Synthesis Framework: From LLM Candidates to Strongly-Typed Patches¶
The Core Problem¶
LLMs produce plausible code, not provably correct code. Their outputs are:
Statistically likely given training distribution
Syntactically valid (usually)
Semantically approximate (often)
Type-correct only by accident
Behaviorally equivalent rarely
The Solution: Treat LLM outputs as seed populations for formal synthesis, not as final artifacts.
Multi-Framework Mutation Architecture¶
Unified Interface Layer¶
┌─────────────────────────────────────────────────────────────────┐
│ Curate-Ipsum Unified Interface │
├─────────────────────────────────────────────────────────────────┤
│ ┌─────────┐ ┌─────────┐ ┌─────────┐ ┌─────────┐ ┌───────┐ │
│ │ Stryker │ │ mutmut │ │cosmic- │ │ poodle │ │univ- │ │
│ │ (JS) │ │ (Py) │ │ ray │ │ (Py) │ │mutator│ │
│ └────┬────┘ └────┬────┘ └────┬────┘ └────┬────┘ └───┬───┘ │
│ │ │ │ │ │ │
│ └────────────┴─────┬──────┴────────────┴───────────┘ │
│ ▼ │
│ ┌───────────────────────┐ │
│ │ Region Assignment │ │
│ │ Engine (implicit │ │
│ │ detection via graph │ │
│ │ spectral analysis) │ │
│ └───────────────────────┘ │
└─────────────────────────────────────────────────────────────────┘
Implicit Region Detection¶
Regions that “no one else is likely to ever detect” emerge from:
class ImplicitRegionDetector:
"""Detect code regions through graph-spectral analysis."""
def detect_orphan_regions(self, graph: CodeGraph) -> List[Region]:
"""Regions with low centrality but high complexity."""
return [
node for node in graph.nodes
if self.centrality(node) < 0.2
and self.cyclomatic_complexity(node) > 10
and self.test_coverage(node) < 0.5
]
def detect_bridge_regions(self, graph: CodeGraph) -> List[Region]:
"""Regions that are graph cut vertices - removal disconnects modules."""
return list(nx.articulation_points(graph))
def detect_spectral_anomalies(self, graph: CodeGraph) -> List[Region]:
"""Regions near Fiedler partition boundaries with high local variance."""
fiedler = self.compute_fiedler_vector(graph)
# Nodes where neighbors have very different Fiedler values
return [
node for node in graph.nodes
if np.std([fiedler[n] for n in graph.neighbors(node)]) > threshold
]
def detect_mutation_resistant(self, history: RunHistory) -> List[Region]:
"""Regions where mutants consistently survive across frameworks."""
# Cross-framework surviving mutant intersection
survivors_by_framework = {
fw: set(get_survivors(history, fw))
for fw in ['stryker', 'mutmut', 'cosmic-ray']
}
# Intersection = resistant to all frameworks
return set.intersection(*survivors_by_framework.values())
Non-Contradictory Framework Orchestration¶
class FrameworkOrchestrator:
"""Coordinate multiple mutation frameworks without contradiction."""
def partition_codebase(self, graph: CodeGraph) -> Dict[str, List[Region]]:
"""Assign regions to frameworks based on strengths."""
assignments = defaultdict(list)
for region in graph.regions:
if region.language == 'javascript':
assignments['stryker'].append(region)
elif region.is_highly_dynamic:
assignments['mutmut'].append(region) # Better with dynamic Python
elif region.needs_custom_operators:
assignments['cosmic-ray'].append(region) # Plugin support
elif region.is_simple:
assignments['poodle'].append(region) # Fast for simple code
else:
assignments['universalmutator'].append(region) # Fallback
# Verify no contradictions
self._verify_disjoint(assignments)
return assignments
def _verify_disjoint(self, assignments: Dict[str, List[Region]]):
"""Ensure no region assigned to contradictory frameworks."""
