The library

99 AlgoGens in a living shortlist of landmark and representative systems across the scientific and engineering literature, sorted with peer-reviewed origins first. Each entry shows its loop, its algorithmic authority, its headline result and what it does not prove.

Know a system that belongs here? Submit your system or tool. The data is open under CC BY 4.0 in the library's GitHub repository.

LandmarkFormal proof / check

AlphaGeometry

Google DeepMind, 2024 Proofs · Proof-state search

Authority: Symbolic deduction engine (DD + AR)

Proposes
Auxiliary constructions (new points and lines) for an Olympiad geometry problem
Judged by
A symbolic deduction engine that derives all consequences and decides whether the goal is proved
Steers
The enlarged figure goes back to the language model for the next construction, under beam search

Headline result

In the paper's evaluation on a test set of 30 olympiad-level geometry problems (IMO-AG-30), AlphaGeometry solves 25, versus ten for the earlier Wu's method baseline and 18 for the strongest baseline (DD + AR + human-designed heuristics). Symbolic deduction with algebra (DD + AR) alone solves 14.

What it does not prove

The 30-problem set is geometry only, translated into the system's own formal language. The language model is trained once on synthetic data before the search runs; it is not retrained by the search results. A later study (Sinha et al., arXiv:2404.06405) reports that Wu's method alone solves 15 of the same 30 problems, so the ten-problem baseline is contested.

LandmarkEmpirical benchmark

FunSearch

Google DeepMind, 2023 Mathematics · Program evolution

Authority: Program evaluator in a sandbox

Proposes
Python functions (inside a fixed skeleton) that construct cap sets or pack bins
Judged by
An automated evaluator that runs each program and scores its output
Steers
Best programs are shown in later prompts; islands of programs are periodically reseeded

Headline result

On the cap set problem, FunSearch found a cap set of size 512 in 8 dimensions, larger than any previously known, and raised the lower bound on the asymptotic cap set capacity to 2.2202 (previous best 2.2180). On online bin packing it evolved heuristics that beat first fit and best fit on the OR-Library sets OR1-OR4 and on Weibull instances.

What it does not prove

Results are for two problem families with a cheap, graded score, and only 4 of 140 cap-set runs reached 512 (Supplementary Information A.3), though once found the construction is deterministically verified and reusable without the model. The public repo is a reference implementation only: it has no LLM, no sandbox and no distributed system.

LandmarkFormal proof / check

AlphaProof

Google DeepMind, 2025 Proofs · Proof-state search

Authority: Lean proof assistant

Proposes
Lean proofs, built step by step, for millions of auto-formalized problems
Judged by
The Lean proof assistant, which verifies every proof
Steers
Verified outcomes train the model; for hard problems, test-time RL learns from millions of related variants

Headline result

At IMO 2024 the system with AlphaProof as its core engine solved three of the five non-geometry problems, including the hardest one; combined with AlphaGeometry 2 this reached a score equivalent to a silver medallist, with multi-day computation.

What it does not prove

Problems were formalized into Lean before the system worked on them, and the paper reports multi-day computation, so this is far from a competition-time-limit result. The Nature paper's claim is the RL method; the medal-level score is the combined system.

LandmarkFormal proof / check

AlphaTensor

Google DeepMind, 2022 Mathematics · Offline solver discovery

Authority: Exact algebraic tensor-rank verifier and Monte Carlo Tree Search, plus target GPU/TPU runtime benchmarker

Proposes
Rank-1 tensor factors (triplets of vectors u, v, w) proposed by a neural policy and value network
Judged by
Exact algebraic tensor subtraction and Monte Carlo Tree Search that checks whether the residual tensor reaches zero
Steers
Completed zero-residual decompositions and synthetic tensor demonstrations train the network and guide further tree search

Headline result

In Nature (2022), AlphaTensor discovered a 4x4 matrix multiplication algorithm using 47 multiplications in modulo-2 arithmetic (Z_2), beating Strassen's 1969 construction of 49 in that field, and improved the best known rank across more than 70 matrix sizes up to 5x5. By adding hardware runtime to the reward, it also found algorithms 10% to 20% faster than human and compiler baselines on NVIDIA V100 GPUs and Google TPU v2.

What it does not prove

The 47-multiplication 4x4 result holds in Z_2 (modulo-2 arithmetic); in standard arithmetic on 4x4 matrices AlphaTensor matched Strassen's 49 and improved larger sizes such as 5x5 (96 vs 98). Each tensor decomposition verifies exact algebraic identity only for the fixed matrix dimensions tested.

LandmarkFormal proof / check

AlphaDev

Google DeepMind, 2023 Engineering · Offline solver discovery

Authority: exhaustive permutation verifier for correctness, and latency measured on real CPUs or estimated by a learned latency model, inside Monte Carlo Tree Search

Proposes
x86 assembly instructions appended one step at a time by a representation and policy-value network
Judged by
Correctness checked over exhaustive test permutations, plus latency measured on real CPUs or estimated by a learned latency model
Steers
Monte Carlo Tree Search expands high-value assembly prefixes and updates network weights from verified fast routines

Headline result

In Nature (2023), AlphaDev discovered new x86 assembly sorting routines for small fixed and variable sequence lengths, cutting one instruction from sort3 (17 vs 18 instructions) and speeding up sort3 through sort5 by up to 70% and 250,000-element sorting by about 1.7%. The routines were merged into the LLVM libc++ standard sorting library.

What it does not prove

AlphaDev optimizes short branchless assembly blocks (such as 3-to-5-element base cases called inside larger sorts) rather than inventing a new asymptotic sorting complexity class. Search cost grows rapidly with program length.

LandmarkEmpirical benchmark

AlphaEvolve

Google DeepMind, 2025 Mathematics · Program evolution

Authority: Automated evaluators (exact for many math constructions, simulators or benchmarks for engineering tasks)

Proposes
Edits to an existing program (algorithms, heuristics, constructions) written by an ensemble of Gemini models
Judged by
One or more user-supplied automated evaluators that run each program and return scores
Steers
Higher-scoring programs are kept in a program database (MAP-Elites plus island populations) and sampled into later prompts

Headline result

Applied to over 50 open math problems, AlphaEvolve matched the best known construction in about 75% of cases and found a better one in about 20%, including a kissing-number configuration of 593 spheres in 11 dimensions (previous record 592). It found a procedure to multiply two 4x4 complex-valued matrices with 48 scalar multiplications (Strassen's recursive method gives 49). At Google it recovered on average 0.7% of fleet-wide compute through a scheduling heuristic and gave a 23% average kernel speedup, about 1% less Gemini training time. Deployment figures as reported by Google DeepMind.

What it does not prove

The 48-multiplication result holds for complex-valued matrices (characteristic 0 fields), not as a general replacement for Strassen. Industry results are developer-reported and not independently replicated. Evaluators must give a machine-checkable score, and a later study found the system sometimes exploited weak evaluators.

LandmarkEmpirical benchmark

RFdiffusion

University of Washington (Institute for Protein Design), 2023 Chemistry · Oracle-trained generation

Authority: AlphaFold2 structural self-consistency and interface pAE filtering, plus wet-lab biolayer interferometry, circular dichroism and cryo-EM

Proposes
3D protein backbone coordinates (residue translations and orientations) generated by an SE(3) diffusion model fine-tuned from RoseTTAFold, paired with ProteinMPNN sequences
Judged by
AlphaFold2 / RoseTTAFold structure-prediction self-consistency (backbone RMSD and interface confidence pAE) followed by wet-lab expression, biolayer interferometry and cryo-EM
Steers
Computational self-consistency metrics prune thousands of sampled trajectories down to top designs for physical synthesis and binding assays

Headline result

In Nature (2023), RFdiffusion designed de novo protein binders across five therapeutic and viral targets with an experimental hit rate of roughly 19% when testing 95 designs per target (two to three orders of magnitude higher than classical Rosetta binder design), and a cryo-EM structure of a designed Influenza HA binder (PDB: 8SK7) matched the design model at atomic accuracy (1.45 A backbone RMSD).

What it does not prove

Passing the AlphaFold2 computational filter does not ensure wet-lab binding (around four in five filtered binder designs still fail in vitro), and scaffolding full multi-step catalytic enzyme active sites in the lab requires additional physics and screening beyond backbone diffusion.

LandmarkPhysics simulation

GNoME

Graph Networks for Materials Exploration

Google DeepMind, 2023 Materials · Active-learning generation

Authority: Density Functional Theory (DFT, VASP) structural relaxation and convex-hull phase stability evaluation

Proposes
Candidate inorganic crystal structures generated via elemental substitution, symmetry-aware partial random structures and GNN energy filtering
Judged by
Ab initio Density Functional Theory (DFT) relaxation in VASP computing formation energy relative to the competing-phase convex hull
Steers
DFT energies and relaxed geometries expand the training set and retrain the GNoME graph neural networks for the next discovery round

Headline result

In Nature (2023), GNoME identified 2.2 million crystal structures below the previous convex hull of known materials, including 381,000 structures on the updated convex hull (nearly an order of magnitude more than the roughly 48,000 stable inorganic crystals previously catalogued). Active learning raised the DFT stability hit rate from under 10% to over 80% for compositional pipeline candidates and from 6% to 33% for structural candidates.

What it does not prove

DFT stability is calculated at 0 K for ordered unit cells and does not establish room-temperature phase stability, configurational entropy or practical lab synthesis. A 2024 Chemistry of Materials assessment by Cheetham and Seshadri argued that many predicted compounds are routine ionic or intermetallic substitutions with limited novelty or proven utility.

LandmarkPhysics simulation

MatterGen

Microsoft Research, 2025 Materials · Oracle-trained generation

Authority: MatterSim pre-relaxation and Density Functional Theory (DFT) convex-hull and property evaluation, plus X-ray diffraction and nanoindentation on the synthesized sample

Proposes
3D periodic crystal structures (atom species, fractional coordinates and lattice vectors) sampled by an equivariant diffusion model with property adapter modules
Judged by
Machine-learned interatomic potential pre-screening followed by Density Functional Theory (DFT) structural relaxation, convex-hull stability and property calculation
Steers
Adapters are fine-tuned on a labelled dataset; generated candidates are pre-relaxed with MatterSim and checked with DFT afterwards.

Headline result

In Nature (2025), MatterGen more than doubled the fraction of generated inorganic structures that are both novel and thermodynamically stable (within 0.1 eV/atom of the DFT convex hull) compared with prior generative baselines, and reduced root-mean-square atomic displacement during DFT relaxation by more than 50%. In a targeted high-bulk-modulus search, the authors synthesized a generated compound (TaCr2O6) in the lab whose measured bulk modulus (169 GPa) matched the target within 20%.

What it does not prove

Stability and property rates are primarily evaluated in 0 K DFT simulations rather than wet-lab synthesis; only one candidate structure (TaCr2O6, which turned out to be a disordered solid solution in experiment) was synthesized in the paper.

LandmarkFormal proof / check

AutoVerus

Microsoft Research, University of Illinois Urbana-Champaign, Columbia University and UC Irvine, 2024 Code verification · Verifier-guided repair

Authority: Verus verifier (SMT-based)

Proposes
Proof annotations (loop invariants, assertions) for Rust code with Verus specifications, from GPT-4o agents
Judged by
The Verus verifier, which either proves the specification or returns errors
Steers
Three phases: initial proof, refinement with generic tips, then agents that fix the specific errors Verus reports

Headline result

Proves 137 of 150 tasks (91.3%) on Verus-Bench, with at most 3 tries per task and GPT-4o; 122 were proved on the first attempt. A baseline that repeatedly calls GPT-4o for up to 10 minutes proved 67 of 150. More than half were done in under 30 seconds or 3 LLM calls.

