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Vijay Ganesh

Publications and source records attributed to Vijay Ganesh.

At least 19 recordsLinked to original sources
arXiv2026

Learning Scattering Amplitudes with Transformer Reinforcement Learning

We introduce a transformer reinforcement learning algorithm that learns to solve high loop-level scattering amplitudes in planar N = 4 Super Yang-Mills theory. Our algorithm improves on previous transformer-only results by incorporating previously derived symmetries and relationships into the learning algorithm. This results in a greatly decreased fraction of the solution that needs to be known a priori to solve the entire problem. An additional benefit is that our algorithm also ensures that every output obeys the set of known relationships. Rather than predicting all coefficients independently, the model proposes assignments that are propagated through exact linear relations, while MCTS searches over assignments when propagation alone is insufficient. This is crucial to the generalization of machine learning approaches to higher loops, as without this, there is no way to overcome the factorially scaling of state sizes and compare to results derived via other methods. Using the symbology representation of the form factor, we frame the problem as learning a mapping between discrete sequences and integer coefficients.

hep-th↗
arXiv2026

Assuring Learning-Enabled Cyber-Physical Systems with Foundation Models: A Neuro-Symbolic Agenda for Testing, Verification, and Runtime Assurance

Learning-enabled cyber-physical systems (CPS), including autonomous vehicles and UAVs, increasingly incorporate foundation models, yet fragmented assurance practices provide insufficient support for system-level guarantees. We argue that restoring engineering control across stochastic neural, symbolic, and physical boundaries requires a shift from model-centric evaluation to system-level assurance. We introduce the Foundation Model Assurance Stack, connecting three directions: (i) neuro-symbolic architectures that constrain foundation models through explicit symbolic world models, reasoning layers, and interfaces; (ii) semantically grounded, conformance-based testing that combines formal specifications and symbolic world models with foundation-model-assisted specification mining, scenario generation, and test synthesis; and (iii) compositional verification and runtime assurance that target formally tractable components and interfaces through modular contracts, equivalence checking, verified synthesis, uncertainty-aware reasoning, and runtime monitoring and enforcement. Symbolic abstractions establish assurance boundaries between stochastic learning and formal reasoning, connecting testing, verification, and runtime evidence across the system lifecycle. Foundation models serve as constrained assistants whose outputs are independently checked wherever correctness can be formally specified. We identify research challenges in specification engineering, abstraction fidelity, evidence composition, and lifecycle certification, and outline a staged agenda toward trustworthy, deployable, and certifiable learning-enabled CPS.

cs.RO↗
arXiv2026

Search-Guided Reasoning Trace Curation for Verilog Generation

Recent advances in large language models (LLMs) have improved Verilog generation. However, most training pipelines distill natural-language reasoning traces whose intermediate design decisions are weakly structured and are primarily selected according to the correctness of the final generated RTL. We present MCTS-SFT, a verifier-guided framework for curating reasoning traces for supervised fine-tuning (SFT) of Verilog generation models. MCTS-SFT formulates trace construction as search over intermediate reasoning states and uses Monte Carlo Tree Search (MCTS) to propagate functional feedback from complete Verilog rollouts to earlier design decisions. Within this framework, we introduce Plan-MCTS, which searches over compact hardware-design plans that explicitly represent interfaces, timing, state, behavior, and update rules. We further study plan-preserving supervision, in which the searched structured state is retained as part of the downstream SFT target. Our experiments find that Plan-MCTS achieves the highest functional trace yield, with 84.2% correct traces compared with 81.9% for one-shot structured planning and 82.0% for unguided tree exploration. For identical Plan-MCTS trajectories and Verilog outputs, retaining the explicit searched plan during SFT raises pass@10 from 52.56% to 57.7%, demonstrating that the representation of intermediate design decisions materially affects downstream learning. In a separate 6,000-trace study, MCTS-curated supervision improves pass@10 over full-response chain-of-thought distillation by up to 3.7 points while reducing downstream generation length by approximately 50% on VerilogEval-v2. These results demonstrate that structured hardware planning, verifier-guided search, and explicit intermediate-state supervision provide complementary mechanisms for constructing more effective and token-efficient training data for Verilog generation.

