02: Mathematical Research Pipeline
Provably Correct by Construction: From Specification to Verified Code
Pillar: #2 of 12 - Supporting Priority
Timeline: 6 Weeks
Dependencies: None
Expected ROI: Zero bugs in critical paths, 50% reduction in debug cycles, automatic proof generation
Executive Summary
Move from statistical code generation (LLMs guessing at correctness) to mathematical code synthesis (provably correct by construction). AI-generated code has a ~30% bug rate. Testing catches some, but bugs slip through. This pillar implements formal methods that make bugs impossible in verified code paths.
Key Insight: The cheapest bug is the one that was never written.
Problem Statement
Current State: Statistical Correctness
Industry data on AI-generated code:
- GitHub Copilot: 40% of suggestions have issues (2023 study)
- ChatGPT/Claude: ~30% of code has bugs (various benchmarks)
- After testing: ~10% of bugs survive to production
- Cost of production bug: 10-100x cost of design-phase bug
The testing → fixing → testing loop:
Generate → Test → Find 30% bugs → Fix → Test → Find 5% more → ...
Time: O(n) iterations
Cost: O(n) LLM calls
Why This Happens
LLM Training Data:
- Learned from human code
- Human code has human bugs
- LLM reproduces those patterns
No Matter How Good the LLM:
- It's statistical, not logical
- It can be confidently wrong
- It has no concept of "correctness"
Current Maestro Evidence
# File: maestro/personas/developer.py
# AI generates code, hopes it works
async def generate_implementation(self, spec: Specification) -> Code:
response = await self.llm.generate(
prompt=self._build_prompt(spec),
# No verification that output satisfies spec
)
return Code(response.content)
Solution Architecture
The Paradigm Shift
Generation: Prompt → LLM → Code (hopefully correct)
"Write a function that..."
No guarantees about correctness
Synthesis: Specification → Solver → Code (provably correct)
"Find code that satisfies these constraints"
Mathematically guaranteed correct
Verification Pipeline
┌─────────────────────────────────────────┐
│ FORMAL METHODS STACK │
├─────────────────────────────────────────┤
│ │
Requirement ───────► │ Level 1: Type-Driven Development │
│ │ - Refinement types (NonZero, etc.) │
▼ │ - Mypy strict checking │
Specification ─────► │ │
│ │ Level 2: Design-by-Contract │
▼ │ - @requires/@ensures decorators │
Contract ──────────► │ - Automatic precondition inference │
│ │ │
▼ │ Level 3: SMT Solver Verification │
Z3 Verification ───► │ - Symbolic execution │
│ │ - Counterexample generation │
▼ │ │
Proof Certificate │ Level 4: Program Synthesis │
│ - Specification to code │
│ - Proof attached to output │
└─────────────────────────────────────────┘
Core Components
class FormalSpecGenerator:
"""
Transform natural language requirements into formal specifications.
Uses LLM to draft, SMT to validate consistency.
"""
async def generate_spec(self, requirement: str) -> FormalSpec:
# Step 1: LLM generates initial formal spec
draft = await self.llm.generate(
prompt=self._spec_prompt(requirement),
examples=self.spec_examples,
)
# Step 2: Parse into formal representation
parsed = self.parser.parse(draft.content)
# Step 3: Check specification consistency with Z3
consistency = await self.z3_checker.check_consistency(parsed)
if not consistency.is_consistent:
# Spec contradicts itself
return await self._fix_spec(parsed, consistency.conflicts)
return FormalSpec(
preconditions=parsed.preconditions,
postconditions=parsed.postconditions,
invariants=parsed.invariants,
)
def _spec_prompt(self, requirement: str) -> str:
return f"""
Convert this requirement to a formal specification:
Requirement: {requirement}
Format:
PRECONDITIONS:
- condition1
- condition2
POSTCONDITIONS:
- condition1
- condition2
INVARIANTS:
- invariant1
- invariant2
"""
2. Design-by-Contract System
from functools import wraps
from z3 import Solver, sat, Not
def requires(condition: str):
"""Precondition decorator with Z3 verification."""
def decorator(func):
func._preconditions = getattr(func, '_preconditions', [])
func._preconditions.append(condition)
@wraps(func)
def wrapper(*args, **kwargs):
# Runtime check (can be disabled in production)
if not eval_condition(condition, args, kwargs):
raise PreconditionViolation(condition)
return func(*args, **kwargs)
return wrapper
return decorator
def ensures(condition: str):
"""Postcondition decorator with Z3 verification."""
