d803bfe2b1
git-subtree-dir: vendor/ruvector git-subtree-split: b64c21726f2bb37286d9ee36a7869fef60cc6900
1361 lines
52 KiB
Markdown
1361 lines
52 KiB
Markdown
# Feature 26: Formally Verified Graph Transformers
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## Overview
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### Problem Statement
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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:
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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.
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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.
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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.
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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).
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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.
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### Proposed Solution
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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:
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1. **Proof-Carrying Graph Transformations**: Every structural graph operation (node add, edge remove, attention reweight) produces a proof that specified invariants are preserved.
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2. **Verified Training Loops**: Each gradient step is accompanied by a proof certificate covering loss monotonicity, conservation law preservation, and equivariance maintenance.
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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.
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4. **Type-Safe Message Passing**: Dependent types ensure message dimensions, aggregation commutativity, and permutation invariance are checked at compile time.
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### Expected Benefits
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- **Certified Safety**: Machine-checked proofs for every GNN operation, enabling deployment in regulated environments (FDA, FAA, SEC)
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- **Adversarial Robustness Certificates**: Provable guarantees that predictions are stable under epsilon-bounded perturbations
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- **Training Correctness**: Proof certificates for each training epoch, enabling auditable model development
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- **Type-Safe GNN Pipelines**: Compile-time elimination of dimension mismatch bugs in message passing
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- **Proof-Carrying Attestations**: 82-byte proof attestations (from `ruvector-verified`) for lightweight verification of GNN outputs
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### Novelty Claim
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**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.
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**Differentiators**:
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1. Lean-agentic dependent types for graph operations (extending `ruvector-verified`)
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2. Proof-carrying training with per-epoch certificates
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3. Exact (not probabilistic) adversarial robustness bounds via interval bound propagation with proof witnesses
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4. Type-safe message passing with compile-time dimension checking
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5. Gated proof routing (from `ruvector-verified/gated.rs`) that allocates verification budget proportional to operation criticality
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---
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## The Verification Gap
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### Current State of GNN Verification
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The verification gap in graph neural networks is severe compared to other software domains:
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| Domain | Verification State | GNN Equivalent |
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|--------|-------------------|----------------|
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| Compilers | Formally verified (CompCert, CakeML) | No verified GNN compiler |
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| Operating Systems | Verified kernels (seL4) | No verified GNN runtime |
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| Cryptography | Machine-checked proofs (HACL*, Fiat-Crypto) | No proved GNN properties |
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| Numerical Libraries | Verified floating-point (VCFloat) | No verified tensor ops |
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| Smart Contracts | Formal verification tools (Certora, Solidity SMT) | No verified graph operations |
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Existing approaches to GNN reliability are insufficient:
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- **Empirical testing**: Tests a finite set of inputs. Cannot prove absence of failure.
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- **Randomized smoothing**: Provides probabilistic certificates. Not deterministic guarantees.
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- **Adversarial training**: Improves empirical robustness. Does not prove robustness.
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- **Model interpretability**: Explains behavior. Does not verify correctness.
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### What Formal Verification Provides
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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:
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- "For ALL graphs with N <= 10000 nodes and epsilon-bounded perturbations, the classifier output is stable" (not "for the 1000 graphs we tested")
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- "For ALL gradient steps, the equivariance error remains below delta" (not "equivariance held on our training run")
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- "For ALL message passing configurations with matching types, no dimension mismatch occurs" (not "our tests passed")
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---
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## Technical Design
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### Architecture Diagram
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```
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Graph Neural Network
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+-----------------+-----------------+
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Graph Ops Attention Training
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(add/del/ (compute (gradient
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reweight) weights) step)
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+----v----+ +----v----+ +-----v-----+
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| Proof | | Proof | | Proof |
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| Witness | | Witness | | Witness |
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| Gen | | Gen | | Gen |
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+----+----+ +----+----+ +-----+-----+
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+--------+--------+--------+--------+
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+-------v-------+ +-----v--------+
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| ProofEnv | | FastTermArena|
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| (ruvector- | | (bump alloc |
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| verified) | | + dedup) |
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+-------+-------+ +-----+--------+
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+-------v-----------------v--------+
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| Gated Proof Router |
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| Reflex | Standard | Deep |
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| <10ns | <1us | <100us |
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+------------------+---------------+
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+--------v--------+
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| Proof |
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| Attestation |
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| (82 bytes) |
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+-----------------+
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Verification Coverage:
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Graph Construction Message Passing Attention Weights
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+---+---+---+ +---+---+ +---+---+---+
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|dim|deg|con| proved |typ|agg| proved |eps|sym|pos| proved
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+---+---+---+ +---+---+ +---+---+---+
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dim = dimension typ = type safety eps = robustness
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deg = degree bounds agg = commutativity sym = symmetry
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con = connectivity pos = positivity
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```
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### Core Data Structures
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```rust
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use ruvector_verified::{
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ProofEnvironment, ProofAttestation, VerifiedOp, VerifiedStage,
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};
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#[cfg(feature = "fast-arena")]
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use ruvector_verified::fast_arena::FastTermArena;
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#[cfg(feature = "gated-proofs")]
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use ruvector_verified::gated::{route_proof, ProofKind, ProofTier, verify_tiered};
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/// A graph with proof-carrying operations.
