) #t]
    ;; project the first n elements of tuples
    [`(project ,(? RA-query? q) to first ,(? nonnegative-integer? n)) #t]
    ;; need closed-world assumption
    [`(- ,(? RA-query? q)) #t]
    [_ #f]))

;; q: query?
;; db: hashmap from relation name ↦ ℘(Tuple)
;; returns a set of tuples
(define (interpret-query q db)
  (match q
    [`(literal-tuple ,es ...) (set es)]
    [`(scan ,R) (hash-ref db R)]
    [`(select from ,q+ where column ,n equals ,k)
     (list->set (filter (lambda (x) (equal? (list-ref x n) k))
                        (set->list (interpret-query q+ db))))]
    [`(,q0 ∪ ,q1) (set-union (interpret-query q0 db)
                             (interpret-query q1 db))]
    [`(,q0 ∩ ,q1) (set-intersect (interpret-query q0 db)
                                 (interpret-query q1 db))]
    [`(reorder ,q ,order ...)
     (let ([ts (set->list (interpret-query q db))])
       (set->list
        (map (λ (t) (map (λ (i) (list-ref t i)) order))
             ts)))]
    [`(project ,q to first ,n)
     (list->set (map (λ (t) (take t n)) (set->list (interpret-query q db))))]
    [`(,q0 ⋈ ,q1 on first ,N)
     (list->set
      (foldl
       (λ (t0 tups)
         (foldl (λ (t1 tups)
                  (if (equal? (take t0 N) (take t1 N))
                      (set-add tups (append (take t0 N) (drop t0 N) (drop t1 N)))
                      tups))
                tups
                (set->list (interpret-query q1 db))))
       (set)
       (set->list (interpret-query q0 db))))]))

(define (RA-rule? r)
  (match r
    [`(,(? relation? R) ∪= ,(? RA-query? q)) #t]
    [_ #f]))

(define (interpret-rule r db)
  (match-define `(,R ∪= ,q) r)
  (hash-set db R (set-union (hash-ref db R (set)) (interpret-query q db))))

(define (interpret-program rules)
  (define (next-iteration db)
    (foldl (λ (r db) (interpret-rule r db)) db rules))
  (define (do-some-more db)
    (if (equal? db (next-iteration db))
        db
        (do-some-more (next-iteration db))))
  (do-some-more (hash)))

;; relations: hash from relation names to their associated arities
;; herbrand-base is a list of atoms 
(define (herbrand-universe relations herbrand-base)
  ;; generate a cartesian product of arity n with atoms from herbrand-base
  (define (h n)
    (if (= n 0)
        '(())
        (apply append (map (λ (lst) (map (λ (x) (cons x lst)) herbrand-base)) (h (- n 1))))))
  (foldl
   (λ (R acc) (hash-set acc R (h (hash-ref relations R))))
   (hash)
   (hash-keys relations)))

(interpret-query
 '(project (reorder ((reorder (scan edge) 1 0) ⋈ (scan path) on first 1) 1 2 0)
           to first 2)
 (hash 'edge (set '(a b) '(b c) '(c d) '(b e))
       'path (set '(a b) '(b c) '(c d) '(b e))))

(interpret-program
 '((edge ∪= (literal-tuple a b))
   (edge ∪= (literal-tuple b c))
   (edge ∪= (literal-tuple c d))
   (path ∪= (scan edge))
   (path ∪= (project (reorder ((reorder (scan edge) 1 0) ⋈ (scan path) on first 1) 1 2 0)
                     to first 2))))

Modern Deduction Post 0: Prologue

Fri, 05 Apr 2024 00:00:00 -0400

Over the past few years, my collaborators and I have been exploring the design of high-performance logic programming engines for a wide variety of tasks, including program analysis (points-to analysis, abstract interpretation), graph analytics (transitive closure, PageRank), and security (binary code similarity, disassembly, and decompilation). We (myself, my collaborators, and our students) have engineered a variety of systems aiming for both (a) state-of-the-art performance at the highest scale (shared-memory parallelism, distribution via MPI, SIMD on GPUs, and GPU cluster computing) and (b) semantic extensions (monotonic aggregation, algebraic data) to Datalog.

I will be chronicling our efforts (mostly the relevant background) in this series of posts, Modern Deduction. I will plan to put out a post once every month or two.

Posts:

Mid-Point Review Materials

Fri, 16 Dec 2022 00:00:00 -0500

This page holds my “year 3 review” materials for evaluation and promotion at Syracuse University.

My detailed personal statement (including teaching and research statements) is here

My current CV is linked here

A (~22MB) zip file linked here gives the full review packet, including CV, papers, and my personal statement.

Certifying Interpreters in Racket

Sun, 14 Aug 2022 00:00:00 -0400

When I began programming, I read a copy of Richard Steven’s “Programming in the UNIX Environment.” Ultimately, my early experimentations with C were a failure; however, I later read David Beazley’s “Python: Essential Reference,” and was quickly able to pick up the UNIX API via it’s much simpler (admittedly, largely due to Beazley’s writing) Python counterpart. After teaching my undergraduate PL courses in Scheme variants these past few years I have wondered if we can understand type theory’s operationalization (via proof objects) using a similar shift in perspective.

