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FORMAL VERIFICATION

Formal Verification Plan

Version: 1.0.0
Last Updated: 2025-11-10
Status: Planning Phase

Executive Summary

This document outlines the formal verification strategy for Mobius Systems, focusing on critical algorithms and protocols that require mathematical proof of correctness. The goal is to provide formal guarantees for safety, liveness, and integrity properties.

Current Status: - ✅ Verification plan defined - ⚠️ TLA+ specifications: In progress - ⚠️ Coq proofs: Planned - ⚠️ Property-based testing: Partial

1. Verification Scope

1.1 Critical Components Requiring Verification

  1. MII Calculation Algorithm
  2. Mathematical correctness
  3. Boundedness (MII ∈ [0, 1])
  4. Monotonicity properties

  5. Stability (damping)

  6. Consensus Protocol (Deliberation Proof)

  7. Safety: Agreement property
  8. Liveness: Termination property

  9. Fault tolerance: Byzantine fault tolerance

  10. Cryptographic Operations

  11. Signature verification correctness
  12. Key rotation safety

  13. Nonce uniqueness

  14. Integrity Verification

  15. Attestation chain validity
  16. Merkle tree verification
  17. Replay attack prevention

2. Verification Methods

2.1 Model Checking (TLA+)

Purpose: Verify temporal properties of distributed protocols.

Tools: - TLA+ (Temporal Logic of Actions) - TLC (TLA+ Model Checker) - Apalache (Symbolic Model Checker)

Target Specifications: 1. Consensus protocol (Deliberation Proof) 2. Integrity verification protocol 3. Attestation chain protocol

2.2 Theorem Proving (Coq)

Purpose: Prove mathematical correctness of algorithms.

Tools: - Coq (Proof Assistant) - Isabelle/HOL (Alternative)

Target Proofs: 1. MII calculation correctness 2. Cryptographic signature verification 3. Merkle tree verification

2.3 Property-Based Testing

Purpose: Discover edge cases and verify properties.

Tools: - fast-check (TypeScript) - Hypothesis (Python)

Target Properties: 1. MII boundedness 2. Consensus safety 3. Cryptographic correctness

3. MII Calculation Verification

3.1 Properties to Verify

Mathematical Properties: 1. Boundedness: MII(S_t) ∈ [0, 1] for all S_t 2. Monotonicity: If S_t' ≥ S_t (component-wise), then MII(S_t') ≥ MII(S_t) 3. Stability: Damping prevents oscillation 4. Composability: MII is composable across subsystems

3.2 TLA+ Specification (Planned)

EXTENDS Naturals, Reals

VARIABLES technical, moral, civic, security, antiGaming, mii

TypeOK == 
    /\ technical \in [0, 1]
    /\ moral \in [0, 1]
    /\ civic \in [0, 1]
    /\ security \in [0, 1]
    /\ antiGaming \in [0, 0.2]
    /\ mii \in [0, 1]

MIIFormula(alpha, beta, gamma, delta, lambda) ==
    LET t == technical IN
    LET m == moral IN
    LET c == civic IN
    LET s == security IN
    LET phi == antiGaming IN
    IN alpha * t + beta * m + gamma * c + delta * s - lambda * phi

Boundedness == 
    /\ mii >= 0
    /\ mii <= 1

Init == 
    /\ technical = 0.95
    /\ moral = 0.90
    /\ civic = 0.92
    /\ security = 0.88
    /\ antiGaming = 0.02
    /\ mii = MIIFormula(0.35, 0.25, 0.25, 0.15, 0.05)
    /\ TypeOK

Next == 
    /\ UNCHANGED <<technical, moral, civic, security, antiGaming>>
    /\ mii' = MIIFormula(0.35, 0.25, 0.25, 0.15, 0.05)
    /\ TypeOK'

Spec == Init /\ [][Next]_<<technical, moral, civic, security, antiGaming, mii>>

BoundednessProperty == []Boundedness

3.3 Coq Proof (Planned)

Require Import Reals.