cs.AR↗
arXiv2026

The Four-Qubit Hyperbolic Quadric Has Contextuality Degree 315

The contextuality degree of a set of Pauli contexts is the least number of contexts that every noncontextual $\pm1$ assignment violates. For the four-qubit hyperbolic quadric $\mathcal{Q}^{+}(7,2)$, with $135$ points and $1575$ lines as contexts, Muller et al. conjectured from heuristic searches that the degree is $315$ and that the violated contexts of every optimal assignment form a dual polar space $DW(5,2)$. We prove both conjectures. The lower bound has a short proof by hand: every doily ($GQ(2,2)$) of the quadric contains ten Peres-Mermin grids covering each of its lines four times, so every assignment violates at least three of its fifteen lines, and averaging over the doilies gives $1575/5=315$. The equality case is local: an assignment is optimal exactly when it violates one ruling of one grid in every doily. Using this as a SAT encoding, we show that there are exactly $240$ optimal violated sets, one orbit of $\mathrm{O}^+(8,2)$, each isomorphic to $DW(5,2)$. The same argument gives the first lower bounds $\lceil L/5\rceil$ for all quadrics with $L$ lines in $N\ge4$ qubits. The degree theorem is kernel-checked in Lean 4, and the completeness of the enumeration is certified by a DRAT proof checked by the verified checker cake_lpr.

quant-ph↗
arXiv2026

AIProver: Agentic Auto-Formalization of Mathematical Research via Certificate-Driven Evolving Harness

Proof auto-formalization translates natural-language (NL) theorems and proofs into a formal language (FL) such as Lean, enabling mechanical verification. Despite rapid progress, research-level proofs often depend on concepts missing from leading proof assistant libraries (e.g., Lean's Mathlib), and successful compilation does not guarantee that a translation preserves the theorem's meaning or the proof's reasoning. Furthermore, aligned NL-FL training data are scarce, and leading agents often rely on costly frontier models and manually engineered harnesses. To address the above issues, we present AIProver, an agentic framework for autonomous proof auto-formalization and proof synthesis (AFPS) that jointly post-trains a 119B open-weight language model and evolves its agentic, tool-calling harness with HarnessEvolve. Verifiers assess type correctness, proof completeness, and semantic correctness, returning rewards and diagnostic certificates that drive model fine-tuning and alternating reinforcement learning via symbolic feedback and HarnessEvolve, a certificate-driven evolutionary search over the whole harness control flow that re-tailors the harness to the updated model. For research-level training and evaluation, we introduce LoCoBench, 58.9k instances from Mathlib, CSLib, Mizar Math Library, and a bounded-arithmetic textbook, with a 771-instance validation split whose theorem-proof pairs have no public Lean formalization. Against 39 frameworks spanning AFPS agents, frontier LLMs, and coding agents, AIProver lifts pass@4 semantic correctness over its Leanstral-1.5 base from 15.7% to 36.7% and outperforms every other open-weight system and Aristotle. As a Claude Code and Codex skill, it lifts their semantic correctness from 41.9% and 34.1% to 79.8% and 62.4%, respectively. Further, it is also 24% cheaper than Numina-Lean-Agent, pushing the accuracy-cost frontier of research-level AFPS.

cs.LO↗
arXiv2026

Towards Certificate-Driven Software Porting: A Self-Improving Agentic Harness for Scientific Program Optimization

