Alexander Feldman's Blog

Tag: applied science

  • Cranks of the First and Second Kind

    Draft. Thinking aloud. This is the beginning of a research statement, not the end of one.

    This morning, Nvidia committed five billion dollars to Safe Superintelligence, the company founded by Ilya Sutskever, in exchange for equity and a promise to multiply its compute tenfold within a year. SSI has, to date, shipped no product, published no research, and demonstrated no commercial traction. This is not a criticism of SSI — it is simply what the reporting says. Five billion dollars and a tenfold increase in compute for a company whose founder went on record last year to say that the age of scaling is ending. One admires the internal consistency of an industry that responds to “more compute will not be enough” by purchasing more compute. Nvidia’s own stock closed down about five percent on the day, which suggests the market has at least a rudimentary sense of irony even when the participants do not.

    Meanwhile Meta is spending seventeen billion dollars a quarter on capital expenditure and promising “personal superintelligence for everyone,” a phrase that has the same relationship to engineering that “subterranean opportunity space” has to a hole in the ground.

    Let us talk about cranks.

    A Taxonomy

    When I was younger and read physics for pleasure, there was a reliable genre of correspondence: the letter, usually typed, usually with underlining, announcing the discovery of a perpetual motion machine. The pleasure was not in the discovery — there was never a discovery — but in the classification. You read the argument, you found the error, and you filed the author.

    A crank of the first kind claims a machine that produces energy from nothing. He violates the first law of thermodynamics: conservation of energy. His diagram has a wheel with weights on it, and if you follow the weights around, they do not come back to where they started.

    A crank of the second kind is more sophisticated. He concedes conservation of energy. He does not claim to make energy; he claims only to extract it — from the ocean, from the ambient air, from the vast reservoir of heat that surrounds us and is admittedly free. He violates the second law. He has invented a machine that takes disorder and returns order, at no cost, and he has not noticed.

    The second kind is always the more interesting letter, because the error is deeper and the author is usually cleverer.

    I propose that the same taxonomy applies, exactly and usefully, to claims about computation. And I propose it in a spirit of genuine intellectual charity, because the classification is not an insult. It is a diagnostic procedure, and I have spent twenty years building diagnostic procedures.

    A crank of the first kind in computer science claims to solve NP-hard problems in polynomial time. He violates P ≠ NP. This is technically a conjecture rather than a law, but it is the best-supported conjecture in the subject, and if you propose to overturn it you had better bring a proof rather than a benchmark. These letters still arrive. They are honest, at least, about what they claim.

    A crank of the second kind concedes the complexity classes entirely. He agrees that travelling salesman is NP-hard, that circuit minimisation sits at the second level of the polynomial hierarchy, that model counting is #P-complete. He agrees to all of it, cheerfully, and then explains that his system does not need to solve the worst case, because it has learned the structure of real instances. It has absorbed, from a corpus, the shape of the answer. It will simply produce the correct output without searching for it.

    This is the machine that extracts work from a single heat bath. He has proposed to obtain a certificate — a guarantee, a proof, a thing you can check — from a process that produces no certificate, and he has not noticed the difference between an answer and an answer you can verify. He has confused a plausible output with a sound one. The energy is free, the ocean is vast, and the machine will run forever.

    There is now a third kind, unknown to classical thermodynamics, who does not claim to solve anything at all and raises five billion dollars regardless. He violates no law of physics. He violates only the conservation of shareholder value, and that quantity was never conserved to begin with.

    I will spend the rest of this post explaining why I believe the second-kind classification is correct, and what I intend to do about it, which is the actual purpose of the exercise.

    Where I Have Been

    Nowhere intellectual. I travelled, I did summer things, I saw people, I did not write. The blog went quiet after Delft and the Cerebras IPO and the AI café ordering eggs for a kitchen with no stove. I recommend the pause. Things settle.

    I am back, and I am back with a specific problem: I am defining projects, looking for gigs, and — most importantly — returning to research. Returning to research requires knowing what your research is, which requires writing it down, which is what this is. Consider it a first draft with the seams showing.