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///
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/// Every structural modification produces a proof that invariants hold.
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/// Invariants are registered at construction time and checked incrementally.
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pub struct VerifiedGraph<V, E> {
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/// Node data indexed by node ID
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nodes: Vec<V>,
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/// Adjacency list with edge data
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adjacency: Vec<Vec<(usize, E)>>,
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/// Registered invariants to maintain
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invariants: Vec<GraphInvariant>,
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/// Proof environment for constructing and caching proofs
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env: ProofEnvironment,
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/// Fast term arena for high-throughput proof construction
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#[cfg(feature = "fast-arena")]
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arena: FastTermArena,
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/// Proof certificates for graph operations
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certificates: Vec<GraphCertificate>,
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}
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/// Graph invariants that must be maintained across operations.
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#[derive(Debug, Clone)]
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pub enum GraphInvariant {
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/// All node feature vectors have dimension d
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UniformDimension(u32),
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/// Maximum node degree
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MaxDegree(usize),
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/// Graph remains connected (single component)
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Connected,
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/// No self-loops
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NoSelfLoops,
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/// Edge weights are non-negative
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NonNegativeWeights,
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/// Total edge weight is conserved (within epsilon)
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WeightConservation { total: f32, epsilon: f32 },
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/// Node count within bounds
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NodeCountBounds { min: usize, max: usize },
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/// Custom invariant with proof obligation
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Custom { name: String, proof_kind: ProofKind },
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}
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/// A machine-checked certificate for a graph operation.
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#[derive(Debug, Clone)]
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pub struct GraphCertificate {
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/// Operation that was verified
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pub operation: GraphOperation,
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/// Proof term IDs for each maintained invariant
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pub invariant_proofs: Vec<u32>,
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/// Proof tier used (from gated routing)
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pub tier: ProofTier,
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/// Compact attestation (82 bytes, serializable)
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pub attestation: ProofAttestation,
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/// Wall-clock verification time
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pub verification_time_ns: u64,
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}
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/// Graph operations that produce proof certificates.
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#[derive(Debug, Clone)]
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pub enum GraphOperation {
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/// Add a node with feature vector
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AddNode { node_id: usize, dim: u32 },
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/// Remove a node
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RemoveNode { node_id: usize },
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/// Add an edge with weight
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AddEdge { src: usize, dst: usize, weight: f32 },
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/// Remove an edge
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RemoveEdge { src: usize, dst: usize },
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/// Reweight an edge
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ReweightEdge { src: usize, dst: usize, old_weight: f32, new_weight: f32 },
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/// Update attention weights for a node
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UpdateAttention { node_id: usize, new_weights: Vec<(usize, f32)> },
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/// Batch operation
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Batch { operations: Vec<GraphOperation> },
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}
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/// Verified message passing configuration.
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///
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/// Uses dependent types to ensure dimension safety at the type level.
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/// The phantom type parameters encode the message dimensions.
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pub struct VerifiedMessagePass<const D_IN: usize, const D_MSG: usize, const D_OUT: usize> {
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/// Message function weights: D_IN -> D_MSG
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message_weights: [[f32; D_MSG]; D_IN],
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/// Aggregation function (must be commutative + associative)
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aggregation: VerifiedAggregation,
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/// Update function weights: D_IN + D_MSG -> D_OUT
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update_weights: [[f32; D_OUT]; { D_IN + D_MSG }],
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/// Proof that aggregation is commutative
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commutativity_proof: u32,
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/// Proof that dimensions are consistent
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dim_proof: u32,
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}
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/// Aggregation functions with verified properties.
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#[derive(Debug, Clone)]
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pub enum VerifiedAggregation {
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/// Sum aggregation (commutative, associative -- trivially provable)
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Sum { commutativity_proof: u32 },
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/// Mean aggregation (commutative -- proved via sum commutativity + division)
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Mean { commutativity_proof: u32 },
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/// Max aggregation (commutative, associative -- proved via total order)
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Max { commutativity_proof: u32, associativity_proof: u32 },
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/// Attention-weighted sum (commutative when weights are symmetric)
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AttentionWeighted {
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symmetry_proof: Option<u32>,
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positivity_proof: u32,
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normalization_proof: u32,
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},
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}
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/// Adversarial robustness certificate for a GNN prediction.