def decorator(func):
func._postconditions = getattr(func, '_postconditions', [])
func._postconditions.append(condition)
@wraps(func)
def wrapper(*args, **kwargs):
old_state = capture_state(args, kwargs)
result = func(*args, **kwargs)
# Check postcondition
if not eval_postcondition(condition, args, kwargs, result, old_state):
raise PostconditionViolation(condition)
return result
return wrapper
return decorator
# Example Usage
@requires("amount > 0")
@requires("account.balance >= amount")
@ensures("account.balance == old(account.balance) - amount")
@ensures("result.success implies destination.balance == old(destination.balance) + amount")
def transfer(account: Account, destination: Account, amount: Money) -> TransferResult:
"""Transfer with verified contracts."""
# Implementation must satisfy contracts
...
3. Z3 SMT Solver Integration
from z3 import *
class Z3Verifier:
"""
Verify function contracts using Z3 SMT solver.
"""
def verify_function(
self,
func_ast: FunctionAST,
preconditions: List[str],
postconditions: List[str]
) -> VerificationResult:
solver = Solver()
# Create symbolic variables for inputs
inputs = self._create_symbolic_inputs(func_ast.parameters)
# Assert preconditions
for pre in preconditions:
solver.add(self._to_z3(pre, inputs))
# Symbolically execute function
output = self._symbolic_execute(func_ast, inputs)
# Check postconditions
for post in postconditions:
solver.push()
# Try to find counterexample (negation of postcondition)
solver.add(Not(self._to_z3(post, inputs, output)))
if solver.check() == sat:
counterexample = solver.model()
return VerificationResult(
verified=False,
counterexample=self._format_counterexample(counterexample),
failed_condition=post,
)
solver.pop()
return VerificationResult(
verified=True,
proof=self._generate_proof(solver),
)
def infer_preconditions(self, func_ast: FunctionAST) -> List[str]:
"""Automatically infer preconditions from code patterns."""
preconditions = []
for statement in func_ast.body:
if isinstance(statement, DivisionExpr):
preconditions.append(f"{statement.divisor} != 0")
elif isinstance(statement, IndexExpr):
preconditions.append(f"0 <= {statement.index} < len({statement.array})")
elif isinstance(statement, UnwrapExpr):
preconditions.append(f"{statement.option}.is_some()")
return preconditions
4. Proof Certificate Generator
class ProofGenerator:
"""
Generate human-readable proof certificates.
"""
def generate_proof(
self,
func: Function,
verification: VerificationResult
) -> ProofCertificate:
if not verification.verified:
return ProofCertificate(
status="FAILED",
counterexample=verification.counterexample,
)
# Build proof steps
steps = []
# Step 1: State preconditions
steps.append(ProofStep(
type="ASSUME",
content=f"Given preconditions: {func._preconditions}",
))
# Step 2: Trace symbolic execution
for trace in verification.execution_trace:
steps.append(ProofStep(
type="DERIVE",
content=f"At line {trace.line}: {trace.state}",
))
# Step 3: Conclude postconditions
for post in func._postconditions:
steps.append(ProofStep(
type="CONCLUDE",
content=f"Therefore: {post} ✓",
))
return ProofCertificate(
status="VERIFIED",
function=func.__name__,
steps=steps,
verifier_version="Z3-4.12",
)
5. Critical Path Identifier
CRITICAL_PATTERNS = [
"authentication", "authorization", "payment",
"encryption", "personal_data", "financial_calculation",
"security_check", "compliance", "transfer", "delete",
]
class CriticalPathIdentifier:
"""
Identify functions that require formal verification.
Not everything needs proofs - focus on critical paths.