    The Universe Is an Optimisation Problem

    Start where physics starts. The principle of least action says that a system evolving from one configuration to another takes the path that makes the action stationary. Fermat’s principle says light takes the path of least time. Soap films find minimal surfaces. The Euler–Lagrange equations are what you get when you set a derivative to zero and solve.

    This is not a metaphor. The universe really does compute, and what it computes is the stationary point of a functional. Newtonian mechanics, electromagnetism, general relativity, the Standard Model — all of it can be written as “here is a Lagrangian, now find where the variation vanishes.” The universe is a very large, very fast, very well-conditioned optimiser, and the method it uses is the derivative.

    Which is to say: the universe does gradient descent.

    And this is precisely the point at which the argument turns, because the second observation follows immediately from the first:

    The universe does gradient descent, and therefore the universe does not solve NP-hard problems either.

    The evidence is everywhere, and it is embarrassing:

    • Soap films do not solve Steiner tree. The folklore claim — dip a frame in soapy water, get the optimal network — is false, and it has been tested rather than argued about. Films get stuck in local minima. They find a minimal surface, not the minimal surface, and which one depends on how you pulled the frame out of the bucket.
    • Proteins misfold. Levinthal’s paradox observes that a protein cannot be searching its conformation space, because the search would outlast the universe, and yet folding happens in microseconds. The resolution is that nature is not searching — it is rolling downhill on a funnel-shaped landscape. It works most of the time. When it does not work you get amyloid plaques and prions, which are what a local minimum looks like from the inside.
    • Spin glasses do not find their ground state. Determining the ground state of a three-dimensional Ising spin glass is NP-hard. A real piece of spin glass, cooled in a real refrigerator, does not find it. It ages, it hystereses, it gets stuck, and it stays stuck for geological time.

    The universe is a superb gradient descender and a hopeless combinatorial optimiser, and these are the same fact stated twice. This is why the “analog computer solves NP-complete problems” genre is a permanent fixture of the crank literature: it is the exact physical analogue of the perpetual motion machine, and it fails for a reason of the same character.

    Integers Are Where It Gets Hard

    I have spent an unreasonable fraction of my professional life on this, so permit me some specificity.

    Gradient descent needs three things: a continuous domain, a differentiable objective, and a landscape on which local information is informative. Given those, it is magnificent. Modern deep learning works not because anyone found a clever algorithm but because, in sufficiently high dimension and with sufficient overparameterisation, the loss landscape turns out to be benign — mostly saddle points rather than bad local minima, mostly connected basins. You take the derivative, you step, you repeat, and it works. Nobody fully knows why it works this well. It works.

    Now require the variables to be integers.

    Every one of the three things breaks at once. The domain is the vertices of a hypercube; there is no gradient on a vertex. The objective is defined only at those vertices; there is nothing to differentiate. And local information stops being informative — flipping one bit can move you anywhere.

    The standard rescue is relaxation: let your binary variables live in [0,1], solve the resulting continuous problem, and hope. This is the foundation of half of operations research and it is genuinely powerful. But notice what it gives you. It gives you a bound, not an answer. The distance between the relaxed optimum and the true integer optimum is the integrality gap, and closing it is the entire discipline: branch and bound, cutting planes, Lagrangian relaxation, column generation, symmetry breaking, forty years of the Mixed Integer Programming literature. The relaxation tells you where the answer cannot be. Finding where it is requires search.

    Nor can you simply round and call it a day. For a large class of problems, no polynomial-time procedure can get you close. Håstad’s result on MAX-3SAT says that beating a 7/8 approximation is itself NP-hard. This is not “we have not found the algorithm yet.” This is a theorem about what rounding can do.

    And where does the guarantee come from, when you have one? From duality and from exhaustion. LP duality gives you a certificate: a dual feasible solution that proves no better primal solution exists. Branch and bound gives you a certificate: a search tree, every branch of which was closed by a bound or a proof of infeasibility. A SAT solver that reports UNSAT can hand you a resolution refutation and you can check it independently with a program you wrote yourself in an afternoon.