Here, I rigorously define an interpreter which produces a certificate of its own correctness—assuming you trust the correctness of our metalanguage, which (in the interests of appeasing the more skeptical among us) we treat as S-expression comparison. Here I use Racket, but any similar dynamic language with matching (or, if in OO, virtual methods) would work to illustrate the key ideas. I do use Racket’s contracts to check certificates, though other implementations could defer this to the end or even elide checking entirely.

The Language

;; Specification of IfArith's syntax as a Racket predicate
(define (expr? e)
  (match e
    [(? nonnegative-integer? n) #t]
    [(? symbol? x) #t]
    [`(let ,(? symbol? x) ,(? expr? e) ,(? expr? e-body))]
    [`(plus ,(? expr? e0) ,(? expr? e1)) #t]
    [`(not ,(? expr? e-guard)) #t]
    [`(if0 ,(? expr? e0) ,(? expr? e1) ,(? expr? e2)) #t]
    [_ #f]))

(define e0 '(plus 2 1)) ;; 3
(define e1 '(plus 1 (if0 0 1 2))) ;; 2
(define e2 '(let x (plus 0 0) (plus x 1))) ;; 1 
(define e3 '(let x (plus 0 (if0 (plus 0 0) 1 0)) (plus x 0))) ;; 1

The Proofs

Now the tricky part: we need to think about what proofs for evaluation of terms in our little language look like. To be precise about it, we need to define a relation: e, ρ ⇓ v, such that e is an expression, ρ is an environment (mapping variables to values) and v is a value. To be constructive about it, we need to algebraically define a structure representing proofs that the evaluation of e in ρ evaluates to v. Following the Scheme tradition of homoiconicity, we represent proofs as S-expressions themselves; inference rules using pattern matching over these S-expressions to mirror the natural deduction (big-step) style in which we would write our semantics on paper. As an example, here’s a derivation of (if0 0 (plus 1 1) 0).

(⇓-if-true
  (⇓-const ----- 
         (0 ∅ ⇓ O))
  (⇓-plus
   (⇓-const -----
        (1 ∅ ⇓ (S O)))
   (⇓-const -----
          (1 ∅ ⇓ (S O)))
   (plus -----
      (= (+ (S O) (S O)) (S (S O))))
   -----
   ((plus 1 1) ∅ ⇓ (S (S O))))
  -----
  ((if0 0 (plus 1 1) 0) ∅ ⇓ (S (S O))))

Racket’s pattern matcher is, of course, very different than a more elaborate matcher centered around higher-order unification in a statically-typed setting. Manually explicating the unification becomes a bit of a chore after a while; certainly a rebuke to the thought of using this as a serious strategy for type theory, but at the same time precisely the reason we’re doing it this way.

Of course it would be possible (especially in such a simple semantics as this) to define e, ρ ⇓ v using substitution to implement variables; I explicitly materialize environments to (a) anticipate closures (later) and (b) present a representational challenge upon which I would like to expand a bit. Of course, using Racket as our metalanguage, we could simply use hashes for environments. But remember—we are trying to appease the pedants among us, thus representation must be as symbolic as possible. Similarly, I represent the naturals symbolically. I do allow myself a serious concession: I internalize plus, as I elide fix in the source language. If this disappoints you, I would say there’s no fundamental barrier; we could easily-enough (via environments) implement application and then a fixed-point combinator.

;; naturals
(define (nat? n)
  (match n ['O #t] [`(S ,(? nat? n)) #t] [_ #f]))

(define/contract (num->nat n)
  (-> nonnegative-integer? nat?)
  (match n
    [0 'O]
    [n `(S ,(num->nat (- n 1)))]))

(define/contract (nat->num n)
  (-> nat? nonnegative-integer?)
  (match n
    ['O 0]
    [`(S ,x) (add1 (nat->num x))]))

(define/contract (nat-add s0 s1)
  (-> nat? nat? nat?)
  (match s0
    ['O s1]
    [`(S ,s0+) `(S ,(nat-add s0+ s1))]))

To represent environments we define (a) a predicate dictating valid structure for environments and (b) a predicate, (env-maps? ρ x v pf), which defines the structure of valid proofs showing that environment ρ maps variable x to value v. Both of these are in the de-facto “trusted computing base,” in the sense that if your definition of proofs is broken (i.e., don’t faithfully capture what you want to prove) then it doesn’t matter that your interpreter produces proofs. And, of course, this is the biggest drawback of using an untyped language to do this—we only get some rough syntactic checking, similar to tools such as Ott but not dependently-typed languages or provers. To avoid completely defeating the purpose of our exercise here, I have followed a bit of a trick to rely minimally on Racket as a metatheory: because many objects internal to the semantics (e.g., numbers, environments, and proofs) are represented purely symbolically (as S-expressions), we primarily rely upon Racket for S-expression equality and dispatch (matching). Aside from plus our semantics uses a very small subset of Racket: I believe essentially either (equivalently) existential fixed-point logic (of a Herbrand base comprising S-expressions) or constrained horn clauses (whose background logic includes S-expression equality and structurally-recursive addition).