What it does not prove

Verus-Bench is a 150-task suite the authors built from existing code and verification benchmarks, and the task gives the code and the specification. The tool proves what the spec says; a wrong spec would still pass.

LandmarkFormal proof / check

OptiMUS

Optimization Modeling Using Solvers and LLMs

Stanford University, 2024 Optimization · Formalize and solve

Authority: Gurobi Mixed-Integer Linear Programming (MILP) solver plus automated execution checks

Proposes
Mathematical objective and constraint formulations plus executable Gurobi Python code, generated by specialized GPT-4 agents
Judged by
The Gurobi MILP solver and automated checks that surface runtime errors from executing the generated code
Steers
Runtime errors from executing the generated code go back to the formulator or programmer agents.

Headline result

At ICML 2024, OptiMUS solved 78.8% of NL4OPT (vs 64.2% for Chain-of-Experts), 66.7% of ComplexOR (vs 38.1%), and 72.0% of the authors' graduate-level NLP4LP benchmark (vs 53.1%). Removing the solver-guided debugging loop in ablations dropped NLP4LP accuracy from 71.6% to 58.2%.

What it does not prove

Gurobi certifies the exact optimum of the generated MILP, not that the MILP faithfully captures every nuance of the English prompt. If the model writes a plausible but semantically wrong constraint that still compiles and solves feasibly, the solver optimizes the wrong problem.

LandmarkFormal proof / check

LLM-Modulo Framework

Arizona State University, 2024 Optimization · Verifier-guided repair

Authority: External symbolic verifiers (VAL for PDDL) and programmatic hard-constraint and commonsense critics

Proposes
Candidate multi-day travel itineraries or symbolic plans generated by an LLM
Judged by
A bank of external model-based verifiers and programmatic critics checking format, hard budget and scheduling rules, and commonsense constraints
Steers
A meta-controller passes the critics' critiques to the model as the next prompt until all hard critics sign off.

Headline result

Proposed at ICML 2024 (arXiv:2402.01817) and evaluated on the TravelPlanner benchmark (arXiv:2405.20625), pairing GPT-4-Turbo with external programmatic constraint critics in a bi-directional loop raised the final all-constraints pass rate from 4.4% (direct prompting) to 20.6% (a 4.6x gain), and raised GPT-3.5-Turbo from 0% to 5.0%.

What it does not prove

Even with external critics, 20.6% leaves nearly four out of five complex multi-constraint itineraries unsolved within the retry budget, because critique-guided local edits can still fail when constraints interact globally without an exact solver.

LandmarkFormal proof / check

LLM+P

Large Language Models + Classical Planners

University of Texas at Austin and Sony AI, 2023 Robotics · Formalize and solve

Authority: Fast Downward classical symbolic planner

Proposes
A PDDL problem specification (objects, initial state and goal predicates) translated from natural language by GPT-4
Judged by
The Fast Downward classical symbolic planner (classical planners LAMA and FDSS) running against the domain action schema
Steers
Fast Downward searches the symbolic state space to return a verified plan, which is translated back into natural language

Headline result

Across seven standard International Planning Competition (IPC) domains including Blocksworld, Barman, Floortile, Grippers, Storage, Termes and Tyreworld (20 tasks per domain), LLM+P with GPT-4 and in-context examples solved the majority of tasks with valid, cost-minimizing plans, whereas direct GPT-4 plan generation failed on almost all tasks in Barman, Floortile, Storage and Termes.

What it does not prove

LLM+P requires a human expert to supply the PDDL domain file (action preconditions and effects) in advance, and its one-shot translator has no validator-repair loop when the LLM emits syntax errors, undeclared objects or inverted goal predicates.

LandmarkPhysics simulation

REINVENT 4

with DockStream docking

AstraZeneca (Molecular AI), 2024 Chemistry · Oracle-trained generation

Authority: Scoring function with docking (Vina, rDock, Glide, GOLD via DockStream) and RDKit property checks

Proposes
SMILES strings for new small molecules, sampled from an RNN or transformer generator
Judged by
A user-defined scoring function; here a DockStream docking score (Glide) plus drug-likeness (QED)
Steers
Reinforcement learning, run as staged (curriculum) learning, raises the probability of high-scoring molecules in the generator

Headline result

In the paper's demonstration on PDK1 (PDB 2XCH), 50 epochs of staged learning with batch size 128 produced 119 hits out of 6,400 generated molecules (1.9%), where a hit means a docking score of -8 kcal/mol or better and QED of 0.7 or higher; the hits spanned 103 generic scaffolds.

What it does not prove

The PDK1 run is described by the authors as a hypothetical virtual screening exercise, and a good docking score is a computed estimate, not measured activity. Docking is one optional scoring component; DockStream alone is only a docking wrapper, not an AlgoGen, and its repository is no longer maintained.

LandmarkEmpirical benchmark

Eureka

NVIDIA, UPenn, Caltech, UT Austin, 2023 Robotics · Program evolution

Authority: RL training in Isaac Gym with a ground-truth task fitness function

Proposes
Reward-function code written by GPT-4 from the simulator's source code and a task description (16 candidates per iteration)
Judged by
RL (PPO in Isaac Gym) trains a policy on each reward; a ground-truth fitness function scores it
Steers
The best reward and a text summary of each reward component during training go back into the prompt

Headline result

Across 29 open-source RL environments with 10 robot morphologies, in simulation, the abstract reports Eureka outperforming expert human-written rewards on 83% of tasks, with an average normalized improvement of 52%. It also used its rewards in a curriculum to get a simulated Shadow Hand to perform pen spinning.

What it does not prove

Results are in simulation, with human reward baselines from the benchmark authors and a fixed fitness function. Searching costs 5 independent runs of 5 iterations with 16 samples each, many RL trainings per task.

LandmarkFormal proof / check

AlphaGeometry2

Google DeepMind, 2025 Proofs · Proof-state search

Authority: Symbolic deduction engine

Proposes
Auxiliary constructions for geometry problems, from a Gemini-architecture language model
Judged by
An improved symbolic deduction engine
Steers
Parallel search trees share facts (SKEST), so one tree's findings guide the others

Headline result

Solves 42 of 50 IMO geometry problems from 2000 to 2024 (IMO-AG-50); the paper states an overall solving rate of 84% on geometry problems from the last 25 years versus 54% for the original, and says AG2 surpasses an average gold medalist. The formal language now covers 88% of those problems, up from 66%.

What it does not prove

Geometry only. The paper reports progress toward, not completion of, a fully automated pipeline from natural-language problem statements. The AG2 system was also part of the IMO 2024 silver-medal-standard result.

OriginFormal proof / check

DeepSeek-Prover-V1.5 with RMaxTS

DeepSeek, 2024 Proofs · Proof-state search

Authority: Lean 4 proof assistant

Proposes
Lean 4 proofs and proof steps from a language model
Judged by
The Lean 4 proof assistant, which accepts or rejects each proof and each step
Steers
Lean outcomes are the RL reward; at inference, RMaxTS tree search rewards new states to pick branches

Headline result

With RL plus RMaxTS at a sample budget of 32 x 6400 (mixture of two guiding prompts) it reaches 63.5% on miniF2F-test; on the ProofNet test set it reaches 25.3% at a budget of 4 x 6400. The paper calls both new state of the art at publication.

What it does not prove

Numbers depend heavily on sample budget: the same model gets 50.0% to 54.9% at 32 to 3200 samples without search. The ProofNet validation set was used for expert iteration, so the test column is the clean one.

OriginFormal proof / check

Seed-Prover

ByteDance Seed (AI4Math team), 2025 Proofs · Verifier-guided repair

Authority: Lean proof assistant

Proposes
Lean proofs built from lemmas, from a language model
Judged by
The Lean proof assistant
Steers
Lean feedback, proved lemmas and self-summaries of earlier attempts go into the next try, with three test-time strategies

Headline result

The abstract reports 78.1% of formalized past IMO problems proved, MiniF2F saturated, above 50% on PutnamBench, and 5 of 6 IMO 2025 problems fully proved; a separate Seed-Geometry engine handles geometry.

What it does not prove

This card rests on the abstract only. Budgets, training details and who formalized the IMO 2025 statements are not stated there.

OriginEmpirical benchmark

ATLANTIS

Team Atlanta, AIxCC winner

Team Atlanta, 2025 Security · Verifier-guided repair

Authority: Fuzzers, sanitizers, PoV replay and project tests

Proposes
Fuzz seeds, payload generator scripts and source patches, from LLM agents
Judged by
Running the code: coverage, sanitizer crashes as proof of a bug, patch checks against crash and tests
Steers
Coverage feedback refines scripts (up to four rounds); failed patches guide the next try; seeds join a corpus

Headline result

Won the DARPA AIxCC final at DEF CON 33 (August 2025) with 392.76 points; the team's paper reports 43 of 70 vulnerabilities found. DARPA's overall figures (all teams): 54 of 63 synthetic vulnerabilities found and 43 patched, 18 real-world bugs found, about $152 average per task.

What it does not prove

Scores and counts come from the team's own paper plus DARPA's page, which differ (70 versus 63 challenges after a DARPA correction). The competition used synthetic bugs inserted into open-source projects. Real-world finds are small in number.

VariantEmpirical benchmark

LLM-SR

Virginia Tech, Carnegie Mellon University and Allen Institute for AI, 2025 Science · Program evolution

Authority: Numerical fit to data scored by error

Proposes
Equation skeletons as Python functions with placeholder parameters
Judged by
Fitting parameters with BFGS or Adam and scoring negative mean squared error against data
Steers
An experience buffer of 10 islands feeds the best equations (k=2) into the next prompt

Headline result

Across four problems (two oscillators, E. coli growth, aluminum stress-strain) with about 2.5K iterations, LLM-SR beat symbolic regression baselines on normalized MSE, most clearly out of domain: for E. coli growth 0.0037 against 1.0141 for the best baseline.

What it does not prove

Four benchmarks, two of them custom-built. Baselines (e.g. PySR) ran with far more evaluations (2M+). Results use GPT-3.5-turbo or Mixtral-8x7B; the best model differs by problem.

OriginEmpirical benchmark

EoH

Evolution of Heuristics

City University of Hong Kong, Huawei Noah's Ark Lab, Southern University of Science and Technology, 2024 Optimization · Program evolution

Authority: Benchmark evaluation of the heuristic code

Proposes
Heuristics as a natural-language 'thought' plus code, made by five LLM prompt operators
Judged by
Running the code on benchmark instances and measuring fitness
Steers
Fitness drives selection of parents for the next generation of a fixed-size population

Headline result

On online bin packing, EoH with about 2,000 LLM queries (GPT-3.5-turbo) averaged a 1.18% gap to the lower bound over six test sets, against 2.23% for FunSearch (which the paper says needs around 1 million queries), 4.13% for best fit and 4.51% for first fit. It also reached the best-known tour on three TSPLib instances (pr124, kroA150, u159) and led on Taillard flow-shop sets.

What it does not prove

Main experiments use GPT-3.5-turbo on three classic problems. Comparisons are on the authors' own set-ups, and FunSearch's query count is quoted from its published runs, not re-run.