    A gradient step produces no certificate. It cannot. It is a local move on a continuous surface, and its output is a slightly better point, not a proof about all other points. Ask it for a proof and it has nothing to give you, because proofs are not the kind of object it manufactures.

    Which Brings Us to the Chatbots

    The training objective of a large language model is cross-entropy on next-token prediction. Gradient descent descends that. There is no theorem — not a weak one, not a conditional one, not one anybody has attempted — connecting low cross-entropy loss to correctness of a mathematical argument. The loss went down. That is what has been established.

    There is also a structural point that deserves more attention than it gets. A transformer forward pass is a fixed-depth circuit. Fixed-depth, polynomial-size, threshold-gated circuits are the class TC0, and TC0 does not contain NP-hard problems unless a great deal of complexity theory is wrong. Each token costs a bounded amount of computation, and no amount of parameter count changes the depth.

    The escape hatch is chain of thought: use the context window as a tape and you recover, in principle, a general computation. Fine — but now you are running a Turing machine on a substrate with no error detection and no error correction, and every step is sampled rather than derived. This is not an incidental defect to be engineered away. Hallucination is what an unverified tape looks like. It is the same phenomenon as a soap film in a local minimum: the process ran, it terminated, it produced something, and nothing in the process was ever obliged to produce the right thing.

    So: chatbots will not do theorem proving. And here I want to be precise, because the imprecise version of this claim is itself crankery, and I would rather not be classified by my own taxonomy.

    What is actually happening in the Lean ecosystem right now is that language models propose and kernel checks. Models generate candidate formalisations and candidate tactic scripts; a small, auditable, deterministic proof kernel accepts or rejects them. Some real results have come out of this, including formalisations of things that were open. This is genuine and I have no interest in denying it.

    But look at where the guarantee lives. It lives in the kernel. It lives in a few thousand lines of code that implement a type theory and can be read by a human being in a week. The model is a proposal distribution — an extremely expensive, occasionally brilliant heuristic for generating candidates. It is playing the role that a branching heuristic plays inside a SAT solver: it makes the search faster and it makes no promises. Take the kernel away and you have a machine that emits confident LaTeX.

    The gradient proposes. The verifier disposes. Anyone who tells you the gradient is doing the proving has confused the two, and the confusion is exactly the second-kind error: mistaking a plausible output for a sound one, and expecting a certificate from a process that emits none.

    So What Am I Actually Doing

    Here is the research statement, in draft.

    The interesting problems live at the boundary between the continuous and the discrete, and almost nobody works there. The continuous side is well served — that is physics, that is simulation, that is the entire deep learning industry. The discrete side is well served by the SAT, SMT, QBF, and MIP communities, who are serious people doing serious work with certificates. The boundary is thinly populated and it is where the real systems are.

    My own work has been sitting on that boundary for twenty years without my having named it properly:

    • Model-based diagnosis is exactly this shape. The fault hypotheses are discrete — this component is healthy or it is not, and there are 2n of them. The behaviour is continuous — ODEs, DAEs, SPICE models, thermal gradients, feedback. Lydia-NG exists because Boolean models are wonderful for switching circuits and useless for anything physical. A hybrid diagnostic engine is a mixed-integer problem wearing a lab coat.
    • Exact synthesis — qbf-designer — is Σ2p-complete and gets exact answers where Vivado and Yosys get heuristic ones. Five LUTs where the heuristic finds six. The point is not that five is smaller than six. The point is that I can prove there is no four.
    • Diagnostic probability is a counting problem, and counting is harder than deciding. The posterior of a diagnosis is a ratio of model counts: how many assignments of the internal signals are consistent with the model, the observation, and this fault hypothesis, over the total. That is #P-complete. LyDiA got this wrong for a decade — it multiplied independent priors together and called the product a probability — and the modern llogic packages fix it. What I still cannot resolve is that every diagnostic probability is therefore a rational, a ratio of two integers, and physical failure rates are not obviously rationals at all. That one keeps me up.
    • Technology mapping for Verilog and VHDL is where I think the near-term commercial value is, and my Delft friends have graciously confirmed it is not entirely a hallucination.