Definition mii (t m c s phi : R) (alpha beta gamma delta lambda : R) : R :=
  alpha * t + beta * m + gamma * c + delta * s - lambda * phi.

Theorem mii_bounded :
  forall t m c s phi alpha beta gamma delta lambda,
    0 <= t <= 1 ->
    0 <= m <= 1 ->
    0 <= c <= 1 ->
    0 <= s <= 1 ->
    0 <= phi <= 0.2 ->
    alpha + beta + gamma + delta = 1 ->
    alpha >= 0 -> beta >= 0 -> gamma >= 0 -> delta >= 0 -> lambda >= 0 ->
    0 <= mii t m c s phi alpha beta gamma delta lambda <= 1.
Proof.
  (* Proof to be implemented *)
Admitted.

4. Consensus Protocol Verification

4.1 Properties to Verify

Safety Properties: 1. Agreement: All correct processes agree on the same value 2. Validity: If a value is decided, it was proposed by some process 3. Integrity: No value is decided twice

Liveness Properties: 1. Termination: All correct processes eventually decide 2. Bounded Deliberation: Deliberation completes within 3 minutes

Fault Tolerance: 1. Byzantine Fault Tolerance: System tolerates f < n/3 Byzantine failures 2. Crash Fault Tolerance: System tolerates f < n/2 crash failures

4.2 TLA+ Specification (In Progress)

EXTENDS Naturals, FiniteSets

CONSTANTS Processes, MaxDeliberationTime

VARIABLES proposals, consensus, deliberationTime

TypeOK == 
    /\ proposals \in [Processes -> STRING]
    /\ consensus \in [Processes -> STRING \cup {NULL}]
    /\ deliberationTime \in [Processes -> Nat]

Init == 
    /\ proposals \in [Processes -> STRING]
    /\ consensus = [p \in Processes |-> NULL]
    /\ deliberationTime = [p \in Processes |-> 0]

Propose(p, value) == 
    /\ proposals[p] = ""
    /\ proposals' = [proposals EXCEPT ![p] = value]
    /\ UNCHANGED <<consensus, deliberationTime>>

ReachConsensus(p, value) == 
    /\ consensus[p] = NULL
    /\ \A q \in Processes : proposals[q] = value \/ proposals[q] = ""
    /\ consensus' = [consensus EXCEPT ![p] = value]
    /\ UNCHANGED <<proposals, deliberationTime>>

BoundedDeliberation == 
    /\ \A p \in Processes : deliberationTime[p] <= MaxDeliberationTime

Agreement == 
    /\ \A p, q \in Processes : 
        consensus[p] # NULL /\ consensus[q] # NULL => consensus[p] = consensus[q]

Next == 
    \/ \E p \in Processes, v \in STRING : Propose(p, v)
    \/ \E p \in Processes, v \in STRING : ReachConsensus(p, v)

Spec == Init /\ [][Next]_<<proposals, consensus, deliberationTime>>

Termination == <>(\A p \in Processes : consensus[p] # NULL)

5. Cryptographic Verification

5.1 Properties to Verify

  1. Signature Correctness: If Verify(pk, msg, sig) = true, then Sign(sk, msg) = sig
  2. Nonce Uniqueness: Nonces are unique within time window
  3. Key Rotation Safety: Old keys are invalidated after rotation

5.2 Property-Based Testing

// tests/property/crypto-properties.test.ts
import fc from 'fast-check';
import { sign, verify, generateKeyPair } from '../../src/crypto/ed25519';

describe('Ed25519 Properties', () => {
  it('should verify signatures correctly', () => {
    fc.assert(
      fc.property(
        fc.string(),
        (message) => {
          const { publicKey, privateKey } = generateKeyPair();
          const signature = sign(privateKey, message);
          return verify(publicKey, message, signature);
        }
      )
    );
  });

  it('should reject invalid signatures', () => {
    fc.assert(
      fc.property(
        fc.string(),
        fc.string(),
        (message1, message2) => {
          if (message1 === message2) return true;

          const { publicKey, privateKey } = generateKeyPair();
          const signature = sign(privateKey, message1);
          return !verify(publicKey, message2, signature);
        }
      )
    );
  });
});