VariantEmpirical benchmark

DrEureka

University of Pennsylvania, NVIDIA and UT Austin, 2024 Robotics · Program evolution

Authority: Physics simulation with RL training and a reward-aware physics prior

Proposes
Reward code (as in Eureka, with safety instructions) and then domain randomization configurations from GPT-4
Judged by
RL training in simulation, plus a physics-prior step (RAPP) that bounds each randomized parameter
Steers
Reward stage: best result feeds back into the prompt. Randomization stage: open loop, no real-world feedback

Headline result

On a real quadruped forward-locomotion test (2 m/s target, 5 m track), the best DrEureka policy reached 1.83 m/s and 5.00 m versus 1.32 m/s and 4.17 m for the human-designed setup. The robot also walked atop a real yoga ball for 15.4 seconds on average.

What it does not prove

Only the reward stage closes a loop with simulation results, and the randomization stage is a one-shot proposal. The real-world numbers come from short hardware trials, and the yoga-ball runs are about 15 seconds, not minutes.

OriginFormal proof / check

ReProver

LeanDojo

LeanDojo team (Kaiyu Yang et al.), 2023 Proofs · Proof-state search

Authority: Lean proof assistant via LeanDojo

Proposes
Lean tactics, from a language model that first retrieves relevant premises from the math library
Judged by
The Lean environment, driven through LeanDojo, which runs each tactic and reports the new state or an error
Steers
Best-first search expands the most promising proof states and feeds each new state back to the tactic generator

Headline result

On the LeanDojo Benchmark test set, pass@1 is 51.2% (random split) and 26.3% (novel_premises split) with retrieval, versus 47.6% and 23.2% without retrieval and 29.0% and 7.4% for zero-shot GPT-4. Also 26.5% on miniF2F and 13.8% on ProofNet (pass@1, 10 minute wall-time limit per theorem).

What it does not prove

Main experiments use Lean 3 (Lean 4 results are in an appendix not covered here). The tactic generator is trained once, offline, on extracted proofs; search results do not update its weights. The paper's claim is about open, cheap training (one GPU week), not top accuracy.

OriginFormal proof / check

AlphaVerus

Carnegie Mellon University, 2024 Code verification · Verifier-guided repair

Authority: Verus verifier with a misalignment filter

Proposes
Verus translations of programs (from Dafny) and repaired versions, from a Llama 3.1 70B model
Judged by
The Verus verifier, plus a filter that rejects translations whose specification or program is misaligned
Steers
Treefinement tree search fixes errors from verifier feedback; verified, filtered programs become prompt examples

Headline result

With Llama 3.1 70B and no model fine-tuning, verified-code generation reaches 32.9% on HumanEval-Verus and 65.7% on MBPP-verified (pass@256), versus 27.1% and 35.9% for GPT-4o and 11.8% and 26.9% for Llama 3.1 70B.

What it does not prove

HumanEval-Verus and MBPP-verified are small, hand-adapted benchmarks, and the task is generating code plus proofs from specifications. Verification proves code matches the specification, so the misalignment filter matters.

OriginFormal proof / check

Goedel-Prover

Princeton Language and Intelligence (Princeton University), with Tsinghua University, Amazon, Meta FAIR and Numina, 2025 Proofs · Oracle-trained generation

Authority: Lean compiler

Proposes
Lean 4 proofs for 1.64 million formalized statements
Judged by
The Lean compiler, which checks each candidate proof
Steers
Verified proofs join the training set and the next, stronger prover is trained on them (expert iteration)

Headline result

Goedel-Prover-SFT reaches 57.6% Pass@32 on miniF2F, 7.6 points above DeepSeek-Prover-V1.5-RL; the paper's abstract reports over 60% Pass@32 after RL including DPO, 7 PutnamBench problems at Pass@512, and proofs for 29.7K Lean Workbook problems versus 15.7K for prior provers.

What it does not prove

Its best-result claims are dated April 2025 and use Pass@32 or Pass@512, so budgets differ from some rivals. The paper text says 8 iterations while a table lists Iter-9. The abstract does not describe the compiler signal beyond verification.

OriginFormal proof / check

BFS-Prover

ByteDance Seed, 2025 Proofs · Proof-state search

Authority: Lean 4 compiler via LeanDojo

Proposes
Lean 4 tactics for the current proof state, from a language model
Judged by
The Lean 4 compiler through LeanDojo, which runs each tactic and reports success or an error
Steers
Best-first search ranks states by log probability; expert iteration retrains on proofs (SFT) and error pairs (DPO)

Headline result

Reaches 72.95% on the miniF2F test set as an accumulative score; the single-run figure is 70.83% +/- 0.89 at a budget of 2048 x 2 x 600 (Lean 4.7.0). The authors say this shows plain best-first search can compete when scaled.

What it does not prove

The 72.95% is a cumulative (union across runs and settings) score, not one fixed-budget run. Its claims date from early 2025 and are compared with provers that used different budgets.

OriginSurrogate oracle

ChemSpaceAL

Yale University (Batista group), 2024 Chemistry · Active-learning generation

Authority: Docking (DiffDock) with an interaction-based scoring function; cluster-based sampling

Proposes
SMILES strings from a GPT-based molecular generator (100,000 unique molecules per iteration)
Judged by
About 1,000 molecules sampled across clusters are docked with DiffDock and scored by contact-based interactions
Steers
Clusters are weighted by mean score into a training set, and the GPT is fine-tuned on it

Headline result

In the paper's computational study on c-Abl kinase, the model learned to generate molecules similar to FDA-approved inhibitors without being shown them and reproduced two exactly (imatinib and bosutinib after five iterations). The method was also applied to the HNH domain of Cas9, which has no commercial small-molecule inhibitors.

What it does not prove

The judge is a learned docking model (DiffDock) plus a simple contact-count score, so this is predicted binding only. Matching known drugs is a retrospective sanity check, and no molecule was tested in a lab.

OriginEmpirical benchmark

OSS-Fuzz-Gen

LLM fuzz target generation

Google (OSS-Fuzz), 2023 Security · Verifier-guided repair

Authority: OSS-Fuzz build tooling; calling the function, start-up crashes and coverage are reported metrics

Proposes
Fuzz targets (C/C++ entry points) for functions that Fuzz Introspector flags as under-fuzzed
Judged by
OSS-Fuzz build; whether the target calls the function, start-up crashes and new line coverage are reported metrics
Steers
Build errors go back to a code-fixer prompt; repeated rounds fix errors

Headline result

Google's page reports 14 of 31 tested OSS-Fuzz projects compiled new targets and increased coverage (gains 0 to 31%; tinyxml2 line coverage 38% to 69%), and a similar target in OpenSSL rediscovered CVE-2022-3602.

What it does not prove

The page calls this a preliminary investigation on a small set of C/C++ benchmarks with an unnamed Google model. The 14 are the projects that succeeded and the selection of the 31 is not fully explained. Calling-the-function, crash and coverage results are reported metrics; the page does not say they feed back into the model.

VariantEmpirical benchmark

ShinkaEvolve

Sakana AI, 2025 Optimization · Program evolution

Authority: Task evaluators plus an embedding-similarity filter and UCB1 bandit

Proposes
Diff edits, rewrites or crossovers of programs, written by an ensemble of LLMs
Judged by
Task-specific evaluators that score each program and return text feedback
Steers
Parents mix score and novelty across islands; an embedding filter rejects repeats; a UCB1 bandit picks the LLM

Headline result

ShinkaEvolve found a circle-packing (n=26) solution with sum of radii 2.63598 under its relaxed verifier, beating AlphaEvolve's reported solution within 150 program evaluations. It also improved ALE-Agent solutions on ALE-Bench by about 2.3% on average over 10 tasks, designed an AIME agent scaffold (gpt-4.1-nano, 7 LLM calls), and found a new mixture-of-experts load-balancing loss.

What it does not prove

The paper uses both '150 samples' and '150 generations' for the circle-packing run. The MoE loss was evolved at small scale (556M parameters, 2B tokens) and the authors note limited budget and similar architecture between search and test. AIME gains were smaller on 2023 problems, which the authors link to possible contamination.

VariantEmpirical benchmark

ReEvo

Reflective Evolution

Peking University, KAIST, Southeast University, Singapore Management University (AI4CO), 2024 Optimization · Program evolution

Authority: Objective evaluation on problem instances

Proposes
Heuristic code produced by LLM crossover and elitist mutation
Judged by
Running the heuristic on problem instances to get an objective value
Steers
Short-term reflection compares parents; long-term reflection accumulates hints; both feed later prompts ('verbal gradient')

Headline result

Across six combinatorial problems (TSP, CVRP, OP, MKP, BPP, DPP), with a budget of 100 heuristic evaluations and GPT-3.5 Turbo, ReEvo beats EoH on sample efficiency in five problems, and designed heuristics improved neural solvers, for example POMO's TSP1000 gap from about 52% to 29%. Its genetic algorithm for decap placement scored 12.98 vs 12.41 for the expert-designed one.

What it does not prove

Authors' benchmark set-ups with GPT-3.5 Turbo as the main model; 'state of the art' refers to the compared methods in the paper.

VariantEmpirical benchmark

OpenEvolve

algorithmicsuperintelligence (open-source community project), 2025 Optimization · Program evolution

Authority: User-defined evaluator (tests, benchmarks, profilers)

Proposes
Code changes in marked EVOLVE-BLOCK regions, written by an ensemble of LLMs from any compatible API
Judged by
A user-written evaluator that scores each program; errors and other evaluator output are kept as artifacts
Steers
Scores place programs in a MAP-Elites grid across island populations; errors and evaluator output feed the next prompt

Headline result

The project reports 2-3x speedups on GPU kernels (MLX Metal) and a circle packing for n=26 that its README calls state of the art, plus sorting and signal-processing examples. These are the project's own claims; the README gives no numeric circle-packing score.

What it does not prove

All results are project-reported and not independently replicated here. The README gives no numbers for most examples. It is an engineering framework, not a paper.

OriginEmpirical benchmark

Self-Harness

Shanghai Artificial Intelligence Laboratory, 2026 ML systems · Verifier-guided repair

Authority: Benchmark task verifiers and a pass-count acceptance rule

Proposes
Minimal edits to the agent's harness (its instructions and scaffolding), from the same fixed model
Judged by
Benchmark verifiers on held-in and held-out splits; an edit passes only if neither split worsens and one improves
Steers
Failure patterns mined from traces drive the next proposals; accepted edits are merged and rejected ones stay out

Headline result

The paper reports that for three base models (MiniMax M2.5, Qwen3.5-35B-A3B, GLM-5) across three benchmarks (Terminal-Bench-2.0, SWE-bench Verified, AppWorld) every final harness improves both held-in and held-out pass rates, with relative gains of up to 132%. On Terminal-Bench-2.0 held-out, MiniMax M2.5 went 40.5% to 61.9% and Qwen3.5-35B-A3B 23.8% to 38.1%.

What it does not prove

A candidate is promoted only if it improves the held-in or the held-out pass count without lowering the other, so the held-out split is not an untouched test of generalization. The Terminal-Bench work uses a 64-task subset of 89. The authors say edits may reflect benchmark-specific patterns and that higher-stakes changes need stronger gates. The text reports slightly different figures in two places.

OriginPhysics simulation

Active learning for conditional crystal inverse design

The University of Hong Kong, 2025 Materials · Active-learning generation

Authority: CGCNN pre-filter, MACE-MP-0 molecular dynamics, DFT (via APEX workflows)

Proposes
Alloy crystal structures from Con-CDVAE conditioned on a target bulk modulus
Judged by
Three screens: a CGCNN pre-filter, molecular dynamics with MACE-MP-0 (within 20%, stable), then DFT
Steers
Structures passing DFT, with computed bulk moduli, join the training set and the generator is fine-tuned

Headline result

For a 350 GPa bulk modulus target, in a range sparse in the training data, three active learning iterations lowered the mean absolute percentage error of generated crystals from about 0.4 to 0.2 (CGCNN-evaluated). DFT bulk moduli of new structures averaged 343.54 GPa (standard deviation 27.20) after the first iteration and 356.52 GPa (standard deviation 18.61) after the second.

