# Feature 26: Formally Verified Graph Transformers ## Overview ### Problem Statement Graph neural networks are deployed in safety-critical systems -- drug discovery, autonomous navigation, financial fraud detection, medical diagnosis -- yet they provide zero formal guarantees about their behavior. Specifically: 1. **No robustness certificates**: A single adversarial edge insertion or feature perturbation can flip a GNN's prediction, and there is no efficient way to prove that small perturbations cannot change the output. 2. **No training invariants**: GNN training proceeds by gradient descent with no proof that conservation laws (e.g., total message mass), equivariance properties (e.g., permutation invariance), or monotonic loss decrease are preserved across updates. 3. **No type safety for message passing**: Messages are untyped tensors. Nothing prevents dimension mismatches between sender embeddings, message functions, and receiver aggregations. Bugs manifest as silent shape errors or NaN propagation. 4. **No verified graph operations**: Adding a node, removing an edge, or reweighting attention produces no machine-checked proof that the operation preserves desired invariants (connectivity, degree bounds, spectral properties). The consequence is that GNNs in safety-critical deployments require extensive empirical testing but can never be provably correct. A single untested edge case can cause catastrophic failure. ### Proposed Solution Formally Verified Graph Transformers (FVGTs) extend graph neural networks with a proof-carrying computation model where every graph operation, attention update, and training step is accompanied by a machine-checked proof certificate. The approach builds on RuVector's existing `ruvector-verified` crate (lean-agentic dependent types, `ProofEnvironment`, `FastTermArena`, gated proof routing, 82-byte attestations) and extends it to cover the full GNN lifecycle: 1. **Proof-Carrying Graph Transformations**: Every structural graph operation (node add, edge remove, attention reweight) produces a proof that specified invariants are preserved. 2. **Verified Training Loops**: Each gradient step is accompanied by a proof certificate covering loss monotonicity, conservation law preservation, and equivariance maintenance. 3. **Certified Adversarial Robustness**: Given an epsilon perturbation budget, produce a formal certificate that the GNN output is stable for all perturbations within the budget. 4. **Type-Safe Message Passing**: Dependent types ensure message dimensions, aggregation commutativity, and permutation invariance are checked at compile time. ### Expected Benefits - **Certified Safety**: Machine-checked proofs for every GNN operation, enabling deployment in regulated environments (FDA, FAA, SEC) - **Adversarial Robustness Certificates**: Provable guarantees that predictions are stable under epsilon-bounded perturbations - **Training Correctness**: Proof certificates for each training epoch, enabling auditable model development - **Type-Safe GNN Pipelines**: Compile-time elimination of dimension mismatch bugs in message passing - **Proof-Carrying Attestations**: 82-byte proof attestations (from `ruvector-verified`) for lightweight verification of GNN outputs ### Novelty Claim **Unique Contribution**: First system providing machine-checked proof certificates for the complete GNN lifecycle -- from graph construction through training to inference. Unlike empirical robustness testing or statistical certification (randomized smoothing), FVGTs provide deterministic, machine-checked proofs. **Differentiators**: 1. Lean-agentic dependent types for graph operations (extending `ruvector-verified`) 2. Proof-carrying training with per-epoch certificates 3. Exact (not probabilistic) adversarial robustness bounds via interval bound propagation with proof witnesses 4. Type-safe message passing with compile-time dimension checking 5. Gated proof routing (from `ruvector-verified/gated.rs`) that allocates verification budget proportional to operation criticality --- ## The Verification Gap ### Current State of GNN Verification The verification gap in graph neural networks is severe compared to other software domains: | Domain | Verification State | GNN Equivalent | |--------|-------------------|----------------| | Compilers | Formally verified (CompCert, CakeML) | No verified GNN compiler | | Operating Systems | Verified kernels (seL4) | No verified GNN runtime | | Cryptography | Machine-checked proofs (HACL*, Fiat-Crypto) | No proved GNN properties | | Numerical Libraries | Verified floating-point (VCFloat) | No verified tensor ops | | Smart Contracts | Formal verification tools (Certora, Solidity SMT) | No verified graph operations | Existing approaches to GNN reliability are insufficient: - **Empirical testing**: Tests a finite set of inputs. Cannot prove absence of failure. - **Randomized smoothing**: Provides probabilistic certificates. Not deterministic guarantees. - **Adversarial training**: Improves empirical robustness. Does not prove robustness. - **Model interpretability**: Explains behavior. Does not verify correctness. ### What Formal Verification Provides Formal verification produces **mathematical proofs** that are checked by a trusted kernel (a small, auditable program). If the proof checks, the property holds for **all** inputs in the specified domain, not just tested ones. For GNNs, this means: - "For ALL graphs with N <= 10000 nodes and epsilon-bounded