The upgrade and rewriting of large scientific codebases has traditionally been a major challenge. While evolutionary search with large language models (LLMs) can port and accelerate legacy code, repair feedback in prompts alone does not prevent subsequent candidates from repeating the same errors. We introduce Certificate-Driven Evolutionary Search (CDES), which extends evolutionary search with enforceable restrictions derived from failed candidates, recorded as certificates of assumptions, checker evidence, and justified restrictions. Its control logic enforces these restrictions through rejection, backtracking, and targeted repair while preserving compatible edits. We apply CDES to CPU-to-GPU translation of two particle-simulation functions from the Geant4 toolkit, evaluated with a harness that goes beyond unit tests to combine formal checks, numerical comparisons, physics checks, and GPU safety tests. Generated implementations achieve 13.78x and 23.54x function-level speedups over CPU code, including data conversion and transfers; for one function, GPU throughput exceeds an expert implementation by 14.9%, reaching 16.1% when complementary components are combined. In an ablation over execution settings, certificate feedback increases the fraction of candidates passing required correctness checks from 55% to 90%.

cs.AI↗
arXiv2026

CDRL: Certification-Driven Reinforcement Learning for Neutrino Flavor Model Discovery

Many scientific discovery problems require searching combinatorial hypothesis spaces under complex domain constraints. Reinforcement learning (RL) offers a promising approach, but existing methods rely on scalar rewards that provide limited information about why candidate solutions fail, leading agents to repeatedly explore invalid regions. We introduce Certification-Driven Reinforcement Learning (CDRL), a framework that leverages structured feedback from symbolic reasoning tools. When a candidate violates domain constraints, these tools produce certificates identifying the actions responsible for failure. CDRL converts these certificates into reusable constraints that eliminate classes of invalid solutions and guide exploration toward valid regions. We evaluate CDRL on neutrino flavor model discovery in theoretical particle physics, where the hypothesis space exceeds $10^{26}$ possible models, and compare it with the state-of-the-art RL approach previously used for this task. Across three theory spaces, CDRL achieves up to 1.95$\times$ higher valid model rates and up to 6.33$\times$ higher neutrino model rates while evaluating up to 4$\times$ fewer candidates. We further extract 40 interpretable rules from search trajectories using a post-hoc decision-tree framework and show that reusing them as soft constraints yields gains of up to 2$\times$ in valid model rates and 3$\times$ in neutrino model discovery across all three theory spaces. These results suggest that CDRL uncovers reusable structure in combinatorial search spaces and provides a general framework for scientific model discovery.

cs.AI↗
arXiv2026

An Exponential Separation between Deterministic CDCL and DPLL Solvers

We prove that there exists a deterministic configuration of Conflict Driven Clause Learning (CDCL) SAT solvers using a variant of the VSIDS branching heuristic that solves instances of the Ordering Principle (OP) CNF formulas in time polynomial in n, where n is the number of variables in such formulas. Since tree-like resolution is known to have an exponential lower bound for proof size for OP formulas, it follows that CDCL under this configuration has an exponential separation with any solver that is polynomially equivalent to tree-like resolution and therefore any configuration of DPLL SAT solvers.

cs.CC↗
arXiv2026

Understanding CDCL Solvers via Scalability Studies and Proofdoors

Over the past several decades, CDCL SAT solvers have proven remarkably effective on large industrial formulas, despite SAT being NP-complete and widely believed to be intractable. While considerable empirical research has been done on solver performance over benchmarks like the SAT competition, as well as scaling studies on random and crafted families, surprisingly little effort has gone into systematic scaling studies over industrial instances. To address this gap, we collect a large benchmark of Bounded Model Checking (BMC) instances (76,600+ across 766 families) and perform a systematic scaling study of solver performance. We observe a spectrum: some families scale linearly, others polynomially or exponentially. Building on this foundation, we study the structural parameters that have been proposed to explain this phenomenon. We first show that previously proposed parameters -- clause-variable ratio, treewidth, and community structure -- fail to discriminate between the linear and exponential regimes. By contrast, the recently proposed \emph{proofdoor} parameter explains this phenomenon well. Informally, a proofdoor is a sequence of interpolants between chunks of a formula, where each interpolant represents the solver's memoization of reasoning effort on chunks it has already analyzed. In support of the proofdoor hypothesis, we make three key contributions. First, we empirically show that CDCL solvers do compute small proofdoors for linearly-scaling BMC instances. Second, we show that for exponentially-scaling instances, sampled proofdoors scale exponentially and are typically not incrementally absorbed. Third, we show that scrambling linearly-scaling instances yields larger proofdoor sizes relative to pre-scrambling, relating poor branching order to larger proofdoor sizes and drop in solver performance.

