Neural Proofs: Hybrid Formal Reasoning

• Neural proofs are a set of methodologies that integrate neural networks with formal reasoning to automate proof discovery and verification across diverse applications.
• They combine data-driven optimization with techniques such as certificate synthesis, interactive protocols, and graph-based representations to enhance scalability and soundness.
• Applications span formal control synthesis, neural network safety verification, and automated theorem proving, pushing the boundaries of traditional proof systems.

Neural proofs constitute a family of methodologies that leverage neural networks not merely as computational primitives but as core components in the discovery, synthesis, and verification of mathematical proofs, program correctness, system invariants, and logical inference procedures. Unlike classical symbolic proof search, neural proofs hybridize data-driven optimization with formal proof rule schemas, probabilistic semantics, graph-based representations, and in certain settings interactive or adversarial game-theoretic frameworks. The aim is to automate or accelerate reasoning and verification tasks that are traditionally intractable due to high dimensionality, expressivity, or non-linearity, while retaining a quantifiable guarantee of soundness. Neural proofs have found applications in formal verification of dynamical systems, control synthesis, scalable proof construction for neural networks, automated theorem proving, program analysis, and beyond.

1. Formalization and Taxonomy

Neural proofs encompass a spectrum of paradigms unified by the explicit participation of neural networks in one or more of the following roles:

• Certificate synthesis: Construction of real- or vector-valued certificates (e.g., Lyapunov functions, barrier certificates) via parameterized networks, with training guided by proof side-condition loss and global soundness enforced by external verification or solver checks (Abate, 20 Dec 2025).
• Inference rule discharge: Encoding formal logical rules as templates where satisfaction of parameteric constraints by a neural function implies the desired property or specification.
• Interaction protocols: Structuring proof tasks as multi-agent games (prover–verifier, multi-prover) where each agent is a neural network, and the overall protocol is judged according to completeness/soundness or other cryptographic criteria (Hammond et al., 12 Dec 2024).
• Neural-guided search: Guiding classical proof search, clause ranking, premise selection, or tactic prediction in symbolic provers using neural networks trained on proof traces or proof graphs (Loos et al., 2017, Sanchez-Stern et al., 2019, Sekiyama et al., 2018).
• Proof extraction and simplification: Employing neural models to mine reusable lemmas, modular substructures, or to compress/simplify large proof scripts while maintaining formal checkability (Zhou et al., 26 Feb 2024, Gu et al., 17 Oct 2025).

This leads to a taxonomy of neural proofs:

• Neural certificates: Parametric real-valued functions (often neural nets) acting as witnesses for system properties, verified via SMT or other formal solvers.
• Neural proof-search agents: Networks predicting proof steps, tactics, or clause priority in automated deduction environments.
• Graph-based neural proofs: Models which represent proof objects/structures as graphs and use GNNs for label prediction, substructure extraction, or proof net construction (Kogkalidis et al., 2020, Moot, 2022).
• Interactive neural proofs: Frameworks in which neural agents play roles akin to provers/verifiers in interactive or zero-knowledge protocols (Hammond et al., 12 Dec 2024).
• Abstraction-based neural proof production: Modular framework combining neural abstraction (intervals, zonotopes) with formal proof outputs for scalability (Elboher et al., 11 Jun 2025).

2. Neural Proofs in Formal Verification and Control

The seminal neural proofs framework for verification and control (Abate, 20 Dec 2025) is grounded in the synthesis of neural certificates capable of discharging semantic proof rules for dynamical, stochastic, or hybrid models:

• Model class: $\mathcal{S}$ with state space $X$; includes deterministic, stochastic, or controlled dynamics.
• Specification: Temporal properties $\psi$ (e.g., invariance, reachability, $\omega$-regular).
• Proof rule schema: Each class of specification gives a proof rule $R$ -- a universally quantified implication relating properties of a certificate function $C:X\to\mathbb{R}$ or $(C_s, C_r)$ (for reach-avoid).
• Synthesis (Learner–Verifier Loop):

1. Sampling: Obtain transitions, simulate rollouts.
2. Neural learner: Parameterize $C_\theta$ (e.g., MLP, ReLU/Tanh) and minimize empirical loss reflecting constraint violations over samples.
3. Verifier: Pose universal side-conditions as SMT queries over $C_\theta$ and (optionally) over the full model dynamics. The SMT returns UNSAT (accept), or provides counterexamples to refine training.
4. Repeat: Continue loop until soundness is verified.