;; environments
(define (environment? ρ)
  (match ρ
    ['∅ #t]
    [`(↦ ,(? environment? ρ+) ,(? symbol? x) ,(? value? v)) #t]
    [_ #f]))

;; environment lookup -- predicate (inductive defn.)
(define/contract (env-maps? pf ρ x v)
  (-> any/c environment? symbol? value? boolean?)
  (match pf
    [`(↦-hit
       ------
       (↦ ,ρ ,x0 ,v0))
     (and (equal? x0 x) (equal? v0 v))]
    [`(↦-miss
       ,next-pf
       -----
       (↦ ,ρ ,x0 ,_))
     (and (not (equal? x0 x))
          (env-maps? next-pf ρ x v))]
    [_ #f]))

The implementation of (⇓ pf e ρ v) follows its natural deduction counterpart (which I have not written down, but try to stylize via my spacing). Our procedure checks the proof, recursively calling itself to check subproofs—mirroring the top-down process you may follow on a whiteboard to check the proof yourself, though Reynolds points out that the order of sub-checks is inherited from the metalanguage (i.e., Racket’s control flow).

;; proofs of evaluation
(define/contract (⇓ pf e ρ v)
  (-> any/c expr? environment? value? boolean?)
  (match pf
    ;; Const
    [`(⇓-const
       -----
       (,(? nonnegative-integer? n) ,ρ+ ⇓ ,v+))
     (and (equal? (num->nat n) v) (equal? v v+) (equal? ρ+ ρ))]
   ...

Checking proofs for constants is easy. The rest of the rules follow their natural deduction counterparts; the tedious part is the administrative overhead required to implement the store-passing construction. Of course my code is a bit smelly here; the ad-hoc use of consequent points to a much cleaner refactoring that elides using pairs of a return value and its proof in favor of projecting the value from the proof, but the point here is to stick to what I think I’d end up with doing it in an interactive proof assistant.

    ;; Var
    [`(⇓-var ,proof-x-in-ρ
       -----
       (,(? symbol? x) ,(? environment? ρ+) ⇓ ,(? value? v+)))
    (and (equal? ρ+ ρ) (equal? v+ v) (env-maps? proof-x-in-ρ ρ x v))]
    ;; Let
    [`(⇓-let
       ,proof-e0
       ,proof-e1
       -----
       ((let ,x ,e0 ,e1) ,(? environment? ρ+) ⇓ ,(? value? v+)))
     (and (equal? v+ v) (equal? ρ+ ρ)
          (⇓ proof-e0 e0 ρ (last (consequent proof-e0)))
          (⇓ proof-e1 e1 `(↦ ,ρ ,x ,(last (consequent proof-e0)))
		                            (last (consequent proof-e1))))]
    ;; Plus
    [`(⇓-plus
       ,proof-e0
       ,proof-e1
       (plus ----- (= (+ ,v0+ ,v1+) ,v-r))
       -----
       ((plus ,e0 ,e1) ,ρ+ ⇓ ,v-r))
     #:when (and (equal? v-r v) (equal? ρ+ ρ))
     (define v0 (last (consequent proof-e0)))
     (define v1 (last (consequent proof-e1)))
     (and (⇓ proof-e0 e0 ρ v0)
          (⇓ proof-e1 e1 ρ v1)
          (equal? v0+ v0)
          (equal? v1+ v1)
          (equal? v-r (nat-add v0 v1)))]
    ;; Not
    [`(⇓-not-0
       ,proof-e0
       -----
       ((not ,e0) ,ρ ⇓ O))
     (define v0+ (match (consequent proof-e0) [`(,_ ,_ ⇓ ,v) v]))
     (and (equal? v0+ v) (not (equal? v 'O)) (⇓ proof-e0 e0 ρ v))]
    [`(⇓-not-1
       ,proof-e0
       -----
       ((not ,e0) ,ρ ⇓ (S O)))
     (define v0+ (match (consequent proof-e0) [`(,_ ,_ ⇓ ,v) v]))
     (and (equal? v0+ v) (equal? v 'O) (⇓ proof-e0 e0 ρ v))]
    ;; If-True
    [`(⇓-if-true
       ,proof-guard-true
       ,proof-e1-v-res
       -----
       ((if0 ,e0 ,e1 ,e2) ,ρ ⇓ ,v))
     (match (consequent proof-guard-true)
       [`(,_ ,_ ⇓ O)
        (and (⇓ proof-guard-true e0 ρ 'O)
             (⇓ proof-e1-v-res e1 ρ v))])]
    ;; If-False
    [`(⇓-if-false
       ,proof-guard-false
       ,proof-e1-v-res
       -----
       ((if0 ,e0 ,e1 ,e2) ,ρ ⇓ ,v+))
     #:when (equal? v+ v)
     (match (consequent proof-guard-false)
       [`(,_ ,_ ⇓ ,n)
        (and (not (equal? n 'O))
             (⇓ proof-guard-false e0 ρ n)
             (⇓ proof-e1-v-res e1 ρ v))])]

The Programs

Compared to the checking, writing the proofs is surprisingly straightforward—even pleasant (okay; maybe not pleasant). A common idiom when writing certified (proof-backed) code is the notion of returning an existential, i.e., a witness alongside its proof. Of course, the key challenge in writing certified code is convincing yourself of its correctness. There are a variety of ways you could do this: you could inspect the output of the evaluator I write and ensure that they match the specification, for example. However, for full clarity, I will use Racket’s contract system, which allows dynamic checking—this allows us to check the correctness of our evaluators as we go, but of course won’t prove correctness across all runs, and using define/contract also interposes the check at every callsite (a serious overhead), though I am sure you can use your imagination about how this could be made faster.