    And the part I need to think harder about: where the learned heuristics legitimately belong. They belong inside the sound framework, not instead of it. Branching order. Variable selection. Restart policy. Which cut to add. Candidate proposal for a kernel to check. All of these are places where a model that is right 70% of the time makes a sound solver dramatically faster and never makes it wrong, because the soundness was never the model’s job. That is a research programme I can defend to a referee, and it is roughly the opposite of the one currently being funded at five billion dollars a pop.

    The universe does gradient descent, and it is very good at it, and it cannot factor a number. The industry has built a magnificent gradient descender and is now surprised that it cannot factor a number either.

    More soon, and with code. As always, everything I discuss is reproducible, which continues to distinguish it from most of what is published within a fifty-mile radius of here.

    Ceterum censeo slopem esse delendam.

    (Cato the Elder ended every speech in the Roman Senate with “Carthage must be destroyed” — regardless of the topic. This is that, but for AI slop.)

  • What’s Actually Broken

    Amazon’s weekly operations meeting in March reportedly focused on a “trend of incidents” characterised by “high blast radius” and “Gen-AI assisted changes.” The Financial Times, which saw the briefing note, reported that AI-generated code had been implicated in a series of outages — including one that took down Amazon’s entire e-commerce website for several hours. Amazon’s response was to deny the problem existed, which is the corporate equivalent of the AI itself: confidently wrong and hoping nobody checks. James Gosling, the creator of Java, who left AWS in 2024, was less diplomatic. He observed that the company’s AI-driven restructuring had “demolished” the teams responsible for infrastructure stability, and that the ROI analysis behind the decision was, in his words, “disastrously shortsighted.” One does not need a diagnostic engine to identify the fault here. A company replaced the engineers who understood its systems with a technology that does not, and the systems fell over. The circuit breaker that the AI removed — the one it classified as “redundant” — had been added after a previous outage. The AI could not distinguish a safety mechanism from dead code, because it had no model of the system. It had statistics. Statistics told it the breaker rarely fired. A model would have told it why.

    This is the difference between machine learning and model-based reasoning, and it is the difference that this post — and the toolchain I am releasing today — is about.

    An Unexpected Reception

    Yesterday’s post announcing qbf-designer, a tool for exact digital circuit synthesis via Quantified Boolean Formula solving, generated rather more attention than I had anticipated. Twenty-two thousand LinkedIn impressions, a hundred-odd reactions, and five hundred profile views in twenty-four hours, for a post about problems at the second level of the polynomial hierarchy and FPGA technology mapping. One concludes that there is an audience for work that produces correct answers, even — or perhaps especially — in an era when the prevailing technology cannot reliably tell you which end of a circuit is up.

    Dusting Off the Arsenal

    To continue with my plans for commercialising formal methods for EDA through Llama Logic Corporation, I have to excavate, modernise, and release the full inventory of tools and concepts I have built over nearly two decades. There are many reusable components in this stack — logic representations, solver bindings, encoding schemes, diagnostic algorithms — and they need to be cleaned up, documented, and made available. The qbf-designer release was the first. Today’s is the second.

    Today I am releasing LyDiA, a language and toolchain for Model-Based Diagnosis. LyDiA was the core of my doctoral research at Delft University of Technology. I will not be using LyDiA itself going forward — the modern llogic packages have fixed all of its imprecise notions and provide a cleaner foundation for everything I am building — but LyDiA was where it all started. It was my first serious work on the diagnosis of circuits, and it contains ideas and algorithms that remain relevant. It deserves to be available.

    Model-Based Diagnosis in 15 Seconds

    The demo takes two inputs. The model (2adder-weak.sys) describes a two-bit full adder — a hierarchical composition of half-adders built from XOR and AND gates. Every gate has a Boolean health variable: true means the gate works correctly, false means it is faulty and its output is unconstrained. We do not specify how a gate fails, only that its output can no longer be trusted. This is called a weak fault model.

    The observation (2adder.obs) records what actually happened: specific values on the inputs and outputs of the circuit that are inconsistent with correct behaviour. Something is broken. We do not know what. The diag command hands both files to the GOTCHA engine — which computes all minimal sets of component failures that explain the discrepancy. Not one guess. Not the most likely answer. Every combination of gate failures that is logically consistent with the model and the observation, with no redundancy.