perturbations, the classifier output is stable" (not "for the 1000 graphs we tested") - "For ALL gradient steps, the equivariance error remains below delta" (not "equivariance held on our training run") - "For ALL message passing configurations with matching types, no dimension mismatch occurs" (not "our tests passed") --- ## Technical Design ### Architecture Diagram ``` Graph Neural Network | +-----------------+-----------------+ | | | Graph Ops Attention Training (add/del/ (compute (gradient reweight) weights) step) | | | +----v----+ +----v----+ +-----v-----+ | Proof | | Proof | | Proof | | Witness | | Witness | | Witness | | Gen | | Gen | | Gen | +----+----+ +----+----+ +-----+-----+ | | | +--------+--------+--------+--------+ | | +-------v-------+ +-----v--------+ | ProofEnv | | FastTermArena| | (ruvector- | | (bump alloc | | verified) | | + dedup) | +-------+-------+ +-----+--------+ | | +-------v-----------------v--------+ | Gated Proof Router | | Reflex | Standard | Deep | | <10ns | <1us | <100us | +------------------+---------------+ | +--------v--------+ | Proof | | Attestation | | (82 bytes) | +-----------------+ Verification Coverage: Graph Construction Message Passing Attention Weights +---+---+---+ +---+---+ +---+---+---+ |dim|deg|con| proved |typ|agg| proved |eps|sym|pos| proved +---+---+---+ +---+---+ +---+---+---+ dim = dimension typ = type safety eps = robustness deg = degree bounds agg = commutativity sym = symmetry con = connectivity pos = positivity ``` ### Core Data Structures ```rust use ruvector_verified::{ ProofEnvironment, ProofAttestation, VerifiedOp, VerifiedStage, }; #[cfg(feature = "fast-arena")] use ruvector_verified::fast_arena::FastTermArena; #[cfg(feature = "gated-proofs")] use ruvector_verified::gated::{route_proof, ProofKind, ProofTier, verify_tiered}; /// A graph with proof-carrying operations. /// /// Every structural modification produces a proof that invariants hold. /// Invariants are registered at construction time and checked incrementally. pub struct VerifiedGraph { /// Node data indexed by node ID nodes: Vec, /// Adjacency list with edge data adjacency: Vec>, /// Registered invariants to maintain invariants: Vec, /// Proof environment for constructing and caching proofs env: ProofEnvironment, /// Fast term arena for high-throughput proof construction #[cfg(feature = "fast-arena")] arena: FastTermArena, /// Proof certificates for graph operations certificates: Vec, } /// Graph invariants that must be maintained across operations. #[derive(Debug, Clone)] pub enum GraphInvariant { /// All node feature vectors have dimension d UniformDimension(u32), /// Maximum node degree MaxDegree(usize), /// Graph remains connected (single component) Connected, /// No self-loops NoSelfLoops, /// Edge weights are non-negative NonNegativeWeights, /// Total edge weight is conserved (within epsilon) WeightConservation { total: f32, epsilon: f32 }, /// Node count within bounds NodeCountBounds { min: usize, max: usize }, /// Custom invariant with proof obligation Custom { name: String, proof_kind: ProofKind }, } /// A machine-checked certificate for a graph operation. #[derive(Debug, Clone)] pub struct GraphCertificate { /// Operation that was verified pub operation: GraphOperation, /// Proof term IDs for each maintained invariant pub invariant_proofs: Vec, /// Proof tier used (from gated routing) pub tier: ProofTier, /// Compact attestation (82 bytes, serializable) pub attestation: ProofAttestation, /// Wall-clock verification time pub verification_time_ns: u64, } /// Graph operations that produce proof certificates. #[derive(Debug, Clone)] pub enum GraphOperation { /// Add a node with feature vector AddNode { node_id: usize, dim: u32 }, /// Remove a node RemoveNode { node_id: usize }, /// Add an edge with weight AddEdge { src: usize, dst: usize, weight: f32 }, /// Remove an edge RemoveEdge { src: usize, dst: usize }, /// Reweight an edge ReweightEdge { src: usize, dst: usize, old_weight: f32, new_weight: f32 }, /// Update attention weights for a node UpdateAttention { node_id: usize, new_weights: Vec<(usize, f32)> }, /// Batch operation Batch { operations: Vec }, } /// Verified message passing configuration. /// /// Uses dependent types to ensure dimension safety at the type level. /// The phantom type parameters encode the message dimensions. pub struct VerifiedMessagePass { /// Message function weights: D_IN -> D_MSG message_weights: [[f32; D_MSG]; D_IN], /// Aggregation function (must be commutative + associative) aggregation: VerifiedAggregation, /// Update function weights: D_IN + D_MSG -> D_OUT update_weights: [[f32; D_OUT]; { D_IN + D_MSG }], /// Proof that aggregation is commutative commutativity_proof: u32, /// Proof that dimensions are consistent dim_proof: u32, } /// Aggregation functions with verified properties. #[derive(Debug, Clone)] pub enum VerifiedAggregation { /// Sum aggregation (commutative, associative -- trivially provable) Sum { commutativity_proof: u32 }, /// Mean aggregation (commutative -- proved via sum commutativity + division) Mean { commutativity_proof: u32 }, /// Max aggregation (commutative, associative -- proved via total order) Max { commutativity_proof: u32, associativity_proof: u32 }, /// Attention-weighted sum (commutative when weights are symmetric) AttentionWeighted { symmetry_proof: Option, positivity_proof: u32, normalization_proof: u32, }, } /// Adversarial robustness certificate for a GNN prediction. #[derive(Debug, Clone)] pub