Let’s start with environments, writing a function (lookup ρ x), which returns a proof that x returns some value v–if it doesn’t, we would just throw a dynamic error due to a match failure–we are posting a proof exists. We can write a contract to ensure that the code we use to do this is correct. Racket’s default arrow contract combinator, ->, is not powerful enough to express dependent contracts, where a property of the result depends on one of the inputs. However, Racket’s more powerful ->i does allow dependent contracts, allowing us to write a version of lookup which both computes what we want and checks to ensure its correctness:

;; decision procedure for lookup -- returns witness and proof of inclusion
(define/contract (lookup ρ x)
        ;; v---- domain ----v
    (->i ([ρ environment?] [x symbol?])
        ;; range -- produces a proof dependent on ρ,x
         [result (ρ x)
                 (match-lambda [`(,(? value? v) . ,pf) (env-maps? pf ρ x v)]
                               [_ #f])])
  (match ρ
    [`(↦ ,ρ1 ,x1 ,v) #:when (equal? x1 x)
                     `(,v . (↦-hit ------ (↦ ,ρ ,x ,v)))]
    [`(↦ ,ρ1 ,x1 ,v) #:when (not (equal? x1 x))
                     (match (lookup ρ1 x)
                       [`(,v . ,pf)
                        `(,v . (↦-miss ,pf ----- (↦ ,ρ1 ,x1 ,v)))])]))

Finally, the interpreter itself—unadorned by the ceremony and pedantics of low-level unification checking—is surprisingly unintimidating and to-the-point.

;; produce a value alongside a proof of its correctness
(define/contract (eval e ρ)
  ;; contract: returns pair of value and proof of its derivation
  (->i ([e expr?] [ρ environment?])
       [result (e ρ)
               (lambda (witness-pf) (match witness-pf
                                      [`(,(? value? v) . ,pf) (⇓ pf e ρ v)]
                                      [_ #f]))])
  (match e
    [(? nonnegative-integer? n)
     (cons (num->nat n)
           `(⇓-const
             -----
             (,n ,ρ ⇓ ,(num->nat n))))]
    [(? symbol? x)
     (match (lookup ρ x)
       [`(,v . ,pf)
        (cons v `(⇓-var
                  ,pf
                  -----
                  (,x ,ρ ⇓ ,v)))])]
    [`(let ,x ,e ,eb)
     (match (eval e ρ)
       [`(,v . ,pf-e)
        (match (eval eb `(↦ ,ρ ,x ,v))
          [`(,v-res . ,pf-v)
           (cons v-res
                 `(⇓-let
                   ,pf-e
                   ,pf-v
                   -----
                   ((let ,x ,e ,eb) ,ρ ⇓ ,v-res)))])])]
    [`(plus ,e0 ,e1)
     (match-define `(,v0 . ,pf-v0) (eval e0 ρ))
     (match-define `(,v1 . ,pf-v1) (eval e1 ρ))
     (define v-res (nat-add v0 v1))
     (cons v-res
           `(⇓-plus
             ,pf-v0
             ,pf-v1
             (plus ----- (= (+ ,v0 ,v1) ,v-res))
             -----
             ((plus ,e0 ,e1) ,ρ ⇓ ,v-res)))]
    [`(not ,e)
     (match (eval e ρ)
       [`(0 . ,pf-e)
        (cons '(S O)
              `(⇓-not-1
                ,pf-e
                -----
                ((not ,e) ρ ⇓ (S O))))]
       [`(,v0 . ,pf-e)
        (cons 'O
              `(⇓-not-0
                ,pf-e
                -----
                ((not ,e) ρ ⇓ O)))])]
    [`(if0 ,e0 ,e1 ,e2)
     (match (eval e0 ρ)
       [`(O . ,pf-e0)
        (match (eval e1 ρ)
          [`(,v . ,pf-v)
           (cons v
                 `(⇓-if-true
                   ,pf-e0
                   ,pf-v
                   -----
                   ((if0 ,e0 ,e1 ,e2) ,ρ ⇓ ,v)))])]
       [`(,n . ,pf-e0)
        (match (eval e2 ρ)
          [`(,v . ,pf-v)
           (cons v
                 `(⇓-if-false
                   ,pf-e0
                   ,pf-v
                   -----
                   ((if0 ,e0 ,e1 ,e2) ,ρ ⇓ ,v)))])])]))

Visualizing our Proofs

Alright, so now our interpreter writes proofs–what do they look like? Thankfully, we used S-expressions.