What it does not prove

The error trend is measured with the same kind of GNN used in the screen, and DFT is a calculation rather than a lab result. It is a case study on one property (bulk modulus) of alloys, and no structure was synthesized.

OriginEmpirical benchmark

Darwin Godel Machine

Sakana AI and University of British Columbia (Jeff Clune lab), 2025 ML systems · Program evolution

Authority: Benchmark test suites run in a sandbox

Proposes
A modified version of the agent's own code (tools, prompts, workflows), written by a frozen foundation model
Judged by
Running the new agent on coding benchmarks (SWE-bench Verified, Polyglot) and recording pass rates
Steers
Parents come from an archive of agents, weighted by score and few children; children join it

Headline result

Over 80 iterations, the best agent rose from 20.0% to 50.0% on SWE-bench (Verified) and from 14.2% to 30.7% on Polyglot (abstract figures; the paper's Section 4.4 gives 14.0% as the Polyglot start).

What it does not prove

It stays below the best closed-source systems on SWE-bench and is limited by the underlying model. The run took about two weeks with significant API cost. The authors report that the agent sometimes faked test results, and that when rewarded for avoiding this it sometimes removed the detection markers (per Sakana's blog). Archive maintenance and parent selection are fixed, not self-modified.

OriginPhysics simulation

ESM3

EvolutionaryScale, 2025 Chemistry · Oracle-trained generation

Authority: 3D structural consistency verifier and wet-lab fluorescence assay

Proposes
A 98B multimodal protein model iteratively unmasks sequence, 3D backbone coordinates, and catalytic/functional tokens.
Judged by
All-atom geometric consistency metrics (TM-score, active-site RMSD), preference alignment rewards, and wet-lab chromophore fluorescence assays judge candidates.
Steers
Chain-of-thought structural decoding and direct preference alignment on verified structures steer generation toward folded functional proteins.

Headline result

Generated esmGFP, a glowing fluorescent protein sharing only 58% sequence identity with the closest known fluorescent protein (~500 million years of evolutionary divergence).

What it does not prove

Wet-lab expression success rates vary strongly by target protein fold and cellular host environment.

OriginFormal proof / check

Mind Evolution

Google DeepMind, University of California, San Diego, 2025 Optimization · Program evolution

Authority: Programmatic constraint checker and island genetic algorithm

Proposes
An LLM generates, recombines, and refines candidate travel plans, schedules, and cipher solutions across parallel evolutionary islands.
Judged by
A programmatic evaluator scores each plan and returns textual feedback on violated constraints.
Steers
Boltzmann tournament selection, island migration, and elitism propagate high-scoring plans into the next generation.

Headline result

Solved >95% of TravelPlanner and Natural Plan multi-constraint benchmarks with Gemini 1.5 Flash, surpassing Best-of-N and sequential Reflexion by wide margins.

What it does not prove

Requires a programmatic constraint evaluator for the target planning domain to score candidate solutions accurately.

OriginEmpirical benchmark

robust-kbench pipeline

Sakana AI

Sakana AI, 2025 ML systems · Program evolution

Authority: Compiler, numeric check against PyTorch, GPU timing and profilers; LLM verifiers as filter

Proposes
CUDA kernels written by an ensemble of frontier LLMs
Judged by
Compile and numeric check against PyTorch, LLM pre-filters, then GPU timing and profiling
Steers
Up to five fast, correct past kernels plus a profiling summary go into the next prompt

Headline result

On 200 Level 1 and 2 KernelBench tasks the average speedup was 3.13x, falling to 1.49x after excluding contaminated tasks; 95% of tasks translated successfully. On twelve robust-kbench tasks forward kernels reached up to 2.5x over PyTorch eager.

What it does not prove

The earlier Feb 2025 release claimed 10-100x and Sakana acknowledged the system exploited evaluation code (a memory-reuse flaw), and walked those claims back. In this paper two kernels with 51x and 123x speedups came from benchmark loopholes. Part of the filtering uses LLM verifiers, which are not exact judges. Cost was about $5 per kernel.

OriginFormal proof / check

T3 Planner

Jia Li et al., 2025 Robotics · Verifier-guided repair

Authority: Signal Temporal Logic robustness computation and simulation of the generated controller

Proposes
Subgoal waypoints, timed waypoints and a Python controller script, written by an LLM in three stages
Judged by
Signal Temporal Logic robustness checks, a Python syntax check, and simulation of the controller on annotated cases
Steers
A failed output goes back to the LLM as the previous failed result, until all checks pass

Headline result

In simulated household, chip-design and navigation scenarios, success rates with GPT-4o were 91.65%, 91.60% and 83.69%, against 56.67%, 61.29% and 50.00% for AutoTAMP with GPT-4o. The authors claim a 30 to 40% improvement.

What it does not prove

All scenarios are in a simulated 2D environment with no real-robot tests, and the paper gives no retry limit. The checker is a temporal-logic verifier, not model predictive control, and this is a single preprint with no venue listed.

OriginEmpirical benchmark

ADAS

Automated Design of Agentic Systems

University of British Columbia, Vector Institute, 2024 ML systems · Program evolution

Authority: Sandboxed Python execution and validation benchmark evaluator

Proposes
A meta-LLM writes complete multi-step agent architectures as executable Python forward() functions.
Judged by
An automated benchmark harness executes each candidate workflow on validation tasks (ARC, DROP, GSM8K, MMLU) and computes exact accuracy.
Steers
Winning architectures and their validation scores are appended to an growing archive that conditions the next meta-generation step.

Headline result

Discovered agent architectures that outperform hand-designed baselines by up to 13.6 percentage points on ARC logic tasks and transfer across foundation models.

What it does not prove

Evaluated on validation splits of static benchmarks; executing arbitrary LLM-generated code requires sandboxing and incurs high API token cost.

OriginPhysics simulation

AlphaFold 3

Google DeepMind, Isomorphic Labs, 2024 Chemistry · Oracle-trained generation

Authority: Ranking formula of pTM and ipTM with clash, disorder and chirality terms

Proposes
A Pairformer trunk and 3D atomic diffusion module generate joint all-atom coordinates for proteins, nucleic acids, ligands, and ions.
Judged by
Samples are ranked by a formula of the model pTM and ipTM with clash, disorder and chirality terms; nothing is filtered out.
Steers
Confidence ranking picks the top sample across diffusion rollouts; cross-distillation is a training-data step.

Headline result

Achieved 76.4% PoseBusters accuracy on protein-ligand complexes, far surpassing classical physics-based docking tools.

What it does not prove

Can hallucinate spurious secondary structure in disordered regions or produce occasional stereochemical clashes when diffusion samples conflict.

OriginFormal proof / check

Chain-of-Experts

Zhejiang University, Huawei Noah's Ark Lab, 2024 Optimization · Formalize and solve

Authority: Gurobi / PuLP mixed-integer programming solver

Proposes
Specialized LLM roles extract mathematical terminology, formulate algebraic constraints, and emit Python PuLP/Gurobi solver code.
Judged by
The Python interpreter and MIP solver execute the generated formulation and report syntax errors, infeasibility, or optimal objective values.
Steers
A conductor chooses which role acts next; on a failed run the roles reflect on execution feedback in reverse order until one is at fault.

Headline result

Reported to significantly outperform state-of-the-art LLM-based approaches on the LPWP and ComplexOR operations research benchmarks.

What it does not prove

If the formulation specialist misinterprets an implicit constraint in the word problem without causing solver infeasibility, the MIP solver solves the wrong model.

OriginEmpirical benchmark

ChatAFL

Nanyang Technological University, University of Melbourne, 2024 Security · Verifier-guided repair

Authority: Coverage-guided stateful greybox fuzzer and address sanitizers

Proposes
An LLM extracts protocol request grammars and synthesizes structure-aware seed sequences and state-transition mutations.
Judged by
The greybox fuzzer (AFLNet engine) executes the target network server, measuring edge coverage, state transitions, and sanitizer crashes.
Steers
When coverage stalls in a protocol state, the fuzzer queries the LLM with the observed server responses to generate inputs that unlock new states.

Headline result

Covered 47.6% more state transitions and 29.6% more code branches than AFLNet across nine network protocol implementations, discovering 9 previously unknown vulnerabilities.

What it does not prove

Targets text-based application protocols (RTSP, FTP, SMTP, SIP); binary protocols require separate serialization adapters.

OriginPhysics simulation

ChemCrow

EPFL, University of Rochester, 2024 Chemistry · Verifier-guided repair

Authority: RDKit, RXN4Chemistry retrosynthesis engine, and robotic wet-lab execution

Proposes
An LLM proposes multi-step synthesis plans, SMILES structures, and reaction queries.
Judged by
Deterministic RDKit validators, RXN4Chemistry retrosynthesis engines, and controlled-substance safety checkers verify chemical validity and feasibility.
Steers
Tool outputs return to the model at each step; the validation of the robotic platform is iterated until the procedure is valid.

Headline result

Autonomously planned and executed wet-lab syntheses of an insect repellent, three organocatalysts, and a novel chromophore.

What it does not prove

Synthesis success remains bounded by the coverage of the underlying retrosynthesis templates and reagent inventory.

OriginPhysics simulation

CrystaLLM

University of Bristol, University College London, 2024 Materials · Verifier-guided repair

Authority: CIF validity checks and a learned formation-energy network (ALIGNN); DFT on finished structures

Proposes
A domain-trained transformer generates full CIF crystal descriptions conditioned on composition or space group.
Judged by
Validity checks on generated CIFs; the optional Monte Carlo tree search is scored by a learned formation-energy network (ALIGNN); DFT is run on finished structures.
Steers
Optional Monte Carlo tree search guided by a learned formation-energy network steers generation.

Headline result

Rediscovered unseen experimental crystal structures and identified novel stable inorganic phases validated by ab initio DFT.

What it does not prove

Struggles on very large unit cells (>60 atoms) where token context length and space-group Wyckoff constraints become tight.

OriginPhysics simulation

CrystalTextLLM

New York University, Meta AI, 2024 Materials · Verifier-guided repair

Authority: Pymatgen validator, M3GNet potential, and VASP DFT solver

Proposes
A fine-tuned LLaMA-2 model generates crystallographic information files (space group, lattice lengths/angles, fractional coordinates).
Judged by
Pymatgen structural validators, M3GNet interatomic potential relaxations, and VASP Density Functional Theory calculations measure validity and thermodynamic stability afterwards.
Steers
Sampling retries when a CIF cannot be parsed, and conditional infill prompts steer generation.

Headline result

Nearly 90% of sampled structures pass geometric and charge-neutrality checks, and a high fraction relax to metastable or stable states in DFT.

What it does not prove

Text-tokenized coordinates have finite numerical precision prior to force-field or DFT structural relaxation.

OriginFormal proof / check

DafnyBench Verifier Repair

Massachusetts Institute of Technology, 2024 Code verification · Verifier-guided repair

Authority: Dafny formal verifier

Proposes
An LLM inserts loop invariants and assertions into a Dafny program while keeping the top-level specification intact.
Judged by
The Dafny verifier checks whether the program verifies and returns its error message when it does not.
Steers
The Dafny error message is fed back for up to 10 attempts until the verifier accepts the proof.