    The fm command lists the results: six double-fault diagnoses, each a minimal set of gates whose simultaneous failure is sufficient to produce the observed misbehaviour. For example, d4 = { !FA.HA1.X.h, !FA.O.h } means the XOR gate in the first half-adder and the OR gate are both broken. There is no single-fault explanation — at least two gates must be faulty, and the engine has proven this by exhaustive enumeration.

    Why Circuits?

    Writing software to diagnose a fabricated IC does not make practical sense. You would use ATPG and scan chains for that. We use digital circuits as benchmarks because they have the properties that matter for diagnosis research: compositional structure, many components, well-defined fault models, and known-correct reference behaviour. These are the same properties that make diagnosis hard in complex engineered systems generally. This is why the ISCAS-85 suite has been the standard MBD benchmark for thirty years.

    Where diagnosis does apply directly in EDA is design verification. Suppose an engineer places a NAND gate instead of an AND gate for the carry computation in the adder above. The circuit passes some tests but fails on specific input vectors. The diagnostic engine, given the intended specification and the observed misbehaviour, will isolate the carry gate as the faulty component — even if the designer has never seen this particular mistake before, even if there are multiple simultaneous design errors. It reasons from the structure of the circuit, not from a database of past bugs.

    The Modelling Problem

    During my early attempts at commercialisation, I encountered a pattern that I suspect anyone in formal methods has seen. People looked at LyDiA diagnosing circuits and said: “Wonderful. Can it diagnose my HVAC system? My chemical plant? My supply chain?” And so they tried to model non-circuits as circuits, and things did not work, because the difficulty of modelling is the hard part.

    Circuit diagnosis is tractable in part because digital circuits have a natural, compositional, Boolean structure. An AND gate is an AND gate. An HVAC system is a tangle of continuous dynamics, feedback loops, thermal gradients, and human behaviour. Cramming that into a Boolean framework requires heroic abstraction, and the resulting models are either too coarse to be useful or too large to be solvable. The aerospace fuel system model included in LyDiA — with its typed fault modes for leaking tanks, stuck sensors, and degraded pumps — hints at what multi-valued modelling can achieve, but it remains a toy compared to the real thing.

    That said, LyDiA was never only about circuits. The distribution includes models of the N-queens problem, map colouring, Sudoku, and SEND+MORE=MONEY — general constraint satisfaction problems expressed in the same language. The diagnostic framework is, at its core, a constraint solver with a notion of health variables. This generality is both its strength and its curse: it can express anything, but making it useful for a specific domain requires domain expertise that no tool can substitute.

    What LyDiA Got Wrong: Probability

    LyDiA assigns fault probabilities to components — each gate gets a prior like 0.99 healthy, 0.01 faulty — but the probabilistic reasoning was never worked out correctly. The probabilities were treated as independent priors, multiplied together to rank diagnoses, with no rigorous account of how observations update beliefs or how correlations between faults propagate through the system.

    The correct formulation turns out to be a #P problem — a counting problem. To compute the exact posterior probability of a diagnosis, you need to count the satisfying assignments of the diagnostic formula: how many ways can the internal signals of the circuit be assigned such that the model, the observation, and a given fault assumption are all consistent? The probability of a diagnosis is the ratio of its satisfying assignment count to the total. This is model counting, and it is #P-complete — harder than NP.

    One consequence is that all diagnostic probabilities are rationals. They are ratios of integers — counts of discrete satisfying assignments. This has some puzzling implications for the relationship between fault probability and physical failure rates that I have not yet fully worked out.

    There is also a quantum angle. Faults are inherently stochastic — a gate either works or it does not, and before you test it, the fault state is indeterminate in precisely the sense that a qubit is indeterminate before measurement. I showed in earlier work that placing health qubits in superposition and propagating them through a quantum circuit that mirrors the classical circuit under diagnosis computes the full probability distribution over all diagnoses simultaneously. This connects to von Neumann’s foundational work on the relationship between logic and probability. The practical implication is Grover’s algorithm: a quadratic speedup for searching the diagnostic space. I need to finish this work and implement a proper Grover-based diagnostic engine. It is on the list.