;; nice latex; comment out the labels by adding a % before \\tiny
(define (pf->tex pf)
  (define name (first pf))
  (define antecedents (reverse (list-tail (reverse (list-tail (cdr pf) 0)) 2)))
  (foldr
   (lambda (k v acc) (string-replace acc k v))
   (format "{{\\tiny \\textsc{ ~a }} \n \\frac{ ~a }{ \\texttt{ ~a } }}"
           name
           (string-join (map pf->tex antecedents) "\\,")
           (consequent pf))
   '("⇓" "∅" "↦")
   '("\\ensuremath{\\Downarrow}" "\\ensuremath{\\emptyset}" 
     "\\ensuremath{\\mapsto}")))

Here’s the complete derivation of ((plus 1 (if0 0 1 2))):

And a longer program where I had to remove the rule names in rendering…

Metatheoretic Concessions for the Working Programmer

Obviously we would never write an interpreter in Racket the way we did above: there’s too much extraneous math (i.e., the proof objects). I think it’s surprising, though, to see how much more direct the code becomes by (a) erasing the proofs, (b) implementing the environment as hashes, and (c) relying upon racket’s built-in representation of numbers. The biggest burden in my implementation is obviously (a), as I have explicated the proofs via cons. Tighter implementations exist that would hide the ugliness via (say) monads. Eliminating proofs also eliminates a lot of administrative matching and in some places allows to make tail calls to eval where we couldn’t before, giving us a textbook metacircular interpreter. I think it is interesting to see just how directly a translation it is: I systematically removed each feature (in the same way an extractor would) to achieve the simpler implementation.

(define (eval-simpl e ρ)
  (match e
    [(? nonnegative-integer? n) n]
    [(? symbol? x) (hash-ref ρ x)]
    [`(let ,x ,e ,eb) (eval-simpl eb (hash-set ρ x (eval-simpl e ρ)))]
    [`(plus ,e0 ,e1) (+ (eval-simpl e0 ρ) (eval-simpl e1 ρ))]
    [`(not ,e)
     (match (eval-simpl e ρ)
       [0 1]
       [_ 0])]
    [`(if0 ,e0 ,e1 ,e2)
     (match (eval-simpl e0 ρ)
       [0 (eval-simpl e1 ρ)]
       [_ (eval-simpl e2 ρ)])]))

Conclusion

Do I think this is the future of dependently typed programming? Of anything? Perhaps both no. Racket is not a particularly appealing implementation language for type theory and its pattern matching is not designed to scale to settings where higher-order unification is of concern to the user. Similarly, this code has serious algorithmic inefficiencies; plenty of the proof checking could be memoized, and perhaps some simple contract trick would achieve this (though other evaluation strategies could, e.g., use tabling in the metatheory). I largely wrote this up to motivate my thoughts on (the potential of) explaining the operationalization of proof objects to students as an extension of operational semantics in the untyped setting I teach in my class. Perhaps this will help someone draw connections between operational semantics and proof objects, though I hesitate to say type theory, broadly.

Ultimately I did manage to find a lot of bugs in my implementation of the interpreter using the contracts. Obviously, though, I was driving the process by generating terms and not using a systematic methodology of proving the correctness–I have tried to say that the interpreter generates certificates, but it is of course not certified. It’s a fun exercise, though–you can read the full code yourself below. I think there are a few follow ups that could be done from here if you are interested in experimenting with the code. Adding closures and application, for example, should not be too hard; I may illustrate that in my course next term.

Thanks to Quinn Wilton for some corrections to the phrasing in this post.

The Code

;; self-certifying interpreters, summer 2022, kris micinski
#lang racket

;; naturals
(define (nat? n)
  (match n ['O #t] [`(S ,(? nat? n)) #t] [_ #f]))

(define/contract (num->nat n)
  (-> nonnegative-integer? nat?)
  (match n
    [0 'O]
    [n `(S ,(num->nat (- n 1)))]))

(define/contract (nat->num n)
  (-> nat? nonnegative-integer?)
  (match n
    ['O 0]
    [`(S ,x) (add1 (nat->num x))]))

(define/contract (nat-add s0 s1)
  (-> nat? nat? nat?)
  (match s0
    ['O s1]
    [`(S ,s0+) `(S ,(nat-add s0+ s1))]))

;; values are nats
(define value? nat?)

;; expressions
(define (expr? e)
  (match e
    [(? nonnegative-integer? n) #t]
    [(? symbol? x) #t]
    [`(let ,(? symbol? x) ,(? expr? e) ,(? expr? e-body)) #t]
    [`(plus ,(? expr? e0) ,(? expr? e1)) #t]
    [`(not ,(? expr? e-guard)) #t]
    [`(if0 ,(? expr? e0) ,(? expr? e1) ,(? expr? e2)) #t]
    [_ #f]))

;; environments
(define (environment? ρ)
  (match ρ
    ['∅ #t]
    [`(↦ ,(? environment? ρ+) ,(? symbol? x) ,(? value? v)) #t]
    [_ #f]))

;; environment lookup -- predicate (inductive defn.)
(define/contract (env-maps? pf ρ x v)
  (-> any/c environment? symbol? value? boolean?)
  (match pf
    [`(↦-hit
       ------
       (↦ ,ρ ,x0 ,v0))
     (and (equal? x0 x) (equal? v0 v))]
    [`(↦-miss
       ,next-pf
       -----
       (↦ ,ρ ,x0 ,_))
     (and (not (equal? x0 x))
          (env-maps? next-pf ρ x v))]
    [_ #f]))

;; decision procedure for lookup -- returns witness and proof of inclusion
(define/contract (lookup ρ x)
        ;; v---- domain ----v
    (->i ([ρ environment?] [x symbol?])
        ;; range -- produces a proof dependent on ρ,x
         [result (ρ x)
                 (lambda (witness-pf) (match witness-pf
                                        [`(,(? value? v) . ,pf) (env-maps? pf ρ x v)]
                                        [_ #f]))])
  (match ρ
    [`(↦ ,ρ1 ,x1 ,v) #:when (equal? x1 x)
                     `(,v . (↦-hit ------ (↦ ,ρ ,x ,v)))]
    [`(↦ ,ρ1 ,x1 ,v) #:when (not (equal? x1 x))
                     (match (lookup ρ1 x)
                       [`(,v . ,pf)
                        `(,v . (↦-miss ,pf ----- (↦ ,ρ1 ,x1 ,v) ))])]))