all_regions = []
for regions in assignments.values():
for r in regions:
if r in all_regions:
raise ConflictError(f"Region {r} assigned to multiple frameworks")
all_regions.append(r)
CEGIS: Counterexample-Guided Inductive Synthesis¶
The CEGIS Loop¶
┌─────────────────────────────────────────────────────────────────┐
│ CEGIS Loop │
│ │
│ ┌──────────┐ candidate ┌──────────┐ │
│ │ Synthesizer│─────────────→│ Verifier │ │
│ │ (LLM + │ │ (SMT/ │ │
│ │ genetic) │←─────────────│ KLEE) │ │
│ └──────────┘ counterexample └──────────┘ │
│ ▲ │ │
│ │ │ verified │
│ │ ▼ │
│ ┌──────────┐ ┌──────────┐ │
│ │ LLM │ │ Strongly │ │
│ │ Candidates│ │ Typed │ │
│ │ (top k) │ │ Patch │ │
│ └──────────┘ └──────────┘ │
└─────────────────────────────────────────────────────────────────┘
Implementation¶
class CEGISEngine:
"""Counterexample-Guided Inductive Synthesis for code patches."""
def __init__(self, verifier: SMTVerifier, synthesizer: HybridSynthesizer):
self.verifier = verifier
self.synthesizer = synthesizer
self.counterexamples: List[Counterexample] = []
def synthesize(
self,
spec: Specification,
llm_candidates: List[CodePatch],
max_iterations: int = 100
) -> Optional[StronglyTypedPatch]:
"""
CEGIS loop: refine candidates until verified or exhausted.
"""
# Initialize population with LLM candidates
population = self.synthesizer.initialize(llm_candidates)
for iteration in range(max_iterations):
# Select best candidate from population
candidate = self.synthesizer.select_best(population, self.counterexamples)
# Verify candidate against specification
result = self.verifier.verify(candidate, spec)
if result.verified:
return self._to_strongly_typed(candidate, result.proof)
# Extract counterexample
counterexample = result.counterexample
self.counterexamples.append(counterexample)
# Refine population using counterexample
population = self.synthesizer.refine(
population,
counterexample,
spec
)
return None # Synthesis failed
def _to_strongly_typed(
self,
candidate: CodePatch,
proof: VerificationProof
) -> StronglyTypedPatch:
"""Convert verified candidate to strongly-typed patch with proof."""
return StronglyTypedPatch(
code=candidate.code,
type_signature=self._infer_types(candidate),
preconditions=proof.preconditions,
postconditions=proof.postconditions,
invariants=proof.invariants,
proof_certificate=proof.certificate
)
CEGAR: Counterexample-Guided Abstraction Refinement¶
Abstraction Hierarchy¶
Concrete Program
↓ (abstract)
Abstract Model (coarse)
↓ (verify)
Spurious Counterexample?
↓ (refine if spurious)
Abstract Model (finer)
↓ (repeat until real CE or verified)
Verified / Real Counterexample
Implementation¶
class CEGAREngine:
"""Counterexample-Guided Abstraction Refinement for patch verification."""
def __init__(self):
self.abstraction_levels: List[AbstractionLevel] = [
TypeAbstraction(), # Coarsest: just types
ControlFlowAbstraction(), # CFG structure
DataFlowAbstraction(), # DFG with abstract values
ConcreteExecution(), # Finest: actual execution
]
def verify(self, patch: CodePatch, spec: Specification) -> VerificationResult:
"""CEGAR loop: start coarse, refine on spurious counterexamples."""
current_level = 0
while current_level < len(self.abstraction_levels):
abstraction = self.abstraction_levels[current_level]
abstract_model = abstraction.abstract(patch)
result = self._verify_abstract(abstract_model, spec)
if result.verified:
return VerificationResult(verified=True, level=current_level)
if self._is_real_counterexample(result.counterexample, patch):
return VerificationResult(
verified=False,
counterexample=result.counterexample
)
# Spurious counterexample - refine abstraction
current_level += 1
# Reached concrete level
return self._concrete_verify(patch, spec)
def _is_real_counterexample(
self,
ce: AbstractCounterexample,
patch: CodePatch
) -> bool:
"""Check if abstract counterexample is realizable in concrete program."""
# Attempt to concretize the counterexample
concrete_ce = self._concretize(ce)
if concrete_ce is None:
return False # Spurious
# Execute concrete counterexample
return self._execute_and_check(concrete_ce, patch)
Genetic Algorithm Population Management¶
Epoch-Based Evolution¶
@dataclass
class Individual:
"""A candidate patch in the population."""
code: str
fitness: float
lineage: List[str] # Parent IDs for tracking
mutations_applied: List[MutationOperator]
generation: int
class GeneticSynthesizer:
"""Genetic algorithm for patch population evolution."""
def __init__(self, config: GAConfig):
self.population_size = config.population_size
self.mutation_rate = config.mutation_rate
self.crossover_rate = config.crossover_rate
self.elite_ratio = config.elite_ratio
self.entropy_threshold = config.entropy_threshold
def evolve_epoch(
self,
population: List[Individual],
counterexamples: List[Counterexample],
spec: Specification
) -> List[Individual]:
"""Evolve population for one epoch."""