cs.LO↗
arXiv2026

Proofdoors and Efficiency of CDCL Solvers

We propose a new parameter called proofdoor in an attempt to explain the efficiency of CDCL SAT solvers over a certain class of formulas derived from circuit (esp., arithmetic) verification applications. Informally, given an unsatisfiable CNF formula F over n variables, a proofdoor decomposition consists of a chunking of the clauses into A_1, ..., A_k together with a sequence of interpolants connecting these chunks. Intuitively, a proofdoor captures the idea that an unsatisfiable formula can be refuted by reasoning chunk by chunk, while maintaining only a summary of the information (i.e., interpolants) gained so far for subsequent reasoning steps. We prove several theorems in support of the proposition that proofdoors can explain the efficiency of CDCL solvers for some class of circuit verification problems. First, we show that formulas with small proofdoors (i.e., where each interpolant is O(n) sized, each chunk A_i has small pathwidth, and each interpolant clause has at most O(log n) backward dependency on the previous interpolant) have short resolution (Res) proofs, and a certain configuration of CDCL solvers can compute such proofs in time polynomial in n. Second, we show that commutativity (miter) formulas over floating-point addition have small proofdoors and hence short Res proofs, even though they have large pathwidth. Third, we identify limits of the proofdoor framework: we show that a poor decomposition of arithmetic miter instances can force exponentially large interpolants, and hence our framework derives exponentially large Res refutations from such a decomposition, even when a different decomposition (i.e., a small proofdoor) yields short proofs. As a byproduct, these interpolant lower bounds imply new lower bounds for the partially ordered resolution proof system.

cs.CC↗
arXiv2026

Making Embodied AI Reliable: A Community Agenda from Testing to Formal Verification

Embodied AI systems are increasingly deployed in open-world environments, yet ensuring their reliability remains a fundamental challenge. Drawing on discussions from the AAAI'26 Bridge Program on "Making Embodied AI Reliable with Testing and Formal Verification", this article argues that reliability in embodied AI is inherently a lifecycle assurance problem arising from uncertainty, human interaction, and emergent behaviors across tightly coupled system components. We identify three complementary directions toward reliable embodied AI: (1) trustworthy scenario-based testing supported by validated specifications and meaningful coverage metrics, (2) compositional verification enabled by structured symbolic representations of system behavior and environmental context, and (3) runtime assurance mechanisms capable of adapting to uncertainty and distribution shifts during deployment. Rather than treating these approaches independently, we advocate integrated assurance workflows that connect testing, verification, and runtime adaptation through shared neuro-symbolic representations and continuous feedback across the system lifecycle. Such integration provides a foundation for building trustworthy embodied AI systems that can operate safely and reliably in complex real-world environments.