; (lookup '(↦ (↦ ∅ x O) y (S O)) 'x)

;; we will now follow the style '(name atecedent0 ... ----- consequent)
(define (consequent judgement) (last judgement))

;; predicate -- inductive definition of evaluation relation
(define/contract (⇓ pf e ρ v)
  (-> any/c expr? environment? value? boolean?)
  (match pf
    ;; Const
    [`(⇓-const
       -----
       (,(? nonnegative-integer? n) ,ρ+ ⇓ ,v+))
     #:when (and (equal? (num->nat n) v) (equal? ρ+ ρ) (equal? v v+))
     #t]
    ;; Var
    [`(⇓-var ,proof-x-in-ρ
       -----
       (,(? symbol? x) ,(? environment? ρ+) ⇓ ,(? value? v+)))
     #:when (and (equal? ρ+ ρ) (equal? v+ v))
     (env-maps? proof-x-in-ρ ρ x v)]
    ;; Let
    [`(⇓-let
       ,proof-e0
       ,proof-e1
       -----
       ((let ,(? symbol? x) ,(? expr? e0) ,(? expr? e1)) ,(? environment? ρ+) ⇓ ,(? value? v+)))
     #:when (equal? v+ v) (equal? ρ+ ρ)
     (and (⇓ proof-e0 e0 ρ (last (consequent proof-e0)))
          (⇓ proof-e1 e1 `(↦ ,ρ ,x ,(last (consequent proof-e0))) (last (consequent proof-e1))))]
    ;; Plus
    [`(⇓-plus
       ,proof-e0
       ,proof-e1
       (plus ----- (= (+ ,v0+ ,v1+) ,v-r))
       -----
       ((plus ,e0 ,e1) ,ρ+ ⇓ ,v-r))
     #:when (and (equal? v-r v) (equal? ρ+ ρ))
     (define v0 (last (consequent proof-e0)))
     (define v1 (last (consequent proof-e1)))
     (and (⇓ proof-e0 e0 ρ v0)
          (⇓ proof-e1 e1 ρ v1)
          (equal? v0+ v0)
          (equal? v1+ v1)
          (equal? v-r (nat-add v0 v1)))]
    ;; Not
    [`(⇓-not-0
       ,proof-e0
       -----
       ((not ,e0) ,ρ ⇓ O))
     (define v0+ (match (consequent proof-e0) [`(,_ ,_ ⇓ ,v) v]))
     (and (equal? v0+ v) (not (equal? v 'O)) (⇓ proof-e0 e0 ρ v))]
    [`(⇓-not-1
       ,proof-e0
       -----
       ((not ,e0) ,ρ ⇓ (S O)))
     (define v0+ (match (consequent proof-e0) [`(,_ ,_ ⇓ ,v) v]))
     (and (equal? v0+ v) (equal? v 'O) (⇓ proof-e0 e0 ρ v))]
    ;; If-True
    [`(⇓-if-true
       ,proof-guard-true
       ,proof-e1-v-res
       -----
       ((if0 ,e0 ,e1 ,e2) ,ρ ⇓ ,v))
     (match (consequent proof-guard-true)
       [`(,_ ,_ ⇓ O)
        (and (⇓ proof-guard-true e0 ρ 'O)
             (⇓ proof-e1-v-res e1 ρ v))])]
    ;; If-False
    [`(⇓-if-false
       ,proof-guard-false
       ,proof-e1-v-res
       -----
       ((if0 ,e0 ,e1 ,e2) ,ρ ⇓ ,v+))
     #:when (equal? v+ v)
     (match (consequent proof-guard-false)
       [`(,_ ,_ ⇓ ,n)
        (and (not (equal? n 'O))
             (⇓ proof-guard-false e0 ρ n)
             (⇓ proof-e1-v-res e1 ρ v))])]
    [_ #f]))

;;
;; Self-certifying interpreter
;;