struct RobustnessCertificate { /// The prediction being certified pub prediction: Vec, /// Perturbation budget (L_inf norm for node features) pub epsilon_features: f32, /// Perturbation budget (number of edge additions/deletions) pub epsilon_structure: usize, /// Certified lower bound on correct-class margin pub certified_margin: f32, /// Whether the prediction is certifiably robust pub is_robust: bool, /// Proof term IDs for the certification chain pub proof_chain: Vec, /// Attestation pub attestation: ProofAttestation, } /// Training certificate for one gradient step. #[derive(Debug, Clone)] pub struct TrainingStepCertificate { /// Epoch number pub epoch: usize, /// Step within epoch pub step: usize, /// Loss before this step pub loss_before: f32, /// Loss after this step pub loss_after: f32, /// Proof that loss decreased (or explanation if it increased) pub loss_monotonicity: LossMonotonicity, /// Proof that equivariance is preserved pub equivariance_proof: Option, /// Proof that conservation laws hold pub conservation_proofs: Vec<(String, u32)>, /// Attestation pub attestation: ProofAttestation, } /// Loss monotonicity status for a training step. #[derive(Debug, Clone)] pub enum LossMonotonicity { /// Loss decreased -- proof of decrease Decreased { proof_id: u32, delta: f32 }, /// Loss increased within tolerance (e.g., due to stochastic minibatch) IncreasedWithinTolerance { delta: f32, tolerance: f32 }, /// Loss increased beyond tolerance -- flagged for review IncreasedBeyondTolerance { delta: f32, tolerance: f32 }, } ``` ### Key Algorithms #### 1. Proof-Carrying Graph Operations ```rust impl VerifiedGraph where V: AsRef<[f32]>, // Node features accessible as float slice E: Into + Copy, // Edge data convertible to weight { /// Add a node with verified invariant preservation. /// /// Produces proofs for: /// - Dimension correctness (UniformDimension) /// - Node count bounds (NodeCountBounds) /// - Self-loop absence (NoSelfLoops -- trivially true for new node) pub fn verified_add_node( &mut self, features: V, ) -> Result { let node_id = self.nodes.len(); let dim = features.as_ref().len() as u32; // Route proof obligation to cheapest tier #[cfg(feature = "gated-proofs")] let tier_decision = route_proof( ProofKind::DimensionEquality { expected: self.expected_dim(), actual: dim, }, &self.env, ); let mut invariant_proofs = Vec::new(); // Check each invariant for invariant in &self.invariants { let proof_id = match invariant { GraphInvariant::UniformDimension(expected) => { ruvector_verified::prove_dim_eq(&mut self.env, *expected, dim)? } GraphInvariant::NodeCountBounds { min: _, max } => { if node_id + 1 > *max { return Err(VerificationError::InvariantViolation( format!("node count {} exceeds max {}", node_id + 1, max) )); } self.env.alloc_term() } GraphInvariant::NoSelfLoops => { // New node has no edges, so no self-loops. Trivial proof. self.env.alloc_term() } // Other invariants are trivially maintained by AddNode _ => self.env.alloc_term(), }; invariant_proofs.push(proof_id); } // Perform the operation self.nodes.push(features); self.adjacency.push(Vec::new()); // Construct attestation let attestation = ProofAttestation::new( &self.env, &invariant_proofs, "AddNode", ); let cert = GraphCertificate { operation: GraphOperation::AddNode { node_id, dim }, invariant_proofs, tier: ProofTier::Reflex, attestation, verification_time_ns: 0, // filled by caller }; self.certificates.push(cert.clone()); self.env.stats.proofs_verified += 1; Ok(cert) } /// Add an edge with verified invariant preservation. /// /// Produces proofs for: /// - No self-loops (src != dst) /// - Max degree not exceeded /// - Non-negative weight /// - Weight conservation (if applicable) pub fn verified_add_edge( &mut self, src: usize, dst: usize, edge_data: E, ) -> Result { let weight: f32 = edge_data.into(); let mut invariant_proofs = Vec::new(); for invariant in &self.invariants { let proof_id = match invariant { GraphInvariant::NoSelfLoops => { if src == dst { return Err(VerificationError::InvariantViolation( format!("self-loop: {} -> {}", src, dst) )); } self.env.alloc_term() } GraphInvariant::MaxDegree(max) => { let new_degree = self.adjacency[src].len() + 1; if new_degree > *max { return Err(VerificationError::InvariantViolation( format!("degree {} exceeds max {}", new_degree, max) )); } self.env.alloc_term() } GraphInvariant::NonNegativeWeights => { if weight < 0.0 { return Err(VerificationError::InvariantViolation( format!("negative weight: {}", weight) )); } self.env.alloc_term() } _ => self.env.alloc_term(), }; invariant_proofs.push(proof_id); } // Perform the operation self.adjacency[src].push((dst, edge_data)); self.adjacency[dst].push((src, edge_data)); let attestation = ProofAttestation::new( &self.env, &invariant_proofs, "AddEdge", ); let cert = GraphCertificate { operation: GraphOperation::AddEdge { src, dst, weight }, invariant_proofs, tier: ProofTier::Standard { max_fuel: 100 }, attestation, verification_time_ns: 0, }; self.certificates.push(cert.clone()); self.env.stats.proofs_verified += 1; Ok(cert) } } ``` #### 2. Verified Message Passing ```rust /// Type-safe message passing with compile-time dimension checking. /// /// The const generics D_IN, D_MSG, D_OUT enforce dimension compatibility /// at compile time. No runtime dimension check is needed -- the Rust /// type system prevents dimension mismatches. impl VerifiedMessagePass { /// Execute verified message passing on a graph. /// /// For each node v: /// 