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#[derive(Debug, Clone)]
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pub struct RobustnessCertificate {
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/// The prediction being certified
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pub prediction: Vec<f32>,
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/// Perturbation budget (L_inf norm for node features)
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pub epsilon_features: f32,
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/// Perturbation budget (number of edge additions/deletions)
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pub epsilon_structure: usize,
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/// Certified lower bound on correct-class margin
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pub certified_margin: f32,
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/// Whether the prediction is certifiably robust
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pub is_robust: bool,
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/// Proof term IDs for the certification chain
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pub proof_chain: Vec<u32>,
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/// Attestation
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pub attestation: ProofAttestation,
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}
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/// Training certificate for one gradient step.
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#[derive(Debug, Clone)]
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pub struct TrainingStepCertificate {
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/// Epoch number
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pub epoch: usize,
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/// Step within epoch
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pub step: usize,
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/// Loss before this step
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pub loss_before: f32,
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/// Loss after this step
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pub loss_after: f32,
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/// Proof that loss decreased (or explanation if it increased)
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pub loss_monotonicity: LossMonotonicity,
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/// Proof that equivariance is preserved
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pub equivariance_proof: Option<u32>,
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/// Proof that conservation laws hold
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pub conservation_proofs: Vec<(String, u32)>,
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/// Attestation
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pub attestation: ProofAttestation,
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}
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/// Loss monotonicity status for a training step.
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#[derive(Debug, Clone)]
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pub enum LossMonotonicity {
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/// Loss decreased -- proof of decrease
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Decreased { proof_id: u32, delta: f32 },
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/// Loss increased within tolerance (e.g., due to stochastic minibatch)
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IncreasedWithinTolerance { delta: f32, tolerance: f32 },
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/// Loss increased beyond tolerance -- flagged for review
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IncreasedBeyondTolerance { delta: f32, tolerance: f32 },
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}
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```
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### Key Algorithms
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#### 1. Proof-Carrying Graph Operations
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```rust
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impl<V, E> VerifiedGraph<V, E>
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where
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V: AsRef<[f32]>, // Node features accessible as float slice
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E: Into<f32> + Copy, // Edge data convertible to weight
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{
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/// Add a node with verified invariant preservation.
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///
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/// Produces proofs for:
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/// - Dimension correctness (UniformDimension)
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/// - Node count bounds (NodeCountBounds)
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/// - Self-loop absence (NoSelfLoops -- trivially true for new node)
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pub fn verified_add_node(
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&mut self,
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features: V,
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) -> Result<GraphCertificate, VerificationError> {
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let node_id = self.nodes.len();
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let dim = features.as_ref().len() as u32;
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// Route proof obligation to cheapest tier
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#[cfg(feature = "gated-proofs")]
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let tier_decision = route_proof(
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ProofKind::DimensionEquality {
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expected: self.expected_dim(),
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actual: dim,
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},
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&self.env,
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);
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let mut invariant_proofs = Vec::new();
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// Check each invariant
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for invariant in &self.invariants {
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let proof_id = match invariant {
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GraphInvariant::UniformDimension(expected) => {
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ruvector_verified::prove_dim_eq(&mut self.env, *expected, dim)?
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}
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GraphInvariant::NodeCountBounds { min: _, max } => {
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if node_id + 1 > *max {
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return Err(VerificationError::InvariantViolation(
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format!("node count {} exceeds max {}", node_id + 1, max)
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));
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}
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self.env.alloc_term()
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}
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GraphInvariant::NoSelfLoops => {
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// New node has no edges, so no self-loops. Trivial proof.
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self.env.alloc_term()
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}
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// Other invariants are trivially maintained by AddNode
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_ => self.env.alloc_term(),
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};
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invariant_proofs.push(proof_id);
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}
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// Perform the operation
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self.nodes.push(features);
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self.adjacency.push(Vec::new());
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// Construct attestation
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let attestation = ProofAttestation::new(
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&self.env,
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&invariant_proofs,
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"AddNode",
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);
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let cert = GraphCertificate {
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operation: GraphOperation::AddNode { node_id, dim },
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invariant_proofs,
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tier: ProofTier::Reflex,
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attestation,
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verification_time_ns: 0, // filled by caller
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};
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self.certificates.push(cert.clone());
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self.env.stats.proofs_verified += 1;
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Ok(cert)
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}
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/// Add an edge with verified invariant preservation.