cs.SE↗
arXiv2026

Symbolic Regression via Latent Iterative Refinement

Symbolic regression (SR) seeks closed-form mathematical expressions that fit observed data. Neural SR methods amortize the search by training an encoder to map observations directly to expressions in a single pass, but this amortized inference leaves a residual amortization gap between its one-shot prediction and the true posterior. We propose Latent Equation Embedding (LEE), a framework that closes this gap through iterative amortized inference in a functionally grounded latent space. LEE learns a shared latent space Z equipped with three components: an encoder f_theta that jointly embeds symbolic tokens and numerical observations into a single latent vector z; an expression decoder g_expr that reconstructs formulas from z; and an evaluation decoder g_eval that predicts function values from z, explicitly grounding the latent space in functional behavior. At inference, LEE performs iterative refinement by re-encoding decoded expressions jointly with observations, progressively improving the latent estimate. LEE uses the encoder itself as a learned inference optimizer: each re-encoding step implicitly computes the mismatch between the candidate and the data. Because g_eval is differentiable in z, we additionally interleave continuous gradient descent with discrete re-encoding, yielding a hybrid iterative and gradient refinement procedure. On SRBench across three noise levels, against 19 baselines spanning genetic programming, symbolic-neural hybrids, and pre-trained Transformers, LEE produces expressions 2--10x simpler than the strongest accuracy-oriented baselines, including Operon, GP-GOMEA, TPSR, RAG-SR, and GenSR, with complexity 8--11 versus 20--90. These results advance the low-complexity region of the accuracy-complexity Pareto frontier and show graceful degradation as noise increases.

cs.LG↗
arXiv2026

Extended Resolution Clause Learning via Dual Implication Points

We present a new extended resolution clause learning (ERCL) algorithm, implemented as part of a conflict-driven clause-learning (CDCL) SAT solver, wherein new variables are dynamically introduced as definitions for {\it Dual Implication Points} (DIPs) in the implication graph constructed by the solver at runtime. DIPs are generalizations of unique implication points and can be informally viewed as a pair of dominator nodes, from the decision variable at the highest decision level to the conflict node, in an implication graph. We perform extensive experimental evaluation to establish the efficacy of our ERCL method, implemented as part of the MapleLCM SAT solver and dubbed xMapleLCM, against several leading solvers including the baseline MapleLCM, as well as CDCL solvers such as Kissat 3.1.1, CryptoMiniSat 5.11, and SBVA+CaDiCaL, the winner of SAT Competition 2023. We show that xMapleLCM outperforms these solvers on Tseitin and XORified formulas. We further compare xMapleLCM with GlucoseER, a system that implements extended resolution in a different way, and provide a detailed comparative analysis of their performance.

cs.LO↗
arXiv2026

MathConstraint: Automated Generation of Verified Combinatorial Reasoning Instances for LLMs

We introduce MathConstraint, a hard, adaptive benchmark for evaluating the combinatorial reasoning capabilities of LLMs. We combine constraint satisfaction problems with rigorous solver-based verification and design an adaptive generator to create instances that remain challenging as the LLMs improve in their reasoning capabilities. Unlike existing benchmarks that quickly saturate on fixed datasets or use LLM-as-a-judge for checking solutions,MathConstraint uses parameterized problem types that enable scalable generation of arbitrarily difficult and automatically verifiable instances. We release MathConstraint-Easy ($266$ instances), on which frontier models achieve between $72.6\%$ (gemini-3.1-flash-lite) and $87.6\%$ (gpt-5.5) accuracy, and MathConstraint ($329$ instances) on which the same models drop to between $18.5\%$ (claude-4.6-sonnet) and $66.9\%$ (gpt-5.5) accuracy, demonstrating the resilience of our benchmark generator against rapid progress in LLM reasoning capabilities. We evaluate 12 frontier and open-weight models with and without access to a sandboxed Python environment that includes generic SAT/SMT solvers. Tool access roughly doubles frontier accuracy on MathConstraint (mean $+28$pp; up to $+52$pp for claude-4.6-sonnet). Further, halving the tool-call budget from $8$ to $4$ rounds erases up to $37$ points -- a sensitivity that most single-budget benchmarks miss. We release the generator, dataset, and evaluation harness as a robust environment for studying combinatorial reasoning and tool-use behavior under adversarially-tunable difficulty.