;; produce a value alongside a proof of its correctness
(define/contract (eval e ρ)
  ;; contract: returns pair of value and proof of its derivation
  (->i ([e expr?] [ρ environment?])
       [result (e ρ)
               (lambda (witness-pf) (match witness-pf
                                      [`(,(? value? v) . ,pf) (⇓ pf e ρ v)]
                                      [_ #f]))])
  (match e
    [(? nonnegative-integer? n)
     (cons (num->nat n)
           `(⇓-const
             -----
             (,n ,ρ ⇓ ,(num->nat n))))]
    [(? symbol? x)
     (match (lookup ρ x)
       [`(,v . ,pf)
        (cons v `(⇓-var
                  ,pf
                  -----
                  (,x ,ρ ⇓ ,v)))])]
    [`(let ,x ,e ,eb)
     (match (eval e ρ)
       [`(,v . ,pf-e)
        (match (eval eb `(↦ ,ρ ,x ,v))
          [`(,v-res . ,pf-v)
           (cons v-res
                 `(⇓-let
                   ,pf-e
                   ,pf-v
                   -----
                   ((let ,x ,e ,eb) ,ρ ⇓ ,v-res)))])])]
    [`(plus ,e0 ,e1)
     (match-define `(,v0 . ,pf-v0) (eval e0 ρ))
     (match-define `(,v1 . ,pf-v1) (eval e1 ρ))
     (define v-res (nat-add v0 v1))
     (cons v-res
           `(⇓-plus
             ,pf-v0
             ,pf-v1
             (plus ----- (= (+ ,v0 ,v1) ,v-res))
             -----
             ((plus ,e0 ,e1) ,ρ ⇓ ,v-res)))]
    [`(not ,e)
     (match (eval e ρ)
       [`(0 . ,pf-e)
        (cons '(S O)
              `(⇓-not-1
                ,pf-e
                -----
                ((not ,e) ρ ⇓ (S O))))]
       [`(,v0 . ,pf-e)
        (cons 'O
              `(⇓-not-0
                ,pf-e
                -----
                ((not ,e) ρ ⇓ O)))])]
    [`(if0 ,e0 ,e1 ,e2)
     (match (eval e0 ρ)
       [`(O . ,pf-e0)
        (match (eval e1 ρ)
          [`(,v . ,pf-v)
           (cons v
                 `(⇓-if-true
                   ,pf-e0
                   ,pf-v
                   -----
                   ((if0 ,e0 ,e1 ,e2) ,ρ ⇓ ,v)))])]
       [`(,n . ,pf-e0)
        (match (eval e2 ρ)
          [`(,v . ,pf-v)
           (cons v
                 `(⇓-if-false
                   ,pf-e0
                   ,pf-v
                   -----
                   ((if0 ,e0 ,e1 ,e2) ,ρ ⇓ ,v)))])])]))

;; nice latex; comment out the labels by adding a % before \\tiny
(define (pf->tex pf)
  (define name (first pf))
  (define antecedents (reverse (list-tail (reverse (list-tail (cdr pf) 0)) 2)))
  (foldr
   (lambda (k v acc) (string-replace acc k v))
   (format "{{\\tiny \\textsc{ ~a }} \n \\frac{ ~a }{ \\texttt{ ~a } }}"
           name
           (string-join (map pf->tex antecedents) "\\,")
           (consequent pf))
   '("⇓" "∅" "↦")
   '("\\ensuremath{\\Downarrow}" "\\ensuremath{\\emptyset}" "\\ensuremath{\\mapsto}")))

(define (derive e)
  (displayln (pf->tex (cdr (eval e '∅)))))

;; examples

;(derive '(let x (plus 0 (if0 (plus 0 0) 1 0)) (plus x 0)))
;(derive '(plus 1 (if0 0 1 2)))
;(derive '(let x (plus 0 0) (plus x 1)))

(define (eval-simpl e ρ)
  (match e
    [(? nonnegative-integer? n) n]
    [(? symbol? x) (hash-ref ρ x)]
    [`(let ,x ,e ,eb)
     (eval-simpl eb (hash-set ρ x (eval-simpl e ρ)))]
    [`(plus ,e0 ,e1) (+ (eval-simpl e0 ρ) (eval-simpl e1 ρ))]
    [`(not ,e)
     (match (eval-simpl e ρ)
       [0 1]
       [_ 0])]
    [`(if0 ,e0 ,e1 ,e2)
     (match (eval-simpl e0 ρ)
       [0 (eval-simpl e1 ρ)]
       [_ (eval-simpl e2 ρ)])]))

Why I hope you'll submit to Scheme Workshop

Wed, 10 Apr 2019 00:00:00 -0400

Link to this year’s Scheme Workshop CFP

This year marks the twentieth anniversary of the workshop on Scheme and Functional Programming. The Scheme Workshop represents a diverse community of hackers, academics, and enthusiasts. The workshop offers a forum to share insights, experience, and technical developments of and within the Scheme family of programming languages. Rather than focus on a specific implementation or community, workshop attendees are united by an appreciation for succinct expression of novel ideas realized via programming. The submission deadline is May 24th, 2019, I hope you’ll help us celebrate by submitting a paper!

What is Scheme Workshop?

The Scheme Workshop website describes it as a…

Yearly meeting of programming language practitioners who share an aesthetic sense embodied by the Algorithmic Language Scheme: universality through minimalism, and flexibility through rigorous design.

Here’s my personal take. I love Scheme Workshop because it brings together a group of people with a truly unique perspective on programming. We represent a diverse background, but a shared vision: to distill insights about programming to their most economic presentation. We appreciate thoughtful, comprehensible solutions to challenging problems. We challenge ourselves to step outside of our boundaries and build fundamentally new paradigms for expressing ideas. Our mission is to build a community where we foster diversity, education, and support each other in pursuit of these ideals.