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///
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/// Produces proofs for:
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/// - No self-loops (src != dst)
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/// - Max degree not exceeded
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/// - Non-negative weight
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/// - Weight conservation (if applicable)
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pub fn verified_add_edge(
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&mut self,
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src: usize,
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dst: usize,
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edge_data: E,
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) -> Result<GraphCertificate, VerificationError> {
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let weight: f32 = edge_data.into();
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let mut invariant_proofs = Vec::new();
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for invariant in &self.invariants {
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let proof_id = match invariant {
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GraphInvariant::NoSelfLoops => {
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if src == dst {
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return Err(VerificationError::InvariantViolation(
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format!("self-loop: {} -> {}", src, dst)
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));
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}
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self.env.alloc_term()
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}
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GraphInvariant::MaxDegree(max) => {
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let new_degree = self.adjacency[src].len() + 1;
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if new_degree > *max {
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return Err(VerificationError::InvariantViolation(
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format!("degree {} exceeds max {}", new_degree, max)
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));
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}
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self.env.alloc_term()
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}
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GraphInvariant::NonNegativeWeights => {
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if weight < 0.0 {
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return Err(VerificationError::InvariantViolation(
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format!("negative weight: {}", weight)
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));
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}
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self.env.alloc_term()
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}
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_ => self.env.alloc_term(),
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};
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invariant_proofs.push(proof_id);
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}
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// Perform the operation
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self.adjacency[src].push((dst, edge_data));
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self.adjacency[dst].push((src, edge_data));
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let attestation = ProofAttestation::new(
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&self.env,
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&invariant_proofs,
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"AddEdge",
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);
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let cert = GraphCertificate {
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operation: GraphOperation::AddEdge { src, dst, weight },
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invariant_proofs,
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tier: ProofTier::Standard { max_fuel: 100 },
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attestation,
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verification_time_ns: 0,
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};
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self.certificates.push(cert.clone());
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self.env.stats.proofs_verified += 1;
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Ok(cert)
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}
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}
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```
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#### 2. Verified Message Passing
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```rust
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/// Type-safe message passing with compile-time dimension checking.
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///
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/// The const generics D_IN, D_MSG, D_OUT enforce dimension compatibility
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/// at compile time. No runtime dimension check is needed -- the Rust
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/// type system prevents dimension mismatches.
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impl<const D_IN: usize, const D_MSG: usize, const D_OUT: usize>
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VerifiedMessagePass<D_IN, D_MSG, D_OUT>
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{
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/// Execute verified message passing on a graph.
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///
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/// For each node v:
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/// 1. For each neighbor u: compute message m_uv = W_msg * h_u
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/// 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<usize>],
|
|
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<u32, VerificationError> {
|
|
// 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<u32>,
|
|
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<const D: usize>(
|
|
&self,
|
|
node_features: &[f32; D],
|
|
layer_weights: &[Vec<Vec<f32>>],
|
|
adjacency: &[Vec<usize>],
|
|
node_idx: usize,
|
|
env: &mut ProofEnvironment,
|
|
) -> Result<RobustnessCertificate, VerificationError> {
|
|
// 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<f32> = 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<f32>],
|
|
env: &mut ProofEnvironment,
|
|
) -> Result<(Vec<f32>, Vec<f32>, 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<ConservationLaw>,
|
|
/// Accumulated certificates
|
|
certificates: Vec<TrainingStepCertificate>,
|
|
}
|
|
|
|
/// 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<Vec<f32>>,
|
|
loss_before: f32,
|
|
gradients: &[Vec<f32>],
|
|
env: &mut ProofEnvironment,
|
|
epoch: usize,
|
|
step: usize,
|
|
) -> Result<TrainingStepCertificate, VerificationError> {
|
|
// Snapshot conserved quantities before update
|
|
let quantities_before: Vec<f32> = 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>]) -> 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<usize>],
|
|
adj2: &[Vec<usize>],
|
|
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<usize> = adj1.iter().map(|a| a.len()).collect();
|
|
let mut degrees2: Vec<usize> = 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<usize>],
|
|
adj2: &[Vec<usize>],
|
|
) -> (Vec<u64>, Vec<u64>, 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<usize>]) -> Vec<u64> {
|
|
colors.iter().enumerate().map(|(i, &c)| {
|
|
let mut neighbor_colors: Vec<u64> = 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<usize>], adj2: &[Vec<usize>]) -> Option<Vec<usize>> {
|
|
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<usize> =
|
|
adj1[i].iter().cloned().collect();
|
|
let mapped_neighbors: std::collections::HashSet<usize> =
|
|
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<usize>,
|
|
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<A, B>` 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<A, B>` 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<V, E>` 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<V, E>` 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.
|