6. Integrity Verification

6.1 Properties to Verify

  1. Attestation Chain Validity: All attestations in chain are valid
  2. Merkle Tree Verification: Merkle proofs are correct
  3. Replay Prevention: Nonces are unique and time-bounded

6.2 TLA+ Specification (Planned)

EXTENDS Naturals, Sequences

VARIABLES attestations, merkleRoot, nonces

TypeOK == 
    /\ attestations \in Seq(Attestation)
    /\ merkleRoot \in STRING
    /\ nonces \in Seq(STRING)

Attestation == [ts: Nat, nonce: STRING, mii: Real, sig: STRING]

ValidAttestation(att) == 
    /\ att.nonce \notin nonces
    /\ att.ts <= Now()
    /\ att.ts >= Now() - TimeWindow
    /\ VerifySignature(att)

MerkleProofValid(att, proof) == 
    /\ ComputeMerkleRoot(att, proof) = merkleRoot

Next == 
    \/ \E att \in Attestation : 
        /\ ValidAttestation(att)
        /\ attestations' = Append(attestations, att)
        /\ nonces' = Append(nonces, att.nonce)
        /\ merkleRoot' = UpdateMerkleRoot(merkleRoot, att)
    \/ UNCHANGED <<attestations, merkleRoot, nonces>>

NoReplay == 
    /\ \A i, j \in DOMAIN attestations : 
        i # j => attestations[i].nonce # attestations[j].nonce

7. Verification Roadmap

7.1 Phase 1: Property-Based Testing (Current)

Timeline: 1-2 months
Status: In progress

  • ✅ MII boundedness properties
  • ✅ Cryptographic signature properties
  • ⚠️ Consensus safety properties
  • ⚠️ Integrity verification properties

7.2 Phase 2: TLA+ Model Checking (Next)

Timeline: 3-6 months
Status: Planned

  • ⚠️ Consensus protocol specification
  • ⚠️ Integrity verification protocol
  • ⚠️ Attestation chain protocol
  • ⚠️ Model checking and property verification

7.3 Phase 3: Coq Theorem Proving (Future)

Timeline: 6-12 months
Status: Planned

  • ⚠️ MII calculation correctness proof
  • ⚠️ Cryptographic operations proof
  • ⚠️ Merkle tree verification proof
  • ⚠️ Consensus algorithm proof

8. Verification Tools Setup

8.1 TLA+ Setup

# Install TLA+ Toolbox
# Download from: https://github.com/tlaplus/tlaplus/releases

# Install TLC (TLA+ Model Checker)
# Included with TLA+ Toolbox

# Install Apalache (Symbolic Model Checker)
# Download from: https://github.com/informalsystems/apalache/releases

8.2 Coq Setup

# Install Coq
# macOS: brew install coq
# Linux: apt-get install coq
# Windows: Download from https://coq.inria.fr/download

# Install CoqIDE (optional GUI)
# Included with Coq installation

8.3 Property-Based Testing

# Install fast-check (TypeScript)
npm install --save-dev fast-check

# Install Hypothesis (Python)
pip install hypothesis

9. Verification Metrics

9.1 Coverage Goals

  • TLA+ Specifications: 100% of critical protocols
  • Coq Proofs: 100% of critical algorithms
  • Property-Based Tests: >90% property coverage

9.2 Current Status

Component Property Tests TLA+ Spec Coq Proof
MII Calculation ✅ Partial ⚠️ Planned ⚠️ Planned
Consensus Protocol ⚠️ In Progress ⚠️ In Progress ⚠️ Planned
Cryptographic Ops ✅ Complete ⚠️ Planned ⚠️ Planned
Integrity Verification ⚠️ In Progress ⚠️ Planned ⚠️ Planned

10. References

Document Status: ✅ Complete
Last Reviewed: 2025-11-10
Next Review: 2026-01-10