    Why Machine Learning Cannot Do This

    In February, a company called Algorhythm Holdings — formerly a manufacturer of karaoke machines, with a market capitalisation of six million dollars — announced that its AI platform could “optimise” freight logistics, scaling volumes by 300–400% without adding staff. The announcement wiped seventeen billion dollars off U.S. transportation stocks in a single day. C.H. Robinson fell 15%. RXO fell 20%. The Russell 3000 Trucking Index dropped 6.6%. DHL, DSV, and Kuehne+Nagel followed in Europe. All of this because a former karaoke company claimed, in effect, to have solved optimal planning — a problem that is PSPACE-complete. If Alan Turing and Stephen Cook could be reached for comment, I suspect they would have questions.

    The same magical thinking pervades “AI for diagnostics.” A machine learning model trained on historical failures will recognise patterns it has seen before. Show it a novel fault — a combination that never appeared in the training data — and it has nothing to generalise from. It will either misclassify the failure or express high confidence in a wrong answer. This is not a limitation that more data or a larger model can fix. It is a structural property of inductive inference: you cannot learn what you have not observed, and complex systems fail in ways that are combinatorially vast and fundamentally unpredictable from examples alone.

    Model-based diagnosis does not have this problem. If you have a model of the system, you can diagnose faults you have never observed, in configurations you have never tested, because the reasoning is deductive rather than inductive. The SAT solver asks: is there an assignment of health variables that is consistent with the model and the observations? The answer is provably correct with respect to the model. This is why NASA uses model-based diagnosis for spacecraft and why the automotive industry uses it for on-board diagnostics. Nobody uses a neural network to diagnose a flight-critical system. The neural network might get it right 95 percent of the time. The other 5 percent is a smoking crater.

    What’s Next

    The modern diagnosis packages in llogic have addressed all of LyDiA’s imprecisions — cleaner encodings, correct probabilistic inference, proper multi-valued support — but those are a story for a separate post.

    There is also Lydia-NG, a framework I built that extends model-based diagnosis to analog systems using a built-in SPICE simulation engine. Rethinking Lydia-NG connects us directly to the analog side of EDA — a domain where formal methods have barely made an appearance and where the tools are, to put it charitably, showing their age.

    And that is the longer ambition. Cadence Virtuoso dates from 1991 — thirty-five years old. Vivado is newer (2012), but its place-and-route lineage descends from NeoCAD, acquired in 1995, and its synthesis from MINC, acquired in 1998. Synopsys Design Compiler has been around since the late 1980s. The EDA industry is running on architectural foundations that predate the web browser. These tools work — in the sense that a 1991 Toyota also works — but the algorithms inside them are heuristic, the interfaces are hostile, and nobody has rethought the fundamentals in decades.

    The goal of Llama Logic Corporation is to challenge this. Modern EDA with proper AI-augmented formal methods — analog, digital, and FPGA. New languages. New solvers. New tools. Not “AI for EDA” in the Silicon Valley sense of wrapping an LLM around Verilog and hoping for the best, but the real thing: algorithms with correctness guarantees, backed by the mathematical foundations that already exist and that the industry has been too comfortable to adopt.

    In the next instalment, I will demonstrate qbf-designer doing FPGA technology mapping — covering a small circuit with k-input Look-Up Tables. The formal methods stack is growing. The software works. It does not hallucinate.

    The repository: LyDiA — language and toolchain for Model-Based Diagnosis.

  • Diagnosing Circuits into Existence

    Cadence recently unveiled ChipStack AI, which El Reg memorably described as “vibe coding for chips.” The idea is that an LLM agent will design your next processor for you, provided you don’t mind the occasional hallucinated transistor. One Reg commenter recalled Jensen Huang’s declaration that nobody needs to learn programming anymore, and suggested he try designing his next GPU with it. Quite. Meanwhile, a water desalination company spent $200,000 on AI-generated engineering advice that turned out to be — and I use the technical term — wrong. They then built a second AI to filter out the nonsense from the first one, which is the silicon valley equivalent of hiring a second drunk to drive the first one home. One does wonder what the industry will achieve once it sobers up. In the meantime, I have been doing something unfashionable: using mathematics to design circuits that are provably correct.