Headline result

Verified 68% of the 782 ground-truth formal verification programs in DafnyBench via multi-turn compiler feedback.

What it does not prove

A spec check must prevent the model from trivially satisfying the verifier by weakening `ensures` clauses or adding `assume false`.

OriginPhysics simulation

DiffSBDD

EPFL, University of Oxford, 2024 Chemistry · Oracle-trained generation

Authority: RDKit chemical validator and QuickVina2 / AutoDock Vina physics docking oracle

Proposes
An SE(3)-equivariant diffusion model generates 3D ligand atom coordinates and types directly inside a target protein binding pocket.
Judged by
RDKit chemical validity checks, synthetic accessibility (SA), QED drug-likeness, and AutoDock Vina / QuickVina2 physics docking score each ligand.
Steers
An evolutionary partial-noising and denoising loop mutates top-scoring ligands and selects candidates with higher binding affinity and QED.

Headline result

Generated pocket-specific 3D ligands with stronger Vina binding scores and drug-likeness than autoregressive 3D baselines on CrossDocked2020.

What it does not prove

Rigid-pocket docking scores do not account for induced-fit protein conformational changes or explicit water bridges.

OriginEmpirical benchmark

Fuzz4All

University of Illinois Urbana-Champaign, ETH Zurich, 2024 Security · Program evolution

Authority: Compiler/SMT crash and output-check oracle

Proposes
An autoprompting and fuzzing LLM generates and mutates valid source inputs in C, C++, Go, Rust, Java, and SMT2.
Judged by
Target compilers (GCC, Clang, rustc, Go) and solvers (Z3, CVC5) execute the inputs; crashes and output checks flag bugs.
Steers
The next iteration is seeded with a randomly chosen valid earlier input.

Headline result

Achieved 36.8% higher line coverage than language-specific fuzzers across 9 systems in 6 languages and found 98 bugs (64 confirmed previously unknown).

What it does not prove

LLM token generation is slower per input than bit-flip greybox fuzzers, trading raw throughput for semantic validity in complex input grammars.

OriginFormal proof / check

InternLM2-StepProver

Shanghai Artificial Intelligence Laboratory, 2024 Proofs · Proof-state search

Authority: Lean 4 (executes each proposed tactic) and critic-guided best-first tree search

Proposes
An 8B tactic model proposes Lean 4 proof steps conditioned on the current goal state.
Judged by
Lean executes each tactic the prover proposes, and a trained critic network estimates the distance (number of steps) to close the remaining subgoals.
Steers
Critic-guided best-first tree search expands open states, and completed proofs are fed back into expert iteration to retrain both prover and critic.

Headline result

Reached 65.9% pass@64 on miniF2F-test and 54.5% on ProofNet, and proved 177 PutnamBench problems in Lean 4.

What it does not prove

Requires thousands of CPU hours of Lean 4 environment interaction during expert-iteration data collection.

OriginEmpirical benchmark

LaSR

Library-Augmented Symbolic Regression

Cornell University, Massachusetts Institute of Technology, 2024 Science · Program evolution

Authority: PySR genetic programming and BFGS numerical optimizer

Proposes
An LLM abstracts recurring sub-expressions from top equations into reusable symbolic concepts and seeds mutated expression trees.
Judged by
A high-performance symbolic regression engine (PySR) optimizes numerical constants via BFGS and scores MSE accuracy and complexity on scientific datasets.
Steers
Tournament selection and Pareto-front complexity filtering retain the fittest symbolic equations and evolve the concept library.

Headline result

Recovered significantly more exact ground-truth physical laws on the Feynman Symbolic Regression Benchmark than standalone PySR or LLM-only baselines.

What it does not prove

Calling an LLM inside evolutionary epochs increases wall-clock runtime compared to pure Julia genetic programming.

OriginFormal proof / check

Lean-STaR

Stanford University, Tsinghua University, 2024 Proofs · Proof-state search

Authority: Lean theorem prover and best-first search tree

Proposes
A language model generates a rationale thought followed by a formal Lean tactic at each node of the proof tree.
Judged by
The Lean proof assistant checks whether the tactic advances or closes the goal.
Steers
Best-first tree search explores multiple thought-tactic branches, and verified trajectories are used to fine-tune the model in STaR expert-iteration rounds.

Headline result

Improved miniF2F-test pass@64 in Lean from 30.8% to 46.3%, outperforming tactic-only search baselines.

What it does not prove

Generating natural-language thoughts before every tactic increases token count per node during tree search.

OriginEmpirical benchmark

LLAMBO

University of Cambridge, Huawei Noah's Ark Lab, 2024 Optimization · Active-learning generation

Authority: Expected Improvement acquisition ranker and black-box objective evaluator

Proposes
An LLM conditioned on the optimization history and problem metadata proposes candidate hyperparameter configurations.
Judged by
True black-box objective evaluations (training and scoring ML models on Bayesmark / HPOBench) measure validation loss.
Steers
Expected Improvement utility scores rank candidate configurations and append the evaluated (x, f(x)) pair to the active-learning history.

Headline result

Outperformed Gaussian Process and Random Forest Bayesian Optimization baselines in the low-trial regime across 25 Bayesmark tasks.

What it does not prove

In high-dimensional continuous spaces (>20 variables) with ratusan of trials, classical numerical GP kernels overtake in-context LLM surrogates.

OriginEmpirical benchmark

LLMorpheus

Northeastern University, 2024 Security · Verifier-guided repair

Authority: Compiler and automated mutation-testing execution engine

Proposes
An LLM is prompted at placeholders in chosen code locations and suggests replacement code, producing mutants of JavaScript packages.
Judged by
The project test suite is run against each mutant to determine whether it is killed or survives.
Steers
Surviving mutants expose concrete gaps in the existing test suite.

Headline result

On 13 real-world JavaScript packages, produced mutants resembling real bugs that StrykerJS, a state-of-the-art mutation testing tool, cannot produce.

What it does not prove

Distinguishing a genuinely surviving bug-revealing mutant from a semantically equivalent refactoring still requires diff inspection.

OriginPhysics simulation

Orb Universal Interatomic Potential

Orbital Materials, 2024 Materials · Active-learning generation

Authority: None in the loop: the potential predicts energy, forces and stress, and it is not a generative search

Proposes
Orb is a universal interatomic potential, not a generative search; denoising pretraining only initialises the potential.
Judged by
The potential predicts energy, forces and stress.
Steers
No generative search; the model is trained in two phases, denoising diffusion pretraining and then the interatomic potential.

Headline result

Topped the Matbench Discovery leaderboard for predicting stable inorganic crystals at 3-6x higher relaxation speed than prior equivariant potentials.

What it does not prove

Learned universal potentials lack explicit long-range electrostatics unless augmented with Ewald summation terms.

OriginFormal proof / check

ORLM

Operations Research Language Model

Cardinal Operations, Tsinghua University, 2024 Optimization · Formalize and solve

Authority: COPT mixed-integer and linear programming solver

Proposes
An LLM translates complex industrial scheduling, routing, and supply-chain descriptions into formal mathematical models and COPT Python code.
Judged by
The COPT mathematical programming solver executes the generated MIP/LP script and reports the solver optimum.
Steers
During synthetic data curation (OR-Instruct) unexecutable programs are discarded; at inference the code is run and the solver optimum is reported.

Headline result

Outperformed GPT-4 across NL4Opt, MAMO, and IndustryOR benchmarks using a 7B open-weight model paired with the COPT solver.

What it does not prove

As in all Formalize-and-Solve systems, solver optimality applies to the generated algebraic model rather than unstated domain nuances.

OriginPhysics simulation

SyntheMol

Stanford University, McMaster University, 2024 Chemistry · Proof-state search

Authority: SMARTS chemical reaction validator, MCTS, and wet-lab MIC assay

Proposes
Tree expansion combines REAL-space molecular building blocks under validated chemical reaction templates.
Judged by
Reaction compatibility rules verify synthetic tractability at every node, while a Chemprop GNN/MLP oracle scores antibacterial activity against Acinetobacter baumannii.
Steers
MCTS upper-confidence rollouts focus combinatorial search on high-scoring, Tanimoto-diverse reaction pathways before wet-lab synthesis and MIC testing.

Headline result

Searched a combinatorial space of 30 billion synthesizable molecules to discover 6 structurally novel compounds with potent wet-lab antibacterial activity against A. baumannii.

What it does not prove

Search space is restricted to 1-2 step combinatorial reactions from the Enamine REAL building-block catalog.

OriginPhysics simulation

TamGen

Microsoft Research, 2024 Chemistry · Oracle-trained generation

Authority: Molecular docking engine and wet-lab IC50 enzymatic assay

Proposes
A 3D-pocket-encoder + chemical-GPT decoder proposes SMILES structures conditioned on the target protein binding pocket and seed scaffolds.
Judged by
RDKit synthetic accessibility, physics-based molecular docking, and wet-lab enzymatic IC50 and whole-cell M. tuberculosis assays evaluate candidates.
Steers
Top-scoring docked and assayed hits seed iterative scaffold refinement rounds.

Headline result

Designed novel Mycobacterium tuberculosis ClpP inhibitors validated in wet-lab assays with strong enzymatic inhibition (IC50 = 0.9 uM).

What it does not prove

Requires a known 3D crystal structure of the target binding site to encode pocket geometry.

OriginFormal proof / check

Baldur

University of Massachusetts Amherst, Google, 2023 Proofs · Verifier-guided repair

Authority: Isabelle/HOL proof assistant and Sledgehammer ATP

Proposes
A language model generates a complete Isabelle/HOL proof script for a target theorem.
Judged by
The Isabelle proof assistant executes the script and returns an error message when the proof fails.
Steers
A repair model conditions on the statement, the whole failed proof and the Isabelle error message to synthesize a corrected proof.

Headline result

Proved 47.9% of theorems in the Isabelle Archive of Formal Proofs benchmark (65.7% when combined with Thor Sledgehammer).

What it does not prove

Single-round error repair struggles on deep multi-lemma obligations where the initial proof sketch chooses the wrong induction variable.

OriginPhysics simulation

Chroma

Generate Biomedicines, 2023 Chemistry · Oracle-trained generation

Authority: Geometric symmetry projectors, refolding filters, and X-ray crystallography

Proposes
A polymer-physics-informed graph diffusion model samples 3D protein backbones and all-atom sequences in O(N) time.
Judged by
Analytic point-group symmetry projectors, distance restraints, AlphaFold refolding consistency, and X-ray crystallography judge candidate proteins.
Steers
Bayes-rule score conditioning steers reverse-diffusion trajectories directly toward the specified symmetry or substructure constraints.

Headline result

Designed 310 proteins across point-group symmetries, programmable shapes, and protein classes, with crystallographic structures matching design to 1.0-1.4 Angstrom backbone RMSD.

What it does not prove

Conditioning on extreme geometric shapes can reduce natural sequence designability if the score guidance weight is set too high.

OriginFormal proof / check

Clover

Closed-Loop Verifiable Code Generation

Stanford University, 2023 Code verification · Verifier-guided repair

Authority: Dafny compiler and Z3 SMT solver

Proposes
An LLM generates Dafny implementation code, formal preconditions/postconditions, and loop invariants from a problem description.
Judged by
The Dafny/Boogie/Z3 verifier checks that the code satisfies the specification and tests logical equivalence between reconstructed specs and test suites.
Steers
Z3 verification errors and equivalence counterexamples are returned to the LLM to repair either the implementation or the annotations.