cs.LG↗
arXiv2026

Report for NSF Workshop on AI for Electronic Design Automation

This report distills the discussions and recommendations from the NSF Workshop on AI for Electronic Design Automation (EDA), held on December 10, 2024 in Vancouver alongside NeurIPS 2024. Bringing together experts across machine learning and EDA, the workshop examined how AI-spanning large language models (LLMs), graph neural networks (GNNs), reinforcement learning (RL), neurosymbolic methods, etc.-can facilitate EDA and shorten design turnaround. The workshop includes four themes: (1) AI for physical synthesis and design for manufacturing (DFM), discussing challenges in physical manufacturing process and potential AI applications; (2) AI for high-level and logic-level synthesis (HLS/LLS), covering pragma insertion, program transformation, RTL code generation, etc.; (3) AI toolbox for optimization and design, discussing frontier AI developments that could potentially be applied to EDA tasks; and (4) AI for test and verification, including LLM-assisted verification tools, ML-augmented SAT solving, security/reliability challenges, etc. The report recommends NSF to foster AI/EDA collaboration, invest in foundational AI for EDA, develop robust data infrastructures, promote scalable compute infrastructure, and invest in workforce development to democratize hardware design and enable next-generation hardware systems. The workshop information can be found on the website https://ai4eda-workshop.github.io/.

cs.LG↗
arXiv2026

SAT + NAUTY: Orderly Generation of Small Kochen-Specker Sets Containing the Smallest State-independent Contextuality Set

We present a search for small Kochen-Specker (KS) sets in dimension 3, specifically targeting extensions of the 13-ray Yu-Oh set, which has been proven to be the minimal witness to state-independent contextuality. To enable this search, we introduce a novel SAT-based orderly generation framework integrating recursive canonical labeling (RCL) with the graph isomorphism tool NAUTY. We demonstrate that previous SAT approaches relying on lexicographical canonicity suffer from exponential scaling on canonical graphs. This limitation renders them intractable on the large instances (25 to 33 vertices) encountered in our search, whereas our RCL check maintains consistent millisecond-level performance, effectively eliminating the bottleneck. Overcoming this bottleneck allows us to perform the first exhaustive enumeration of all KS sets with up to 33 rays containing the complete 25-ray state-independent contextuality (SI-C) set obtained by rigid extensions of the Yu-Oh set in 1,641 CPU hours. We found and verified that the 33-ray set discovered by Schütte is the smallest three-dimensional KS set containing the complete 25-ray SI-C set. All non-existence results are backed by independently verifiable proof certificates via an extension of the DRAT proof format.

cs.LO↗
arXiv2026

Towards AI-assisted Neutrino Flavor Theory Design

Particle physics theories, such as those which explain neutrino flavor mixing, arise from a vast landscape of model-building possibilities. A model's construction typically relies on the intuition of theorists. It also requires considerable effort to identify appropriate symmetry groups, assign field representations, and extract predictions for comparison with experimental data. We develop an Autonomous Model Builder (AMBer), a framework in which a reinforcement learning agent interacts with a streamlined physics software pipeline to search these spaces efficiently. AMBer selects symmetry groups, particle content, and group representation assignments to construct viable models while minimizing the number of free parameters introduced. We validate our approach in well-studied regions of theory space and extend the exploration to a novel, previously unexamined symmetry group. While demonstrated in the context of neutrino flavor theories, this approach of reinforcement learning with physics software feedback may be extended to other theoretical model-building problems in the future.

hep-ph↗
arXiv2026

Symbolic Density Estimation: A Decompositional Approach

We introduce AI-Kolmogorov, a novel framework for Symbolic Density Estimation (SymDE). Symbolic regression (SR) has been effectively used to produce interpretable models in standard regression settings but its applicability to density estimation tasks has largely been unexplored. To address the SymDE task we introduce a multi-stage pipeline: (i) problem decomposition through clustering and/or probabilistic graphical model structure learning; (ii) nonparametric density estimation; (iii) support estimation; and finally (iv) SR on the density estimate. We demonstrate the efficacy of AI-Kolmogorov on synthetic mixture models, multivariate normal distributions, and three exotic distributions, two of which are motivated by applications in high-energy physics. We show that AI-Kolmogorov can discover underlying distributions or otherwise provide valuable insight into the mathematical expressions describing them.

cs.LG↗