Formal Verification AI

Category: Phase 3 Core - Correctness Guarantees Status: Skeleton Implementation Dependencies:

categorical-composition

(correctness as functoriality)

Overview

Integrates formal verification methods with AI systems: theorem proving for correctness guarantees, interval arithmetic for certified bounds, and categorical proofs for compositional correctness.

Capabilities

• Theorem Proving: Automated verification of AI properties
• Interval Arithmetic: Certified bounds on network outputs
• Categorical Correctness: Functorial preservation guarantees
• Adversarial Robustness: Verified defense certificates

Core Components

1. Theorem Prover Interface (

   theorem_proving.jl

   )
   • Integration with Z3, Lean, or Coq
   • Encode neural networks as logical formulas
   • Automated proof search

2. Interval Arithmetic (

   interval_arithmetic.jl

   )
   • Interval propagation through networks
   • Certified bounds on outputs
   • Robustness verification

3. Categorical Proofs (

   categorical_correctness.jl

   )
   • Verify functor laws for compositional networks
   • Natural transformation diagrams
   • Commutativity checking

4. Verification Examples (

   verification_examples.jl

   )
   • Adversarial robustness proofs
   • Fairness guarantees
   • Safety-critical system verification

Integration Points

• Input from: All Phase 3 skills (provides verification layer)
• Output to:

  categorical-composition

  (verified transformations)
• Coordinates with:

  oriented-simplicial-networks

  (topological invariants)

Usage

using FormalVerificationAI

# Define neural network
network = SimpleNN([Dense(10, 20, relu), Dense(20, 2)])

# Verify robustness using interval arithmetic
input_interval = Interval([0.0, 0.0], [1.0, 1.0])
output_bounds = propagate_intervals(network, input_interval)

# Prove categorical correctness
F = network_to_functor(network)
@assert verify_functor_laws(F)

# Automated theorem proving
property = "∀x. ||x - x'|| < ε ⟹ ||f(x) - f(x')|| < δ"
proof = prove_property(network, property, timeout=60)

References

• Katz et al. "Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks" (2017)
• Singh et al. "An Abstract Domain for Certifying Neural Networks" (POPL 2019)
• Fong & Spivak "Hypergraph Categories" (2019)

Implementation Status

• Basic interval arithmetic
• Z3 interface skeleton
• Full neural network encoding
• Categorical correctness verification
• Benchmark on standard verification tasks