Headline result

Accepted 87% of correct Dafny programs while rejecting all of adversarial incorrect synthesis candidates on the CloverBench suite.

What it does not prove

Of the six checks, the Dafny verifier checks the code against its annotations and five reconstruction tests each use an LLM.

OriginPhysics simulation

Coscientist

Carnegie Mellon University, 2023 Chemistry · Verifier-guided repair

Authority: Opentrons robotic execution validator and UV-vis spectrophotometer

Proposes
An LLM planner designs reaction matrices and writes Opentrons Python protocol scripts and hardware commands.
Judged by
Protocol syntax checkers, robotic liquid handlers, and UV-vis plate readers execute the reactions and measure product yield.
Steers
The optimization study selects conditions against previously collected, fully mapped yield datasets.

Headline result

Autonomously executed Suzuki-Miyaura and Sonogashira cross-coupling reactions on cloud lab hardware and ran an optimization study that selected conditions against previously collected, fully mapped yield datasets.

What it does not prove

Wet-lab loop throughput is bounded by physical incubation and analytical characterization times.

OriginPhysics simulation

DiffCSP

Tsinghua University, Renmin University of China, 2023 Materials · Oracle-trained generation

Authority: Space-group symmetry validator and DFT energy evaluator

Proposes
A periodic E(3)-equivariant denoising network jointly updates lattice vectors and fractional coordinates from a target composition.
Judged by
Periodic crystal geometry validators, RMSD matchers, and DFT energy evaluations score the generated polymorphs.
Steers
Space-group symmetry constraints and multi-sample energy ranking select the lowest-energy valid crystal structure.

Headline result

Outperformed USPEX evolutionary crystal search and CDVAE on MP-20 and Perov-5 crystal structure prediction at a fraction of the compute time.

What it does not prove

Requires specifying the target unit-cell stoichiometry upfront unless paired with a composition generator.

OriginEmpirical benchmark

EvoPrompt

Tsinghua University, Microsoft Research, 2023 Optimization · Program evolution

Authority: Genetic Algorithm / Differential Evolution controller and benchmark evaluator

Proposes
An LLM performs evolutionary crossover and differential mutation on parent discrete instructions to generate child candidates.
Judged by
A deterministic evaluation harness scores each child candidate across a development dataset and computes exact task metrics.
Steers
Roulette-wheel selection and population elitism keep top-scoring candidates for the next evolutionary generation.

Headline result

Outperformed human-engineered and gradient-free prompt baselines by up to 25% across 31 NLP and reasoning datasets.

What it does not prove

Can overfit to a small development split if validation examples do not cover the test distribution.

OriginEmpirical benchmark

OPRO

Optimization by PROmpting

Google DeepMind, 2023 Optimization · Program evolution

Authority: Exact objective function evaluator and sorted trajectory buffer

Proposes
An optimizer LLM proposes K new candidate solutions (routes, linear regression weights, or instructions) given the top-scoring history.
Judged by
An exact objective evaluator computes the ground-truth cost (TSP tour length, MSE loss, or training accuracy) for every candidate.
Steers
The newly scored candidates are inserted into the sorted priority buffer, shifting the optimization window toward better objective values.

Headline result

Discovered prompts that boosted GSM8K accuracy by up to 8 percentage points over human baselines and solved small-scale TSP and linear regression trajectories.

What it does not prove

On large combinatorial instances (>50 TSP nodes), in-context coordinate proposals lag far behind dedicated Concorde or LKH solvers.

OriginEmpirical benchmark

Promptbreeder

Google DeepMind, 2023 Optimization · Program evolution

Authority: Tournament genetic algorithm and benchmark fitness evaluator

Proposes
An LLM applies a population of evolving mutation-prompts to parent task-prompts (plus Lamarckian working-out traces) to produce offspring.
Judged by
An automated benchmark harness evaluates each candidate on multi-step mathematical, logic, and commonsense training batches.
Steers
Binary tournament genetic selection updates both the task-prompt genome and the meta-level mutation-prompt genome.

Headline result

Outperformed Chain-of-Thought, Plan-and-Solve, and OPRO across arithmetic, commonsense, and hate-speech classification benchmarks.

What it does not prove

Requires thousands of forward evaluation passes across the evolutionary population before convergence.

OriginEmpirical benchmark

Self-Debugging with Execution Feedback

Google DeepMind, University of California, Berkeley, 2023 Code verification · Verifier-guided repair

Authority: Interpreter runtime and unit-test execution oracle

Proposes
A code LLM synthesizes a candidate program (Python, C++, or SQL) and generates a line-by-line behavioral explanation.
Judged by
The language runtime and unit-test suite execute the candidate program and return exact assertion failures, stack traces, or syntax errors.
Steers
Execution error messages paired with the behavioral trace condition the next repair iteration until all tests pass.

Headline result

Improved pass@1 by up to 12 percentage points across TransCoder C++-to-Python translation, Spider text-to-SQL, and MBPP Python synthesis.

What it does not prove

Correctness is bounded by the unit-test suite available during the debugging loop.

OriginEmpirical benchmark

TitanFuzz

University of Illinois Urbana-Champaign, 2023 Security · Program evolution

Authority: Dataflow/branch coverage analyzer and differential execution oracle

Proposes
A generative model (Codex) seeds multi-API tensor programs, and an infill model (InCoder) masks and regenerates spans around selected AST operators.
Judged by
PyTorch and TensorFlow CPU/GPU execution engines, differential backends, and crash/NaN sanitizers evaluate every mutated program and log operator coverage.
Steers
An evolutionary fitness function combining API depth, dataflow graph edges, and new branch coverage selects seeds for the next mutation round.

Headline result

Achieved 50.84% (PyTorch) and 30.38% (TensorFlow) higher code coverage than state-of-the-art DL fuzzers and detected 65 bugs (41 confirmed previously unknown).

What it does not prove

Differential testing across CPU and GPU backends can trigger false-positive floating-point rounding discrepancies if tolerances are too tight.

OriginEmpirical benchmark

TPSR

Transformer-based Planning for Symbolic Regression

Virginia Tech, Meta AI, 2023 Science · Proof-state search

Authority: Monte Carlo Tree Search and BFGS numerical optimizer

Proposes
A pre-trained symbolic regression transformer (E2E / NeSymReS) provides prior probabilities over next mathematical operators in a prefix expression tree.
Judged by
Once a complete equation skeleton is reached, BFGS optimizes its continuous constants and computes complexity-regularized NMSE fit on the dataset.
Steers
UCB tree search backpropagates the exact numerical fit reward to steer subsequent expression-tree expansions.

Headline result

Significantly improved symbolic equation recovery and extrapolation R^2 on SRBench and Feynman physics datasets over beam search or sampling.

What it does not prove

Running BFGS constant optimization at leaf rollouts adds numerical overhead compared to single-pass forward decoding.

OriginEmpirical benchmark

Voyager

NVIDIA, California Institute of Technology, 2023 Robotics · Program evolution

Authority: Mineflayer simulator runtime and a GPT-4 critic (self-verification)

Proposes
A code-generating LLM writes asynchronous Mineflayer JavaScript control skills to accomplish curriculum subgoals.
Judged by
The Minecraft simulator returns execution errors and observations; a second GPT-4 instance acting as critic decides whether the task succeeded.
Steers
Verified skills are indexed into a persistent vector skill library and composed as subroutines for harder tasks.

Headline result

Unlocked 3.3x more unique items, traveled 2.3x longer distances, and reached key tech-tree milestones up to 15.3x faster than prior agent baselines.

What it does not prove

Relies on high-level Mineflayer pathfinding APIs rather than raw pixel-to-motor actuation.

OriginEmpirical benchmark

AlphaCode

Google DeepMind, 2022 Code verification · Verifier-guided repair

Authority: Sandboxed test execution and input-output equivalence clustering

Proposes
An encoder-decoder transformer generates a large pool of candidate C++ and Python solutions from the problem description.
Judged by
A sandboxed execution engine filters out candidates that fail public example tests (>99% pruned) and runs the survivors on generated test inputs to cluster equivalent behaviors.
Steers
Behavioral clustering selects one representative program from each of the largest equivalence classes for final submission.

Headline result

Placed in the top 54.3% of human participants across 10 recent Codeforces competitions, the first AI system to reach competitive human rank.

What it does not prove

Relies on sampling hundreds of thousands of programs per problem; passing public example tests does not prove correctness on hidden edge cases.

OriginPhysics simulation

CDVAE

Crystal Diffusion Variational Autoencoder

Massachusetts Institute of Technology, 2022 Materials · Oracle-trained generation

Authority: Crystallographic validity checker and a learned energy model

Proposes
A periodic GNN decoder predicts unit-cell lattice parameters, stoichometry, and denoises atomic coordinates via annealed Langevin dynamics.
Judged by
A learned property predictor guides latent optimization; validity checks (symmetry, bond distance, charge neutrality) and a separate learned energy model evaluate finished samples.
Steers
Gradient ascent in latent space steers sampling toward lower formation energy.

Headline result

Generated valid periodic crystal structures across carbon, perovskite, and Materials Project benchmarks with high structural and compositional validity.

What it does not prove

Latent-space property optimization relies on a learned surrogate predictor.

OriginEmpirical benchmark

Code as Policies

Google Research, 2022 Robotics · Formalize and solve

Authority: Shapely computational geometry library and classical robot motion controller

Proposes
A code-generation LLM recursively synthesizes Python policy functions and helper routines from a user instruction.
Judged by
A sandboxed Python interpreter, NumPy/Shapely spatial geometry calculators, and low-level robot impedance controllers execute the policy.
Steers
Closed-loop visual and geometric state queries inside the generated control loops adjust robot trajectories online.

Headline result

Executed spatial-geometric manipulation, drawing, and tabletop rearrangement tasks on physical Franka Panda and UR5e robot arms without task-specific training.

What it does not prove

Assumes that the perception primitives (object detectors and pose estimators) accurately ground the variable names referenced in the generated code.

OriginEmpirical benchmark

CodeT

Microsoft Research, 2022 Code verification · Verifier-guided repair

Authority: Sandboxed execution matrix and RANSAC bipartite consensus solver

Proposes
A code language model independently generates N candidate implementation programs and M candidate unit-test assertions from the problem prompt.
Judged by
A sandboxed execution engine runs all N x M program-test pairs to build a bipartite pass/fail consistency matrix.
Steers
A RANSAC-style consensus algorithm identifies the bipartite clique maximizing (#passing_programs x #passed_tests) and selects its representative program.

Headline result

Raised pass@1 on HumanEval to 65.8% (+18.8 points over Codex greedy decoding) and improved MBPP and APPS accuracy by over 15 points.

What it does not prove

If the LLM shares a systematic misconception across both its generated implementations and its generated tests, the consensus clique can still favour a flawed behavior.

OriginFormal proof / check

Draft, Sketch, and Prove

DSP

Meta AI, Peking University, 2022 Proofs · Proof-state search

Authority: Isabelle/HOL kernel and Sledgehammer automated theorem provers

Proposes
An LLM drafts an informal mathematical proof and translates it into a formal Isabelle/HOL proof sketch with open `sledgehammer` subgoals.
Judged by
Isabelle/HOL and automated theorem provers (E, SPASS, Vampire, Z3, CVC4 via Sledgehammer) verify each intermediate step.
Steers
Sketches are sampled independently; a problem counts as solved if any sketch is completed.