Soundness is established by theorem: if the certified side-conditions are universally valid (witnessed by the SMT check), then the model $\mathcal{S}$ satisfies the desired temporal specification $\psi$ with mathematical certainty.

The methodology extends to simultaneous controller and certificate synthesis, enabling provably correct state-feedback policies $\kappa:X\to U$ together with neural control certificates (e.g., control-Lyapunov functions), all formally validated (Abate, 20 Dec 2025).

3. Neural Proofs for Neural Network Verification

3.1 Proof-producing Verifiers, Abstraction, and Shared Certificates

Neural proofs have enabled new verification workflows for deep neural networks (DNNs):

• Proof-producing LP/SMT methods: Enhanced simplex-based or SMT-based verifiers compute formal unsatisfiability certificates (e.g., Farkas vectors, explicit proof-trees) for safety properties, enabling independent checking and assurance (Isac et al., 2022). Proofs track every variable elimination, case-split, and bound tightening, producing a certificate that can be replayed without division or floating-point instability.
• Abstraction-based Proof Production: Frameworks modularize the task into (A) verifying an abstract, simplified network and (B) proving that the abstraction soundly overapproximates the real network. Proof composition leverages interval/hyperrectangle abstractions and their soundness rules, reducing proof size and enabling scalable, machine-checkable verification (Elboher et al., 11 Jun 2025).
• Shared certificates: Exploiting redundancy in proof obligations by constructing reusable intermediate templates ("shared certificates") at hidden layers—significantly reducing the end-to-end cost via proof subsumption and inclusion checking (Fischer et al., 2021).

3.2 Zero-Knowledge Proofs and Verifiable Execution

Protocols such as SafetyNets and zkDL generalize neural proofs to cryptographic, zero-knowledge settings:

• Arithmetic circuit representation: Neural networks are encoded as low-degree arithmetic circuits over finite fields, with sum-check-based interactive-proof protocols ensuring that the inference (or training) was executed as prescribed (Ghodsi et al., 2017, Sun et al., 2023).
• Zero-knowledge assurance: Advanced circuit gadgets (e.g., zkReLU) and circuit flattening techniques (FAC4DNN) provide privacy and succinctness, with per-batch proof times well below 1 second for 10M+ parameter networks.
• Soundness and completeness: Probability of undetected cheating is provably negligible in the field size, and privacy is information-theoretic.

4. Neural Proofs in Theorem Proving, Proof Synthesis, and Proof Compression

Neural proofs profoundly impact the practice of automated and interactive theorem proving:

• Neural-guided proof search: Supervised neural networks (CNN, TreeLSTM, GNN) guide clause selection, premise relevance, or tactic prediction, often within a hybrid search relying on classical ATP heuristics (Loos et al., 2017, Sanchez-Stern et al., 2019, Sekiyama et al., 2018). Measurable improvements in proof completion rates are observed.
• Graph-based neural proof nets: Sentence-to-proof architectures (e.g., Sinkhorn networks for axiom matching or GNN-based graph construction) yield differentiable pipelines mapping text or semantic forms to formal proof graphs, with high correctness for complex type-logical grammars (Kogkalidis et al., 2020, Moot, 2022).
• Proof extraction and modularity: Neural models (e.g., REFACTOR) are trained to locate theorems and reusable subtrees within large proof DAGs; this automates lemma discovery and proof refactoring, resulting in substantial proof length reductions and improved prover performance (Zhou et al., 26 Feb 2024).
• Proof compression and simplification: Large transformer models (ProofOptimizer) trained with RL and expert iteration are capable of simplifying massive Lean proofs by up to 87% in tokens, substantially reducing checking time and improving downstream neural-prover statistics—all with formal guarantees of completeness via Lean verification (Gu et al., 17 Oct 2025).
• Fine-grained proof augmentation: Techniques such as ProofAug embed LLMs within interactive proof state models, performing granular gap-filling by ATPs or heuristic tactic replay at any failed proof block, and recursively refining partial proofs for maximal compatibility and sample efficiency (Liu et al., 30 Jan 2025).