Scheme Workshop is unique in that it is:

Open to everyone who cares about Scheme-ish ideas. Instead of focusing on promoting the most cutting edge results, Scheme Workshop centers around thoughtful discussions relating to unique ideas that embody the ethos of Scheme. You don’t have to be an academic to attend; in fact, we have a rich history of participation from hobbyists and industrial attendees.
Non-archival and open access. We aim to support and foster unique ideas. Naturally, some of these ideas may grow into work eventually submitted to research conferences. Scheme Workshop provides a platform for researchers to get perspective from the Scheme community on their work at all stages of development. While we publish a technical report of submitted papers, submission to Scheme Workshop does not preclude subsequent submission to archival research conferences. In fact, we see Scheme Workshop as an ideal proving ground for groundbreaking research ideas. We hope that the discussions generated by the presentation of those ideas at Scheme Workshop will productively inform the authors’ scholarship.
Supportive of off-the-wall ideas. There are some ideas that are really interesting, kind of zany, and just inspire us in a deep way we can’t quite put our finger on. These ideas aren’t necessarily the kind that would fit well at a programming-oriented conference, but they might not be a good fit for research conferences either. Scheme Workshop is a perfect venue for these kinds of ideas. You’ll find like-minded people who enjoy challenging their perspectives on what programming even means.
A great place for your first paper. We place an emphasis on giving thoughtful, high-quality, and friendly reviews. We also have a strong appreciation for “half-baked” ideas. When I was a beginning PhD student, submitting to so-called top conferences often felt overwhelming. Scheme Workshop wants to help budding researchers develop their writing and presentation. Similarly, we realize that great ideas can come from people in industry, the open-source community, or the broader hobbyist community. We work hard to ensure that their ideas are taken seriously and give them the feedback they deserve.

What about Racket, Haskell, Clojure, SML, C++, etc…?

The language isn’t what’s important, it’s the idea. We want the workshop to be filled with exciting talks that broaden our horizons and challenge us to think about programming in a way we wouldn’t have before. I could easily imagine relevant papers dealing with template metaprogramming in C++, hygiene in OCaml, or domain-specific languages in Clojure.

If you think your idea might be relevant to the workshop, please don’t hesitate to contact Tom and myself. We’ll gladly give you an honest assessment of whether your idea would be a good fit. If not, we’ll do our best to direct you towards an appropriate venue for your ideas.

Who should submit?

Traditionally, submissions to Scheme Workshop come from those within or adjacent to academia. However, this is not a requirement. We welcome submissions from everyone who thinks they have something thoughtful to say to the Scheme community. This year, I want to specifically encourage submissions from:

Undergraduate or graduate students who are looking for practice writing papers on their own. It can be a challenge to develop your own sense of research style. Scheme Workshop is a great place to get feedback on your work in a low-stakes environment. Rather than judge the quality of your work against a high technical bar, we want to help build your ideas and presentation so that you can be proud of the way you’re articulating them.
People within the tech industry who have never written an academic paper before. Scheme Workshop leans a bit more academic than conferences such as RacketCon and StrangeLoop. However, this doesn’t mean we don’t value ideas from industry. If you’re in industry and would like to write a paper but aren’t sure how, please reach out to me: I’m happy to Skype with you and speak frankly about how the process works.
Seasoned Scheme wizards. Scheme Workshop has a long history of insightful talks from key contributors to the Scheme community. These community members shape our vision, and guide newcomers (such as myself!) by helping us understand the key principles that embody the Scheme community.

Some random Scheme Workshop papers I liked

There’s truly been some amazing work at Scheme Workshop over the years. I can’t hope to summarize it all here. Instead, I picked a random collection of papers to help you get an idea of the workshop’s direction.

A Self-Hosting Evaluator using HOAS, by Eli Barzilay. This paper discusses how–when you’re implementing a Scheme interpreter in Scheme–you can avoid implementing substitution by reflecting lambdas from the source language into the metalangauge (which still gives you means of abstracting, e.g., control structure). This is a neat idea that is the perfect size for Scheme Workshop.
Interaction-Safe State for the Web by Jay McCarthy and Shriram Krishnamurthi. This paper discusses new linguistic mechanisms for building abstractions for how we understand navigation within web applications. I think this idea nicely combines the practice of building web applications correctly with the linguistic abstraction mechanisms afforded by Scheme.
Well-typed programs can’t be blamed by Philip Wadler and Robby Findler. This paper from 2007 appears to be an earlier version of the highly influential 2009 ESOP paper of the same name.
Implementing Continuations Marks in JavaScript by John Clements, Ayswarya Sundaram, and David Herman. This paper discusses how continuation marks can be implemented in JavaScript. I see this as an exciting paper exploring a novel idea in a new space, and the authors mention some foundational challenges in scaling up to languages that allow tail calls with conventional “return”-style semantics.
A pattern matcher for miniKanren–or–How to get into trouble with CPS macros by Andy Keep, Michael Adams, Lindsey Kuper, Will Byrd, and Dan Friedman. This paper reveals an interesting ramification of writing macros in a CPS-based style alongside conditional macro expansion. They build a pattern matcher for the miniKanren programming language, demonstrating the issue along the way. I like that this paper reveals a foundational problem in macro engineering.

Help us build a community!

The goal of Scheme Workshop is to build a community that celebrates the fun in presenting beautifully minimal ideas. We often masquerade these ideas as programs. But we’re all bound together by the passion we share for articulating our ideas in an elegant way. Programming languages give us the ability to rapidly interact with our ideas, breaking and rebuilding our realization of those ideas at a pace never before seen by humanity. Scheme celebrates this opportunity by allowing us unprecedented flexibility in the way we articulate and execute these ideas.

I hope that you share my excitement. If you do, I sincerely hope you’ll consider submitting a paper to the twentieth workshop on Scheme and Functional Programming!