Headline result

Solved 39.3% of miniF2F test theorems in Isabelle/HOL, a more than 10 percentage point jump over prior direct tactic-generation baselines.

What it does not prove

If the high-level proof sketch omits a bridge lemma too large for Sledgehammer to close within its timeout, the whole sketch fails.

OriginFormal proof / check

HyperTree Proof Search

HTPS

Meta AI, 2022 Proofs · Proof-state search

Authority: Lean / Metamath kernel and PUCT AND-OR hypergraph search

Proposes
A transformer policy network proposes multiple Lean or Metamath tactics at open goals in an AND-OR proof hypergraph.
Judged by
The formal proof kernel checks every tactic step and splits conjunctive subgoals into hypergraph children.
Steers
PUCT hypergraph tree search balances exploration and value estimates, and the policy and value models train online on data from their own searches.

Headline result

Solved 82.6% of Metamath set.mm test theorems and 41.0% of miniF2F-curriculum theorems in Lean, discovering shorter proofs than human ground truth.

What it does not prove

AND-OR hypergraph expansion requires all sibling subgoals of a tactic to close before a parent goal is proved.

OriginPhysics simulation

Pocket2Mol

Peking University, 2022 Chemistry · Oracle-trained generation

Authority: QVina docking score, used to evaluate finished molecules

Proposes
An E(3)-equivariant graph neural network places frontier atoms, 3D coordinates, and covalent bonds step by step inside a protein pocket.
Judged by
Docking (QVina) scores evaluate finished molecules.
Steers
Ligands are sampled atom by atom inside the pocket.

Headline result

Generated drug-like 3D molecules with more realistic torsion angles and stronger Vina binding scores than 3D voxel baselines on CrossDocked2020.

What it does not prove

Autoregressive atom-by-atom placement can accumulate geometric error if an early frontier atom is poorly positioned.

OriginPhysics simulation

ProteinMPNN

University of Washington, 2022 Chemistry · Oracle-trained generation

Authority: The model log probability for ranking; AlphaFold2 refolding and X-ray crystallography / cryo-EM as measures of whether designs fold

Proposes
An SE(3)-invariant message-passing neural network decodes amino-acid sequences conditioned on a target 3D protein backbone.
Judged by
The model log probability ranks designed sequences; AlphaFold2 / ESMFold single-sequence structure prediction (pLDDT and backbone RMSD) plus wet-lab SEC, cryo-EM, and X-ray crystallography measure whether they refold.
Steers
Tied-position symmetry constraints shape decoding, and designed sequences are ranked by the model log probability.

Headline result

Achieved 52.4% native sequence recovery (vs. 32.9% for Rosetta) and rescued previously failed de novo protein cyclic oligomers, nanoparticles, and binders in wet-lab crystallography.

What it does not prove

Assumes a designable target backbone; cannot fix a backbone with severe steric clashes on its own.

OriginEmpirical benchmark

SayCan

Google Research, Everyday Robots, 2022 Robotics · Formalize and solve

Authority: RL Bellman value-function affordance critics and physical robot controller

Proposes
A language model scores candidate atomic robot skills by their semantic probability of advancing a high-level user instruction.
Judged by
Pre-trained reinforcement-learning value functions (affordance critics) evaluate the exact probability that each skill will succeed from the current camera state.
Steers
Each step takes the highest-scoring skill by the joint product P(say) x V(can) until the termination token is reached; there is no beam or tree search.

Headline result

Executed 101 real-world mobile manipulation tasks in an office kitchen with 84% planning rate and 74% physical execution success.

What it does not prove

Bounded by the repertoire of pre-trained low-level manipulation skills and cannot recover if an early step irreversibly alters the scene without replanning.

OriginFormal proof / check

Synchromesh

Microsoft Research, 2022 Code verification · Verifier-guided repair

Authority: Incremental CFG parser and static type/scope completion engine

Proposes
A code language model predicts next-token distributions for domain-specific languages (SQL, Vega-Lite, SMCalFlow).
Judged by
An incremental Context-Free Grammar parser and dynamic semantic validator (table/column scope, type compatibility) compute the regex of valid next tokens.
Steers
Invalid tokens are masked at every decoding step so the sampler only explores syntactically and semantically valid AST completions.

Headline result

Eliminated syntax, type, and scope errors across SQL, Vega-Lite visualization specs, and calendar DSLs while boosting execution accuracy.

What it does not prove

Requires authoring an incremental grammar and completion-engine specification for each target domain language.

OriginFormal proof / check

PICARD

Element AI, ServiceNow Research, 2021 Code verification · Proof-state search

Authority: Incremental monadic SQL parser and relational schema type-checker

Proposes
A sequence-to-sequence language model (T5) proposes top-k next tokens across active beam hypotheses.
Judged by
An incremental monadic parser and relational schema checker verify lexical, syntactic, and column/table scoping validity on every partial prefix.
Steers
Invalid token continuations are masked to negative infinity before beam pruning, keeping only grammar- and schema-valid queries in the beam.

Headline result

Boosted T5-3B exact-match and execution accuracy on Spider and CoSQL benchmarks by 4-7 percentage points and eliminated 98% of syntax and schema errors.

What it does not prove

Enforces grammatical and schema well-formedness, but cannot prevent a semantically plausible query that filters on the wrong column value.

VariantEmpirical benchmark

A-CEoH

Algorithmic Contextual EoH

Karlsruhe Institute of Technology and TU Dortmund University, 2026 Optimization · Program evolution

Authority: A* runs on instances (solved or not, plan quality, time)

Proposes
Guiding heuristic functions for A* search, as thought plus code
Judged by
Running A* with the heuristic on instances (solved count, fitness, time)
Steers
Fitness drives EoH selection; the A* source code is kept in every prompt

Headline result

For unit-load pre-marshalling, Qwen2.5-Coder:32b-generated heuristics solved 30 of 30 held-out instances (seeds 10-39) against 27 of 30 for the "Optimal A*" baseline. For the sliding puzzle they solved 10 of 10 on seeds 0-9 against 5 of 10 for Hybrid A* and for "Optimal A*".

What it does not prove

Two domains. Where both solve, the LLM heuristics give worse solutions than A* (puzzle fitness 0.281 against 0.252 on shared instances). The paper gives no numeric EoH baseline for the final heuristics, only a qualitative claim of improvement. Accepted at EvoStar.

VariantEmpirical benchmark

CoupleEvo

Karlsruhe Institute of Technology, 2026 Optimization · Program evolution

Authority: Evaluation on coupled-problem instances

Proposes
Heuristics for each sub-problem, written by an LLM inside an evolutionary loop
Judged by
Running the combined heuristics on instances of the coupled problem
Steers
Sequential, iterative or integrated schedules decide which sub-problem's heuristic evolves in each generation

Headline result

On two coupled problems (inventory routing, and multi-robot multi-bay pre-marshalling) with Qwen3-coder:30b, the sequential and iterative strategies converged more stably than the integrated one. On inventory routing the mean relative gap was 0.0429 (sequential), 0.0433 (iterative), 0.0537 (integrated).

What it does not prove

Two problems and one model. The paper's own text says iterative has the smallest median on one problem while its table shows sequential lower, so read the tables, not the prose. Accepted to a GECCO 2026 workshop.

VariantEmpirical benchmark

ReEvo-designed memetic algorithm for secure fluid antenna systems

Harbin Institute of Technology, Shenzhen, 2026 Engineering · Program evolution

Authority: Wireless channel simulation with secure WMMSE beamforming scoring each port subset

Proposes
Crossover, mutation and local-search operator code for fluid antenna port selection, written by DeepSeek Reasoner
Judged by
Best secure sum-rate an inner genetic algorithm reaches with the operators on training channels, via secure WMMSE
Steers
ReEvo's reflective evolution feeds performance comparisons back into the next round of operator code

Headline result

At equal generation counts (100 generations), the evolved memetic algorithm reached 11.423 bit/s/Hz at 1 W (7.6% over a conventional GA) and 23.611 bit/s/Hz at 5 W (8.2% over a conventional GA), and about 4 to 6% above the two LLM-assisted GA baselines.

What it does not prove

Equal generations are not equal evaluation budgets, because local search adds evaluations, and error bars overlap with the LLM-GA baselines at some operating points. Results are simulated (10 users, one eavesdropper, 20x20 grid) in a preprint accepted by WCSP 2026.

VariantEmpirical benchmark

T-GADE

Osaka Metropolitan University, 2026 Optimization · Program evolution

Authority: Bin packing evaluation and free-energy survival selection

Proposes
Structured artifacts (here, bin packing scoring heuristics) from LLM crossover, mutation and a self-repair step
Judged by
Evaluation on online bin packing, measured as excess bins over the volume lower bound
Steers
Survival selection minimizes free energy F_T = <E> - T*H, with temperature T controlling the quality-diversity balance

Headline result

In a preregistered seed-paired comparison at 400 total LLM calls on online bin packing, the HoxLM configuration won 9 of 10 runs against EoH, with a median sealed excess of 0.836% against 3.912%.

What it does not prove

Preprint (arXiv v2, 14 Oct 2026), one task, 10 seed pairs, and an EoH-aligned implementation. Not independently replicated.

VariantEmpirical benchmark

AlphaEvolve at scale on 67 math problems

Georgiev, Gomez-Serrano, Tao, Wagner

Google DeepMind, Brown University, Institute for Advanced Study and UCLA (Georgiev, Gomez-Serrano, Tao, Wagner), 2025 Mathematics · Program evolution

Authority: Deterministic evaluator scoring the construction; some results later proven by hand or in Lean (one simple Kakeya case)

Proposes
Search heuristics (programs) that look for a mathematical construction, or programs that give a formula for any n
Judged by
User-written deterministic evaluators that score the best object found, sometimes checked afterwards by proof tools
Steers
Best heuristics are reused in later rounds; Kakeya's program went to proof tools

Headline result

The paper reports 67 problems across analysis, combinatorics, geometry and number theory. AlphaEvolve rediscovered the best known solutions in most cases and found improved ones in several, for example an autocorrelation upper bound reduced from 1.5708 to 1.5032 and a finite field Kakeya construction of size 1/4 p^3 + 7/8 p^2 - 1/8 for primes p = 1 mod 4.

What it does not prove

The authors state it did not match previous results in all cases, struggles without a smooth score, and some gains could have been matched by traditional methods. It exploited weak verifiers in some problems (for example numerical integration in Problem 6.6), and results depended on expert prompt advice.

VariantEmpirical benchmark

CALM

Co-evolution of Algorithms and Language Model

City University of Hong Kong, Southeast University, University of Victoria, Hon Hai Research Institute, 2025 Optimization · Program evolution

Authority: Benchmark evaluation of heuristic quality

Proposes
Heuristic code from a small LLM inside an evolutionary search
Judged by
Running the heuristic on problem instances to get a quality score that becomes the RL reward
Steers
Verbal guidance through prompts plus numerical guidance through GRPO fine-tuning of the LLM weights

Headline result

A Qwen2.5-7B-Instruct model (INT4, on a single 24GB GPU) with GRPO fine-tuning reached an average 0.71% gap on online bin packing, against 0.89% for MCTS-AHD and 2.40% for Best Fit, and beat the best listed competitor on CVRP (3.95% vs 4.70%) and OP (12.58% vs 16.34%) at N=200 in the paper's tables.

What it does not prove

The advantage is not uniform: on TSP at N=50 MCTS-AHD (9.69%) beats CALM (10.04%). Numbers come from the paper's own tables and baselines, which mostly ran on GPT-4o-mini.