5. Interactive and Game-Theoretic Neural Proof Protocols

The emerging domain of neural interactive proofs frames proof tasks as multi-round games between neural agents:

• Formal structure: Prover and verifier networks exchange messages over $T$ rounds, with the protocol transcript evaluated with respect to completeness and soundness error rates (Hammond et al., 12 Dec 2024). Nash or Stackelberg equilibrium strategies correspond to valid proof protocols.
• Generalization of classical IPs: By parameterizing both sides with neural networks, this approach subsumes debate, adversarial, and multi-prover settings, and supports formal reductions to cryptographically significant protocols (e.g., zero-knowledge by adding simulators).
• Empirical illustration: Neural interactive proofs have shown robust advantage on toy graph isomorphism and competitive programming code validation benchmarks, outperforming conventional QA and static debate by leveraging dynamic, adaptive interaction and role specialization.
• Safety and oversight: By structuring oversight of powerful (possibly untrusted) AI agents as a neural interactive proof game, the protocol ensures verifiability, robustness (worst-case error rates), and transparency, with all decisions traceable to the transcript.

6. Logical Foundations and Elementary Proofs of Expressivity

• Universal approximation results: Foundational work establishes that neural networks with entire non-polynomial activations are dense in the space of polynomials (Stone–Weierstrass, Mergelyan), and under minimal structural constraints, fully connected or Residual networks with polynomial activations can realize all polynomials to arbitrary degree, giving "elementary" neural proofs for the functional approximation power of neural networks (Park et al., 2022).
• Harmonic and analytic networks: Parallel results exist for networks with harmonic and analytic activations, indicating that universality is a generic property for broad classes of activations, with constructive, stepwise proofs based on difference quotients and series truncation.

7. Limitations, Open Problems, and Future Directions

Neural proofs, though rapidly advancing, are constrained by several open fronts:

• Scalability: Constraints imposed by SMT/SAT-based verifiers limit the tractable system dimension or network depth, while aggressive abstraction or statistical verification can compromise pure soundness (Abate, 20 Dec 2025, Elboher et al., 11 Jun 2025).
• Expressivity: While safety, reachability, and some $\omega$-regular properties are amenable to neural proof methods, completeness for arbitrary temporal logic, complex program synthesis, or higher-order reasoning remains incomplete.
• Optimization bias and representation: Neural proof synthesis may founder on architecture-dependent inductive biases ("template bias"), non-convexity, or lack of suitable training data for rare or subtle proof patterns.
• Integration with classical symbolic tools: Full automation and performance gains in complex formal systems depend on tight coupling of neural and symbolic reasoning (e.g., robust interaction between LLMs, ATPs, and the ITP kernel), currently a domain of active engineering and algorithmic progress.
• Formal guarantees and trust: For critical applications (safety, cryptography, AI oversight), robust end-to-end soundness proofs, independent certificate checkers, and resistance to adversarial manipulation are essential. Approaches that combine zero-knowledge, interactive, and statistical guarantees may form a future backbone.

Across all dimensions, the neural proofs paradigm connects machine learning, formal logic, control theory, verification, and cryptography, and continues to generate new algorithms, formal models, and practical tools that advance the synthesis, verification, and understanding of high-dimensional formal reasoning.