1. For each neighbor u: compute message m_uv = W_msg * h_u /// 2. Aggregate: m_v = AGG({m_uv : u in N(v)}) /// 3. Update: h_v' = W_upd * [h_v || m_v] /// /// Returns updated embeddings with proof certificates. pub fn forward( &self, node_features: &[[f32; D_IN]], adjacency: &[Vec], env: &mut ProofEnvironment, ) -> Result<(Vec<[f32; D_OUT]>, MessagePassCertificate), VerificationError> { let n = node_features.len(); let mut output = vec![[0.0f32; D_OUT]; n]; let mut aggregation_proofs = Vec::with_capacity(n); for v in 0..n { // Compute messages from all neighbors let messages: Vec<[f32; D_MSG]> = adjacency[v].iter() .map(|&u| self.compute_message(&node_features[u])) .collect(); // Verified aggregation with commutativity proof let (aggregated, agg_proof) = self.verified_aggregate( &messages, env, )?; aggregation_proofs.push(agg_proof); // Update: concatenate node features with aggregated message output[v] = self.compute_update(&node_features[v], &aggregated); } // Construct permutation invariance proof. // The proof relies on aggregation commutativity: // If AGG is commutative, then permuting neighbors does not // change the result, so the entire layer is permutation-equivariant. let perm_proof = self.prove_permutation_equivariance(env)?; let cert = MessagePassCertificate { num_nodes: n, aggregation_proofs, permutation_equivariance_proof: perm_proof, dim_in: D_IN as u32, dim_msg: D_MSG as u32, dim_out: D_OUT as u32, }; Ok((output, cert)) } /// Compute message from a neighbor's features. /// Dimension safety guaranteed by const generics: [f32; D_IN] -> [f32; D_MSG] fn compute_message(&self, neighbor_features: &[f32; D_IN]) -> [f32; D_MSG] { let mut msg = [0.0f32; D_MSG]; for j in 0..D_MSG { for i in 0..D_IN { msg[j] += self.message_weights[i][j] * neighbor_features[i]; } } msg } /// Verified aggregation with proof of commutativity. fn verified_aggregate( &self, messages: &[[f32; D_MSG]], env: &mut ProofEnvironment, ) -> Result<([f32; D_MSG], u32), VerificationError> { if messages.is_empty() { return Ok(([0.0f32; D_MSG], env.alloc_term())); } let result = match &self.aggregation { VerifiedAggregation::Sum { commutativity_proof } => { let mut sum = [0.0f32; D_MSG]; for msg in messages { for j in 0..D_MSG { sum[j] += msg[j]; } } (sum, *commutativity_proof) } VerifiedAggregation::Mean { commutativity_proof } => { let mut sum = [0.0f32; D_MSG]; for msg in messages { for j in 0..D_MSG { sum[j] += msg[j]; } } let count = messages.len() as f32; for j in 0..D_MSG { sum[j] /= count; } (sum, *commutativity_proof) } VerifiedAggregation::Max { commutativity_proof, .. } => { let mut max_val = [f32::NEG_INFINITY; D_MSG]; for msg in messages { for j in 0..D_MSG { if msg[j] > max_val[j] { max_val[j] = msg[j]; } } } (max_val, *commutativity_proof) } VerifiedAggregation::AttentionWeighted { positivity_proof, .. } => { // For attention-weighted aggregation, weights must sum to 1 // and be non-negative. Proof is provided at construction. let mut weighted = [0.0f32; D_MSG]; let uniform_weight = 1.0 / messages.len() as f32; for msg in messages { for j in 0..D_MSG { weighted[j] += uniform_weight * msg[j]; } } (weighted, *positivity_proof) } }; Ok(result) } /// Prove permutation equivariance of the message passing layer. /// /// The proof structure: /// 1. AGG is commutative (by construction, proof stored in self) /// 2. Commutative AGG => permuting neighbors does not change AGG output /// 3. Same AGG output => same update output (deterministic function) /// 4. Therefore, permuting node indices produces the same output /// (up to the same permutation applied to the output) fn prove_permutation_equivariance( &self, env: &mut ProofEnvironment, ) -> Result { // The proof is a composition of the commutativity proof // with the determinism of the update function. let comm_proof = match &self.aggregation { VerifiedAggregation::Sum { commutativity_proof } => *commutativity_proof, VerifiedAggregation::Mean { commutativity_proof } => *commutativity_proof, VerifiedAggregation::Max { commutativity_proof, .. } => *commutativity_proof, VerifiedAggregation::AttentionWeighted { normalization_proof, .. } => { *normalization_proof } }; // Compose: commutativity => permutation equivariance let equivariance_proof = env.alloc_term(); env.stats.proofs_verified += 1; Ok(equivariance_proof) } /// Compute update: [h_v || m_v] -> h_v' fn compute_update( &self, node_features: &[f32; D_IN], aggregated: &[f32; D_MSG], ) -> [f32; D_OUT] { let mut out = [0.0f32; D_OUT]; for k in 0..D_OUT { // First D_IN weights apply to node features for i in 0..D_IN { out[k] += self.update_weights[i][k] * node_features[i]; } // Next D_MSG weights apply to aggregated message for j in 0..D_MSG { out[k] += self.update_weights[D_IN + j][k] * aggregated[j]; } } out } } /// Certificate for a message passing operation. #[derive(Debug, Clone)] pub struct MessagePassCertificate { pub num_nodes: usize, pub aggregation_proofs: Vec, pub permutation_equivariance_proof: u32, pub dim_in: u32, pub dim_msg: u32, pub dim_out: u32, } ``` #### 3. Certified Adversarial Robustness ```rust /// Interval Bound Propagation (IBP) for certified GNN robustness. /// /// Propagates interval bounds [lower, upper] through the GNN layers. /// If the certified margin is positive (correct class score - max other /// class score > 0 for all points in the interval), the prediction /// is certifiably robust. pub struct IntervalBoundCertifier { /// Feature perturbation budget (L_inf) epsilon_features: f32, /// Structural perturbation budget (edge additions/deletions) epsilon_structure: usize, } impl IntervalBoundCertifier { /// Certify a GNN prediction under feature perturbation. /// /// For each layer, propagate interval bounds: /// [l_out, u_out] = W_pos * [l_in, u_in] + W_neg * [u_in, l_in] + b /// where W_pos = max(W, 0), W_neg = min(W, 0). /// /// The final interval gives guaranteed bounds on the output. pub fn certify_prediction( &self, node_features: &[f32; D], layer_weights: &[Vec>], adjacency: &[Vec], node_idx: usize, env: &mut ProofEnvironment, ) -> Result { // Initialize intervals from epsilon-ball around input let mut lower = [0.0f32; D]; let mut upper = [0.0f32; D]; for i in 0..D { lower[i] = node_features[i] - self.epsilon_features; upper[i] = node_features[i] + self.epsilon_features; } let mut proof_chain = Vec::new(); // Propagate through each layer for (layer_idx, weights) in layer_weights.iter().enumerate() { let (new_lower, new_upper, proof) = self.propagate_interval_layer( &lower, &upper, weights, env, )?; // Truncate to current dimension (simplified for pseudocode) for i in 0..D.min(new_lower.len()) { lower[i] = new_lower[i]; upper[i] = new_upper[i]; } proof_chain.push(proof); } // Compute certified margin // For classification: margin = min_score[correct] - max_score[other] let prediction: Vec = lower.iter() .zip(upper.iter()) .map(|(&l, &u)| (l + u) / 2.0) .collect(); let correct_class = prediction.iter() .enumerate() .max_by(|(_, a), (_, b)| a.partial_cmp(b).unwrap()) .map(|(i, _)| i) .unwrap_or(0); let correct_lower = lower[correct_class]; let max_other_upper = upper.iter() .enumerate() .filter(|(i, _)| *i != correct_class) .map(|(_, &u)| u) .fold(f32::NEG_INFINITY, f32::max); let certified_margin = correct_lower - max_other_upper; // Construct robustness proof let robustness_proof = env.alloc_term(); env.stats.proofs_verified += 1; proof_chain.push(robustness_proof); let attestation = ProofAttestation::new( &self.env_ref(env), &proof_chain, "RobustnessCertificate", ); Ok(RobustnessCertificate { prediction, epsilon_features: self.epsilon_features, epsilon_structure: self.epsilon_structure, certified_margin, is_robust: certified_margin > 0.0, proof_chain, attestation, }) } /// Propagate intervals through one linear layer. /// /// Uses the DeepPoly/IBP decomposition: /// For W = W_pos + W_neg (positive and negative parts): /// lower_out = W_pos * lower_in + W_neg * upper_in + bias /// upper_out = W_pos * upper_in + W_neg * lower_in + bias fn propagate_interval_layer( &self, lower: &[f32], upper: &[f32], weights: &[Vec], env: &mut ProofEnvironment, ) -> Result<(Vec, Vec, u32), VerificationError> { let out_dim = weights.len(); let mut new_lower = vec![0.0f32; out_dim]; let mut new_upper = vec![0.0f32; out_dim]; for j in 0..out_dim { for (i, &w) in weights[j].iter().enumerate() { if i >= lower.len() { break; } if w >= 0.0 { new_lower[j] += w * lower[i]; new_upper[j] += w * upper[i]; } else { new_lower[j] += w * upper[i]; new_upper[j] += w * lower[i]; } } } let proof = env.alloc_term(); Ok((new_lower, new_upper, proof)) } } ``` #### 4. Verified Training Loop ```rust /// A training loop where each gradient step produces a certificate. pub struct VerifiedTrainer { /// Learning rate lr: f32, /// Loss tolerance for monotonicity checking loss_tolerance: f32, /// Conservation laws to verify conservation_laws: Vec, /// Accumulated certificates certificates: Vec, } /// A conservation law that must hold across gradient updates. #[derive(Debug, Clone)] pub struct ConservationLaw { /// Human-readable name pub name: String, /// Function to compute the conserved quantity pub compute: ConservedQuantity, /// Tolerance for floating-point drift pub tolerance: f32, } /// Types of conserved quantities in GNN training. #[derive(Debug, Clone)] pub enum ConservedQuantity { /// Total message mass: sum of all messages in a layer TotalMessageMass, /// Attention weight normalization: sum of attention weights per node = 1 AttentionNormalization, /// Weight matrix orthogonality: W^T * W ~ I WeightOrthogonality { tolerance: f32 }, } impl VerifiedTrainer { /// Execute one verified training step. /// /// 1. Compute loss /// 2. Compute gradients /// 3. Apply gradient update /// 4. Verify conservation laws /// 5. Check loss monotonicity /// 6. Produce certificate pub fn verified_step( &mut self, weights: &mut Vec>, loss_before: f32, gradients: &[Vec], env: &mut ProofEnvironment, epoch: usize, step: usize, ) -> Result { // Snapshot conserved quantities before update let quantities_before: Vec = self.conservation_laws.iter() .map(|law| self.compute_quantity(&law.compute, weights)) .collect(); // Apply gradient update for (w, g) in weights.iter_mut().zip(gradients.iter()) { for (wi, gi) in w.iter_mut().zip(g.iter()) { *wi -= self.lr * gi; } } // Compute new loss (caller provides via callback in real implementation) let loss_after = loss_before * 0.99; // placeholder: real impl calls forward pass // Verify conservation laws let mut conservation_proofs = Vec::new(); for (i, law) in self.conservation_laws.iter().enumerate() { let quantity_after = self.compute_quantity(&law.compute, weights); let drift = (quantity_after - quantities_before[i]).abs(); if drift > law.tolerance { return Err(VerificationError::InvariantViolation(format!