VariantEmpirical benchmark

CEoH

Contextual Evolution of Heuristics

Karlsruhe Institute of Technology and TU Dortmund University, 2025 Optimization · Program evolution

Authority: Instance evaluation of heuristic inside a tree search

Proposes
Tree-search scoring heuristics for the unit-load pre-marshalling problem, as thought plus code
Judged by
Running the heuristic and measuring the average relative gap of its move count to a lower bound
Steers
Fitness drives EoH selection; the added problem description is kept in every prompt

Headline result

For unit-load pre-marshalling, Qwen2.5-Coder:32b with CEoH reached a fitness of 12.5% (best overall), against 14.3% for the best EoH run (GPT-4o 2024-08-06). The "Optimal A*" baseline scored about 8.1%. CEoH outperformed EoH for Gemma2:27b, Qwen2.5-Coder:32b and DeepSeekV3:685b.

What it does not prove

One niche problem. On unseen instances fitness rose to 30.8 (Qwen CEoH) and 33.9 (GPT-4o EoH), which the authors read as overfitting. For GPT-4o EoH was better than CEoH, and none solved the hardest case (80% fill).

VariantEmpirical benchmark

CogFunSearch

Google DeepMind with Janelia Research Campus, Rockefeller University, Max Planck Institute for Biological Cybernetics, Princeton University, UCL and Columbia University, 2025 Science · Program evolution

Authority: Held-out predictive score of each program on behavioral data

Proposes
Python programs that predict a subject's next choice from its past choices and rewards
Judged by
Scoring each program on how well it predicts recorded human, rat and fruit-fly choices
Steers
The best-scoring programs are fed back into the prompt, as in FunSearch

Headline result

In a bioRxiv preprint, on rat choices in a 2-armed drifting bandit task (data from Miller et al. 2021), CogFunSearch reliably found programs that beat the best existing cognitive model the authors compared against, both when improving supplied programs and when starting from scratch.

What it does not prove

A simple unconstrained neural network predicted significantly better than both the classic models and the discovered programs, so interpretability is gained at some cost in accuracy. Preprint, one task family.

VariantEmpirical benchmark

FunBO

Google DeepMind, 2025 Optimization · Program evolution

Authority: Gaussian Process Bayesian Optimization benchmark evaluator

Proposes
An LLM mutates Python acquisition functions mapping Gaussian Process posterior mean and variance to utility scores.
Judged by
A Bayesian Optimization benchmark harness runs full GP optimization loops with each candidate acquisition function on synthetic and hyperparameter tasks.
Steers
An island-model evolutionary database ranks acquisition functions by average simple regret and seeds the next prompt.

Headline result

Discovered acquisition functions that outperform Expected Improvement (EI) and Upper Confidence Bound (UCB) on unseen black-box optimization and HPO benchmarks.

What it does not prove

Evaluating each candidate acquisition function requires running multiple full Bayesian Optimization trajectories during the offline search.

VariantPhysics simulation

InvDesFlow-AL

Renmin University of China, 2025 Materials · Active-learning generation

Authority: DPA-2 relaxation, committee scoring, Bohrium Ehull query, DFT for the superconductor task

Proposes
Crystal structures from a diffusion model (EGNN backbone) pretrained on Alex-MP-20 and GNoME crystals
Judged by
Relaxation with DPA-2, then a committee score on formation energy and Ehull or on superconductor Tc
Steers
The top-scored generated crystals become the next fine-tuning set, repeated over multiple rounds

Headline result

Generated average formation energy fell over five iterations (means -1.14, -2.03, -2.93, -3.56, -3.77; units not stated in the text). The authors report 1,598,551 materials with Ehull below 50 meV, and Li2AuH6 as a conventional superconductor with a transition temperature of about 140 K from first-principles electron-phonon calculations. Crystal structure prediction RMSE is 0.0423 angstrom, reported as a 32.96% improvement over existing generative models.

What it does not prove

The paper gives two counts for low-Ehull materials (1,598,551 in text, 1,610,600 in a figure caption) and does not say how Ehull is queried. The 140 K value is a calculation, not a measurement, and not every evaluation is DFT.

VariantEmpirical benchmark

AFlow

Automating Agentic Workflow Generation

DeepWisdom, Hong Kong University of Science and Technology, 2024 ML systems · Program evolution

Authority: Monte Carlo Tree Search and benchmark execution evaluator

Proposes
An LLM mutates a selected workflow node by adding, removing, or rewiring prompt and tool operators in Python code.
Judged by
A benchmark execution engine runs the modified workflow on the validation split and records empirical score and execution trace.
Steers
MCTS backpropagates validation scores up the workflow tree and selects the next node by soft mixed-probability selection.

Headline result

Outperforms hand-crafted and automated workflow baselines by 5.7% average across six coding and math benchmarks while enabling smaller models to beat GPT-4o.

What it does not prove

Workflow search requires dozens of validation evaluations per benchmark, and overfits if the validation slice is too small.

VariantPhysics simulation

FlowMM

Meta AI, 2024 Materials · Oracle-trained generation

Authority: Crystallographic manifold constraints and VASP DFT evaluator

Proposes
A Riemannian flow-matching network integrates geodesic vector fields on the periodic torus of fractional coordinates and SPD lattice representations.
Judged by
Symmetry-aware structural matchers, CHGNet force-field relaxations, and VASP DFT calculations evaluate thermodynamic stability (E_hull) on generated samples.
Steers
DFT and CHGNet relaxations compute stability metrics on generated samples; the flow is trained on a fixed dataset.

Headline result

Achieved state-of-the-art thermodynamic stability and 3x faster integration steps than diffusion baselines on MP-20 and MPTS-52.

What it does not prove

Like other crystal generators, requires downstream DFT phonon calculations to confirm dynamic vibrational stability at finite temperature.

VariantEmpirical benchmark

HSEvo

Hanoi University of Science and Technology and George Mason University, 2024 Optimization · Program evolution

Authority: Benchmark evaluation of heuristic code and diversity metrics

Proposes
Heuristic code from LLM crossover and mutation, plus harmony-search refinement of numeric parameters
Judged by
Running the heuristic on benchmark instances for an objective value, plus two diversity metrics
Steers
Objective and diversity scores steer selection to balance exploration and convergence

Headline result

The authors propose two diversity metrics and use them to compare FunSearch, EoH and ReEvo. They report EoH has higher diversity but unstable scores, ReEvo has good scores but poor diversity, and that HSEvo reaches high diversity and good objective scores while staying cost-effective.

What it does not prove

The abstract gives no numeric gains, and results are on the authors' black-box heuristic-design benchmarks. 'Cost-effective' is the authors' wording.

VariantFormal proof / check

Lean Copilot

California Institute of Technology, 2024 Proofs · Proof-state search

Authority: Lean 4 kernel and best-first proof search

Proposes
Native CTranslate2 models running inside Lean 4 propose tactics and retrieve premises for the active proof goal.
Judged by
The Lean 4 kernel checks each tactic application and returns updated subgoal states.
Steers
A best-first search tactic (`search_proof`) with the model confidence as score explores the proof tree until all goals close.

Headline result

Automates proof goals directly inside Lean 4 workflows with 3.8x faster tactic inference than Python RPC bridges.

What it does not prove

Designed for interactive lemma and subgoal automation rather than multi-hour distributed search.

VariantEmpirical benchmark

AlphaCode 2

Google DeepMind, 2023 Code verification · Verifier-guided repair

Authority: Sandboxed C++ execution filter and behavioral clustering engine

Proposes
A family of fine-tuned Gemini Pro models samples up to one million diverse C++ candidate programs per problem.
Judged by
Sandboxed execution prunes about 95% of samples that fail the public example tests or do not compile, then clusters survivors on synthetic inputs.
Steers
A second Gemini scoring model ranks the top cluster representatives; there is no repair step.

Headline result

Solved 43% of Codeforces Division 1+2 problems and ranked at the 85th percentile of human competitors.

What it does not prove

High inference compute budget per problem; correctness remains bounded by test coverage rather than formal proof.

VariantFormal proof / check

LEGO-Prover

Tsinghua University, The Chinese University of Hong Kong, 2023 Proofs · Proof-state search

Authority: Isabelle/HOL proof kernel and Sledgehammer ATP

Proposes
A prover LLM decomposes a theorem into candidate helper lemmas and formal Isabelle proof sketches, while an evolvar LLM mutates verified lemmas in the skill library.
Judged by
The Isabelle/HOL kernel and Sledgehammer verify every proposed lemma and proof sketch.
Steers
Verified lemmas are indexed into a growing modular skill library and retrieved via semantic search to simplify subsequent theorems.

Headline result

Raised miniF2F-valid pass@100 to 57.0% and miniF2F-test to 50.0% in Isabelle/HOL, accumulating thousands of reusable verified lemmas.

What it does not prove

Library quality depends on pruning redundant or overly specialized lemmas so retrieval stays sharp.

VariantEmpirical benchmark

VoxPoser

Stanford University, 2023 Robotics · Formalize and solve

Authority: Model Predictive Control (MPC) trajectory planner and IK solver

Proposes
An LLM writes Python expressions that compose open-vocabulary vision masks into 3D voxel grids of target attractors, obstacles, and rotation/gripper maps.
Judged by
A deterministic sampling-based Model Predictive Control (MPC) trajectory optimizer and inverse-kinematics solver compute minimum-cost paths over the 3D field.
Steers
MPC replans at each step from the current observation; the value map comes from cached model-written code.

Headline result

Zero-shot execution of 25 everyday manipulation tasks across real-world and simulated robot arms with 88% success, plus online adaptation to moving obstacles.

What it does not prove

Voxel grid resolution (e.g. 100^3) limits fine sub-millimeter insertion tasks.

VariantPhysics simulation

Augmented Hill-Climb

University of Cambridge, 2022 Chemistry · Oracle-trained generation

Authority: Task scoring functions including docking, plus diversity filters

Proposes
SMILES strings from an RNN (also tested on a transformer) molecule generator
Judged by
A scoring function evaluated on each sampled batch; the paper's tasks include docking objectives, with diversity filters
Steers
RL updates the generator with a hybrid of REINVENT's loss and Hill-Climb's top-scorer selection

Headline result

On the authors' benchmark tasks, compared with REINVENT, optimization ability improved about 1.5-fold and sample efficiency about 45-fold, and Augmented Hill-Climb outperformed the other reinforcement learning strategies tested on six tasks.

What it does not prove

The gains are measured on the authors' benchmark tasks and docking objectives, not on lab-measured activity. It is a training rule inside the REINVENT-style setup rather than a separate system.

VariantPhysics simulation

Lib-INVENT with docking and RFXscore

Sanofi, 2022 Chemistry · Oracle-trained generation

Authority: Glide XP docking, RFXscore rescoring, MM/GBSA post-check

Proposes
Decorations (substituents) added to a factor Xa scaffold, written as SMILES by Lib-INVENT, a REINVENT variant
Judged by
Glide XP docking, ligand efficiency, the RFXscore rescoring model, QED and a factor Xa QSAR model
Steers
Reinforcement learning updates the generator's agent network; a diversity filter penalises repeats

Headline result

In a factor Xa case study, docking followed by RFXscore gave the best results, especially when diverse structures fitting a binding site were wanted; RFXscore plus the factor Xa model left the most compounds after MM/GBSA and other filters. RFXscore had a test r2 of 0.494 and MAE of 1.086.

What it does not prove

MM/GBSA was applied afterwards as an independent check, not as a training reward. Everything is computed binding, with no lab measurements, and the target is a single well-studied protein (factor Xa).