"""
def should_verify(self, function: Function) -> bool:
# Check function name/docstring
if any(p in function.name.lower() for p in CRITICAL_PATTERNS):
return True
# Check called functions (transitive criticality)
if any(self.should_verify(called) for called in function.calls):
return True
# Check data types handled
if any(self._is_sensitive_type(t) for t in function.types_used):
return True
return False
def get_verification_strategy(self, function: Function) -> str:
if self.should_verify(function):
return "FORMAL" # Full Z3 verification
elif function.complexity > 10:
return "PROPERTY_BASED" # Hypothesis testing
else:
return "UNIT" # Standard unit tests
Maestro Persona Integration
Current:
"Users should be able to reset their password via email"
Formal:
SPECIFICATION: PasswordReset
PRECONDITIONS:
- user.exists(email)
- user.account_status != SUSPENDED
POSTCONDITIONS:
- email_sent(user.email, reset_token)
- reset_token.expires_at == now() + 1.hour
- user.password unchanged until reset_token.used
INVARIANTS:
- at most 3 reset attempts per hour per user
- reset_token.length >= 32
- reset_token is cryptographically random
Developer → Verified Implementation
# Before: Hope-based development
def process_payment(order_id, card_info):
order = get_order(order_id)
charge = stripe.charge(card_info, order.amount)
order.status = 'paid'
save_order(order)
return charge
# After: Verified development
@requires("order.exists(order_id)")
@requires("order.status == 'pending'")
@requires("card_info.is_valid")
@ensures("order.status == 'paid' implies charge.success")
@ensures("not charge.success implies order.status == 'pending'")
@atomic
def process_payment(order_id: OrderId, card_info: CardInfo) -> Charge:
order = get_order(order_id)
charge = stripe.charge(card_info, order.amount)
if charge.success:
order.status = 'paid'
save_order(order)
return charge
# PROOF: postconditions verified via symbolic execution
QA Engineer → Proof Verification
THEOREM: PaymentConsistency
GIVEN:
- order.exists(order_id)
- order.status == 'pending'
- card_info.is_valid
- stripe.charge succeeds
PROVE:
- After process_payment: order.status == 'paid'
PROOF:
1. By preconditions, order exists and is pending
2. stripe.charge(card_info, order.amount) called
3. Given: charge succeeds
4. By line 8: order.status = 'paid'
5. By line 9: save_order(order) persists change
6. By postcondition: order.status == 'paid' ✓
QED
Integration Points
Files to Modify
Verification in Phase Executor
# File: phases/phase_integrated_executor.py
async def execute_implementation_phase(self, spec: Specification) -> Implementation:
# Generate formal specification
formal_spec = await self.spec_generator.generate_spec(spec)
# Generate implementation
code = await self.developer.generate(formal_spec)
# Verify critical paths
critical_functions = self.critical_identifier.find_critical(code)
for func in critical_functions:
result = await self.z3_verifier.verify_function(
func,
formal_spec.preconditions,
formal_spec.postconditions,
)
if not result.verified:
# Regenerate with counterexample guidance
code = await self.developer.fix_with_counterexample(
func, result.counterexample
)
return Implementation(code=code, proofs=results)
Implementation Plan
Week 1: Contract System Foundation
- [ ] Implement @requires/@ensures decorators
- [ ] Runtime contract checking
- [ ] Contract inference from code patterns
- [ ] Unit tests for contract system
Week 2: Z3 Integration
- [ ] Z3 Python wrapper
- [ ] Symbolic execution engine
- [ ] Precondition verification
- [ ] Postcondition verification
Week 3: Proof Generation
- [ ] Proof certificate format (JSON + human-readable)
- [ ] Counterexample formatting
- [ ] Integration with verification pipeline
- [ ] Proof display in Maestro output
Week 4: Maestro Persona Integration
- [ ] FormalSpecGenerator for Requirement Analyst
- [ ] Contract generation for Developer
- [ ] Proof verification for QA Engineer
- [ ] Critical path identification
Week 5: Property-Based Testing
- [ ] Hypothesis integration
- [ ] Type-based test generation
- [ ] Invariant-based testing
- [ ] Integration with CI
Week 6: Production Deployment
- [ ] Performance optimization (incremental verification)
- [ ] Caching for verified functions
- [ ] Monitoring and metrics
- [ ] Documentation and training
Expected Results
Bug Reduction
Development Efficiency
Risks & Mitigations
Success Criteria
Alternatives Considered
Hybrid chosen: Formal verification for critical paths, property testing for medium, unit tests for the rest.
- #01 AI-Driven SDLC: Formal specs enable autonomous verification
- #07 Formal Verification: This pillar feeds into comprehensive testing
- #08 Observability: Proof certificates in telemetry
Status: Ready for Implementation
Key Insight: Testing finds bugs; mathematics prevents them
Owner: Quality & Verification Team