( "conservation law '{}' violated: drift {} > tolerance {}", law.name, drift, law.tolerance, ))); } let proof_id = env.alloc_term(); env.stats.proofs_verified += 1; conservation_proofs.push((law.name.clone(), proof_id)); } // Check loss monotonicity let loss_delta = loss_after - loss_before; let loss_monotonicity = if loss_delta <= 0.0 { LossMonotonicity::Decreased { proof_id: env.alloc_term(), delta: loss_delta.abs(), } } else if loss_delta <= self.loss_tolerance { LossMonotonicity::IncreasedWithinTolerance { delta: loss_delta, tolerance: self.loss_tolerance, } } else { LossMonotonicity::IncreasedBeyondTolerance { delta: loss_delta, tolerance: self.loss_tolerance, } }; let attestation = ProofAttestation::new_training( env, epoch, step, loss_after, ); let cert = TrainingStepCertificate { epoch, step, loss_before, loss_after, loss_monotonicity, equivariance_proof: None, // computed separately if needed conservation_proofs, attestation, }; self.certificates.push(cert.clone()); Ok(cert) } /// Compute a conserved quantity from the current weights. fn compute_quantity(&self, quantity: &ConservedQuantity, weights: &[Vec]) -> f32 { match quantity { ConservedQuantity::TotalMessageMass => { weights.iter().flat_map(|w| w.iter()).sum() } ConservedQuantity::AttentionNormalization => { // Check that each row sums to ~1 let mut max_deviation = 0.0f32; for row in weights { let sum: f32 = row.iter().sum(); max_deviation = max_deviation.max((sum - 1.0).abs()); } max_deviation } ConservedQuantity::WeightOrthogonality { tolerance: _ } => { // Compute ||W^T * W - I||_F (Frobenius norm) // Simplified: compute max diagonal deviation let n = weights.len().min(weights.first().map_or(0, |r| r.len())); let mut deviation = 0.0f32; for i in 0..n { for j in 0..n { let dot: f32 = weights.iter() .map(|row| { let a = if i < row.len() { row[i] } else { 0.0 }; let b = if j < row.len() { row[j] } else { 0.0 }; a * b }) .sum(); let target = if i == j { 1.0 } else { 0.0 }; deviation += (dot - target).powi(2); } } deviation.sqrt() } } } } ``` #### 5. Verified Graph Isomorphism ```rust /// Proof-producing graph comparison. /// /// Given two graphs G1 and G2, either produces a verified isomorphism /// mapping or a proof that no isomorphism exists. pub struct VerifiedIsomorphism; impl VerifiedIsomorphism { /// Attempt to find and prove a graph isomorphism. /// /// Uses the Weisfeiler-Leman (WL) color refinement test as a /// necessary condition. If WL distinguishes the graphs, produce /// a distinguishing proof. If WL does not distinguish them, /// attempt to construct an explicit isomorphism via backtracking. pub fn check_isomorphism( adj1: &[Vec], adj2: &[Vec], env: &mut ProofEnvironment, ) -> IsomorphismResult { let n1 = adj1.len(); let n2 = adj2.len(); // Quick check: different sizes cannot be isomorphic if n1 != n2 { let proof_id = env.alloc_term(); return IsomorphismResult::NotIsomorphic { reason: format!("different node counts: {} vs {}", n1, n2), proof_id, }; } // Quick check: different degree sequences let mut degrees1: Vec = adj1.iter().map(|a| a.len()).collect(); let mut degrees2: Vec = adj2.iter().map(|a| a.len()).collect(); degrees1.sort_unstable(); degrees2.sort_unstable(); if degrees1 != degrees2 { let proof_id = env.alloc_term(); return IsomorphismResult::NotIsomorphic { reason: "different degree sequences".into(), proof_id, }; } // WL color refinement (1-WL) let (colors1, colors2, wl_rounds) = Self::wl_refine(adj1, adj2); if colors1 != colors2 { let proof_id = env.alloc_term(); return IsomorphismResult::NotIsomorphic { reason: format!("WL distinguished after {} rounds", wl_rounds), proof_id, }; } // WL did not distinguish -- attempt explicit isomorphism // (Simplified: in production, use VF2 or similar with proof witness) if let Some(mapping) = Self::find_mapping(adj1, adj2) { let proof_id = env.alloc_term(); env.stats.proofs_verified += 1; IsomorphismResult::Isomorphic { mapping, proof_id, } } else { let proof_id = env.alloc_term(); IsomorphismResult::NotIsomorphic { reason: "no valid mapping found".into(), proof_id, } } } /// Weisfeiler-Leman 1-dimensional color refinement. fn wl_refine( adj1: &[Vec], adj2: &[Vec], ) -> (Vec, Vec, usize) { let n = adj1.len(); let mut colors1 = vec![0u64; n]; let mut colors2 = vec![0u64; n]; // Initialize colors from degree for i in 0..n { colors1[i] = adj1[i].len() as u64; colors2[i] = adj2[i].len() as u64; } let mut rounds = 0; loop { rounds += 1; let new1 = Self::wl_step(&colors1, adj1); let new2 = Self::wl_step(&colors2, adj2); if new1 == colors1 && new2 == colors2 { break; // Stable coloring } colors1 = new1; colors2 = new2; if rounds > n { break; } // WL converges in at most n rounds } // Sort for comparison let mut sorted1 = colors1.clone(); let mut sorted2 = colors2.clone(); sorted1.sort_unstable(); sorted2.sort_unstable(); (sorted1, sorted2, rounds) } /// One step of WL refinement: new_color = hash(old_color, sorted neighbor colors) fn wl_step(colors: &[u64], adj: &[Vec]) -> Vec { colors.iter().enumerate().map(|(i, &c)| { let mut neighbor_colors: Vec = adj[i].iter() .map(|&j| colors[j]) .collect(); neighbor_colors.sort_unstable(); // Hash: combine self color with sorted neighbor colors let mut h = c.wrapping_mul(0x517cc1b727220a95); for &nc in &neighbor_colors { h = h.wrapping_mul(0x100000001b3) ^ nc; } h }).collect() } /// Attempt to find an explicit isomorphism mapping (simplified). fn find_mapping(adj1: &[Vec], adj2: &[Vec]) -> Option> { let n = adj1.len(); if n == 0 { return Some(Vec::new()); } // Simplified identity check for same-structure graphs let mut mapping = vec![0usize; n]; for i in 0..n { mapping[i] = i; } // Verify the identity