    Today I am releasing qbf-designer, a tool for exact digital circuit synthesis from arbitrary component libraries via Quantified Boolean Formula solving. It is the top of a dependency stack that has also been modernised and released: llogic for logic representation, transformation, and solving; lcfgen for generating circuit primitives used both in the QBF encoding itself and as benchmark specifications; and a collection of solver bindings — pydepqbf, pylgl, pyllq, and pycudd — that connect the Python layer to the C/C++ solvers doing the actual heavy lifting. The software works. It synthesises provably minimal circuits from specifications. It found a five-gate full-subtractor that improves on the seven-gate textbook design. It does not hallucinate.

    Now, explaining what “provably minimal circuit design” actually means turns out to require rather more than a single blog post — so this is the first in a series. The short version: given a functional specification (“I want a circuit that adds two numbers”) and a bag of components (“here are some AND, OR, and XOR gates”), find the smallest circuit that does the job. The practical application is technology mapping for FPGAs, where you need to cover a circuit with k-input Look-Up Tables using as few LUTs as possible. The silicon is already on the chip and you have already paid for it — every LUT you save is space freed for more logic, letting you fit a larger design onto the same device. Current industry tools — Vivado, Yosys — use heuristics for this. qbf-designer gives you the exact answer, at least for sub-circuits small enough to chew on. Early results are promising: on a 2-bit comparator mapped to 3-LUTs, the solver finds a 5-LUT implementation where heuristic methods produce 6. One does not need to be a venture capitalist to notice that 5 is fewer than 6.

    There is a fundamental difference between this work and the “AI for chip design” circus currently touring Silicon Valley. Circuit synthesis — the problem of finding a minimum-size circuit equivalent to a specification — sits at the second level of the polynomial hierarchy (Σ₂ᵖ-complete, for those keeping score). This is not a problem you can solve by autocompleting Verilog. It has a precise computational complexity classification, a formal proof of correctness, and a guarantee of optimality. In other words, it is science, the kind that involves theorems rather than pitch decks, and where “it works” means something more rigorous than “the demo didn’t crash during the investor meeting.”

    My interest in circuit synthesis comes from an unexpected direction: breaking things. I spent years working on Model-Based Diagnosis of digital circuits — the problem of figuring out which component in a circuit has failed, given observed misbehaviour. My colleague Johan de Kleer, who has been thinking about this sort of thing since before most AI entrepreneurs were born, used to describe synthesis as “diagnosing a circuit into existence.” The idea is beautifully perverse: start with an empty circuit, treat the absence of every gate as a “fault,” and ask the diagnostic engine what collection of fixes would make the circuit behave like, say, a 32-bit ALU. It turns out that the mathematical machinery for diagnosis and synthesis is nearly identical — the same ∃∀ quantifier structure, the same miter-based equivalence checking, the same PSPACE-hard satisfaction problems. The only difference is whether you are looking for what went wrong or what should be there in the first place.

    The theoretical foundations and experimental results have been written up and submitted to Constraints, a journal that publishes work reviewed by people who can tell the difference between a proof and a press release. The paper covers the QBF encoding, the universal component cell, the configurable interconnection fabric, symmetry breaking, and extensive benchmarks on arithmetic circuits, 74XXX integrated circuits, and exact synthesis function sets. I mention this not to boast but to draw a gentle contrast with the prevailing approach to AI research in the Valley, where the peer review process consists of checking whether the blog post got enough likes on Twitter X and the replication methodology is “we lost the weights.” Should the reviewers find fault with the work, they will at least be able to point to a specific equation rather than gesturing vaguely at a loss curve and muttering about emergence.

    In the next instalment, I will demonstrate qbf-designer doing FPGA technology mapping — taking a circuit, mapping it to k-input Look-Up Tables, and producing a result that uses fewer LUTs than Xilinx Vivado. Not by a little. Not by accident. By mathematics.