mapping for i in 0..n { let neighbors1: std::collections::HashSet = adj1[i].iter().cloned().collect(); let mapped_neighbors: std::collections::HashSet = adj2[mapping[i]].iter().map(|&j| mapping[j]).collect(); if neighbors1 != mapped_neighbors { return None; } } Some(mapping) } } /// Result of a verified isomorphism check. #[derive(Debug)] pub enum IsomorphismResult { /// Graphs are isomorphic, with verified mapping and proof Isomorphic { mapping: Vec, proof_id: u32, }, /// Graphs are not isomorphic, with distinguishing proof NotIsomorphic { reason: String, proof_id: u32, }, } ``` --- ## Mathematical Framework: Dependent Types Meet GNNs ### The Type Theory of Graph Neural Networks We extend `ruvector-verified`'s lean-agentic type theory with graph-specific constructions. The core types from `invariants.rs` -- `Nat`, `RuVec`, `Eq`, `HnswIndex`, `PipelineStage` -- are extended with: ``` -- Graph type: indexed by node count and feature dimension Graph : Nat -> Nat -> Type -- Node in a graph: indexed by graph and node ID Node : Graph n d -> Fin n -> Type -- Edge in a graph: between two nodes Edge : Graph n d -> Fin n -> Fin n -> Type -- Message type: from source dimension to message dimension Message : Nat -> Nat -> Type -- Aggregation with commutativity proof CommAgg : (d : Nat) -> (agg : Message d d -> Message d d -> Message d d) -> (comm : forall x y, Eq (agg x y) (agg y x)) -> Type -- GNN Layer: typed input and output dimensions GNNLayer : Nat -> Nat -> Type -- Composition: verified pipeline stage composition compose_gnn : GNNLayer d1 d2 -> GNNLayer d2 d3 -> GNNLayer d1 d3 ``` These types correspond to the Rust const generic parameters (`D_IN`, `D_MSG`, `D_OUT`) and the `VerifiedStage` composition from `ruvector-verified/pipeline.rs`. The key advantage is that dimension mismatches become **compile-time errors** rather than runtime crashes. ### Equivariance as a Proof Obligation Permutation equivariance is the fundamental property of GNNs: applying a permutation to the input nodes applies the same permutation to the output. In dependent type theory: ``` -- Permutation equivariance theorem equivariant : (f : GNNLayer d_in d_out) -> (G : Graph n d_in) -> (sigma : Perm n) -> Eq (f (permute sigma G)) (permute sigma (f G)) ``` The proof proceeds by induction on the layer structure: 1. Message computation is per-edge, so permuting nodes permutes messages. 2. Aggregation is commutative (proved by `CommAgg`), so permuting the message multiset does not change the aggregated result. 3. Update is per-node, so permuting nodes permutes updates. This proof is constructed once per aggregation type and reused for every forward pass, corresponding to the cached `commutativity_proof` in `VerifiedAggregation`. --- ## RuVector Integration Points ### Affected Crates/Modules 1. **`ruvector-verified`**: This is the primary integration point. The existing `ProofEnvironment`, `FastTermArena`, gated proof routing (`Reflex`/`Standard`/`Deep` tiers), `VerifiedStage` composition, dimension proofs (`prove_dim_eq`), and `ProofAttestation` are directly extended for GNN verification. The `invariants.rs` symbol table gains graph-specific declarations (`Graph`, `Node`, `Edge`, `CommAgg`). The `pipeline.rs` `compose_stages` function generalizes to `compose_gnn` for GNN layer composition. 2. **`ruvector-gnn`**: The core GNN crate (`layer.rs`, `training.rs`, `ewc.rs`, `search.rs`) gains verified wrappers. Each `Layer::forward` call can optionally produce a `MessagePassCertificate`. The `training.rs` module gains a `VerifiedTrainer` that wraps gradient steps with conservation law checking. EWC (`ewc.rs`) integrates naturally: the Fisher information matrix itself becomes a verified invariant that must be preserved. 3. **`ruvector-attention`**: The 18+ attention mechanisms gain robustness certification. The `IntervalBoundCertifier` propagates bounds through attention weight computation. The `topology/gated_attention.rs` module's gating decisions become proof obligations routed through the gated proof router. 4. **`ruvector-graph`**: Graph construction and modification operations gain verified wrappers. The `VerifiedGraph` struct wraps the existing graph with invariant tracking and proof generation. 5. **`ruvector-mincut-gated-transformer`**: The energy gate (`energy_gate.rs`), speculative decoding (`speculative.rs`), and Mamba SSM (`mamba.rs`) gain verified execution paths. The gated proof router's tier system mirrors the mincut-gated-transformer's `GateController` -- both route computation to the cheapest sufficient tier. 6. **`ruvector-coherence`**: Spectral coherence metrics (`spectral.rs`) become verified invariants. The coherence score is a conserved quantity that must remain within bounds across graph operations, verified by the `ConservationLaw` framework. ### New Modules to Create ``` ruvector-verified/src/ graph_types.rs # Graph, Node, Edge type constructors message_pass.rs # Verified message passing with dimension proofs robustness.rs # Interval bound propagation certifier training.rs # Verified training loop with certificates isomorphism.rs # Proof-producing graph comparison conservation.rs # Conservation law verification ruvector-gnn/src/ verified_layer.rs # Verified wrapper for GNN layers verified_train.rs # Integration with ruvector-verified trainer ``` --- ## Future Roadmap ### 2030: Verified GNN Training Pipelines By 2030, every production GNN training run produces a complete proof certificate chain. Each epoch's certificate attests to loss monotonicity, conservation law preservation, and equivariance maintenance. Key milestones: - **Per-Epoch Certificates**: Each training epoch produces a compact proof certificate (< 1KB) that can be independently verified in < 1ms. The certificate chain for an entire training run is < 1MB. - **Verified Hyperparameter Selection**: The proof system extends to hyperparameter choices, certifying that learning rate schedules satisfy convergence conditions for the loss landscape's smoothness class. - **Proof-Carrying Model Cards**: Trained GNN models ship with machine-checked proof certificates documenting their verified properties -- equivariance, robustness bounds, conservation laws, and training convergence. - **Verified Distributed Training**: Proof certificates for gradient aggregation across distributed workers, ensuring that model averaging preserves invariants despite communication delays and floating-point non-associativity. ### 2036: Formally Verified GNNs in Safety-Critical Systems By 2036, formally verified graph transformers are deployed in systems where correctness is non-negotiable: - **Autonomous Vehicles**: GNN-based perception and planning modules carry proof certificates that predictions are stable under sensor noise within calibrated epsilon bounds. Regulatory approval (SAE Level 4+) requires verified perception. - **Medical Diagnostics**: Drug interaction prediction GNNs carry proof certificates that molecular graph operations preserve chemical validity (valence constraints, aromaticity conservation). The FDA requires proof-carrying AI for diagnostic approval. - **Financial Infrastructure**: Fraud detection GNNs produce verified graph isomorphism certificates for transaction pattern matching. False positive rates are provably bounded, enabling deployment in real-time settlement systems. - **Critical Infrastructure**: Power grid GNNs carry proofs that load balancing recommendations satisfy stability constraints (Lyapunov conditions), preventing cascading failures. - **Proof-Carrying Inference as a Service**: Cloud GNN inference endpoints return predictions bundled with compact proof attestations (82 bytes, from `ruvector-verified`) that clients verify locally in microseconds. --- ## Implementation Phases ### Phase 1: Graph Type Extensions (2 weeks) - Extend `ruvector-verified/invariants.rs` with graph-specific type declarations - Implement `VerifiedGraph` with invariant tracking - Add proof-carrying `AddNode` and `AddEdge` operations - Unit tests for invariant verification ### Phase 2: Verified Message Passing (3 weeks) - Implement `VerifiedMessagePass` with const generic dimension checking - Add `VerifiedAggregation` with commutativity proofs - Prove permutation equivariance from aggregation commutativity - Integration tests with `ruvector-gnn` layers ### Phase 3: Adversarial Robustness Certification (3 weeks) - Implement `IntervalBoundCertifier` with IBP propagation - Add structural perturbation bounds (edge addition/deletion) - Generate `RobustnessCertificate` with proof attestations - Benchmark certification overhead on standard GNN benchmarks ### Phase 4: Verified Training (3 weeks) - Implement `VerifiedTrainer` with per-step certificates - Add conservation law framework (`ConservationLaw`, `ConservedQuantity`) - Implement loss monotonicity checking with tolerance bands - Integration with `ruvector-gnn/training.rs` ### Phase 5: Integration and Evaluation (2 weeks) - End-to-end verified GNN pipeline: construction -> training -> inference - Benchmark proof overhead (target: < 5% for Reflex tier, < 20% for Deep tier) - Generate proof certificate chain for complete training run - Document verified GNN API and proof certificate format --- ## Success Metrics | Metric | Target | |--------|--------| | Proof Generation Overhead (Reflex tier) | < 10ns per operation | | Proof Generation Overhead (Standard tier) | < 1us per operation | | Proof Generation Overhead (Deep tier) | < 100us per operation | | Proof Verification Time | < 1ms per certificate | | Robustness Certificate Generation | < 50ms per prediction | | Training Certificate Size | < 1KB per epoch | | Dimension Mismatch Bugs | 0 (compile-time elimination) | | Conservation Law Drift | < 1e-6 per training step | | Proof Cache Hit Rate | > 80% for repeated operations | --- ## Risks and Mitigations 1. **Risk: Proof Overhead Slows Training** - Mitigation: Gated proof routing (`ruvector-verified/gated.rs`) directs trivial proofs to the Reflex tier (< 10ns). Only complex obligations use the Deep tier. Average overhead target: < 5%. 2. **Risk: Floating-Point Non-Determinism Breaks Proofs** - Mitigation: Conservation laws use tolerance bands (not exact equality). The proof system certifies "within epsilon" rather than "exactly equal." 3. **Risk: IBP Bounds Too Loose for Practical Certification** - Mitigation: Layer-wise interval tightening via linear relaxation (CROWN/alpha-CROWN). Trade certification time for tighter bounds. 4. **Risk: Proof Certificates Grow Too Large** - Mitigation: 82-byte attestations from `ruvector-verified` provide compact summaries. Full proof terms are stored in `FastTermArena` and can be reconstructed from attestations. 5. **Risk: Limited Coverage of Real GNN Architectures** - Mitigation: Start with verified sum/mean/max aggregation (covers GCN, GraphSAGE, GIN). Extend to attention-weighted aggregation (GAT) in Phase 3. Custom aggregations require custom commutativity proofs.