# Evaluate fitness against counterexamples
for individual in population:
individual.fitness = self._evaluate_fitness(
individual, counterexamples, spec
)
# Selection: tournament + elitism
elite = self._select_elite(population)
parents = self._tournament_select(population)
# Crossover: combine successful patches
offspring = self._crossover(parents)
# Mutation: directed by counterexamples
mutated = self._mutate_directed(offspring, counterexamples)
# Maintain entropy (diversity)
if self._population_entropy(mutated) < self.entropy_threshold:
mutated = self._inject_diversity(mutated)
return elite + mutated
def _evaluate_fitness(
self,
individual: Individual,
counterexamples: List[Counterexample],
spec: Specification
) -> float:
"""
Fitness = (counterexamples avoided) + (spec conditions met) - (complexity)
"""
ce_score = sum(
1 for ce in counterexamples
if not self._triggers_counterexample(individual, ce)
) / len(counterexamples) if counterexamples else 1.0
spec_score = self._spec_satisfaction_ratio(individual, spec)
complexity_penalty = self._ast_complexity(individual.code) / 100
return (0.4 * ce_score) + (0.5 * spec_score) - (0.1 * complexity_penalty)
def _crossover(self, parents: List[Individual]) -> List[Individual]:
"""AST-aware crossover between parent patches."""
offspring = []
for i in range(0, len(parents) - 1, 2):
p1, p2 = parents[i], parents[i + 1]
if random.random() < self.crossover_rate:
# AST subtree exchange
child1, child2 = self._ast_crossover(p1, p2)
offspring.extend([child1, child2])
else:
offspring.extend([p1, p2])
return offspring
def _mutate_directed(
self,
population: List[Individual],
counterexamples: List[Counterexample]
) -> List[Individual]:
"""Apply mutations directed by counterexample analysis."""
mutated = []
for individual in population:
if random.random() < self.mutation_rate:
# Analyze which counterexample is closest to being avoided
closest_ce = self._find_closest_counterexample(
individual, counterexamples
)
# Apply mutation that addresses the counterexample
mutation_op = self._select_mutation_for_ce(closest_ce)
new_individual = self._apply_mutation(individual, mutation_op)
mutated.append(new_individual)
else:
mutated.append(individual)
return mutated
Entropy-Aware Diversity Maintenance¶
class EntropyManager:
"""Maintain population diversity through entropy monitoring."""
def compute_population_entropy(self, population: List[Individual]) -> float:
"""
Compute Shannon entropy over population's structural features.
High entropy = diverse population
Low entropy = convergence (potentially premature)
"""
# Extract structural features
features = [self._extract_features(ind) for ind in population]
# Cluster features
clusters = self._cluster_features(features)
# Compute entropy over cluster distribution
cluster_counts = Counter(clusters)
total = len(population)
entropy = -sum(
(count / total) * math.log2(count / total)
for count in cluster_counts.values()
if count > 0
)
return entropy
def inject_diversity(
self,
population: List[Individual],
target_entropy: float
) -> List[Individual]:
"""Inject diverse individuals to reach target entropy."""
current_entropy = self.compute_population_entropy(population)
while current_entropy < target_entropy:
# Generate structurally novel individual
novel = self._generate_novel_individual(population)
population = self._replace_most_similar(population, novel)
current_entropy = self.compute_population_entropy(population)
return population
def _generate_novel_individual(
self,
population: List[Individual]
) -> Individual:
"""Generate individual maximally different from current population."""
# Compute centroid of current population in feature space
features = [self._extract_features(ind) for ind in population]
centroid = np.mean(features, axis=0)
# Generate in opposite direction from centroid
direction = -centroid / np.linalg.norm(centroid)
# Convert feature direction to code generation constraints
constraints = self._features_to_constraints(direction)
# Generate novel code satisfying anti-centroid constraints
return self._constrained_generation(constraints)
LLM Integration: Top-K Candidates as Seed Population¶
The Pipeline¶
User Query / Failing Test / Surviving Mutant
↓
┌──────────────────────┐
│ LLM (Claude/GPT) │
│ Generate top-k │
│ candidate patches │
└──────────────────────┘
↓
[k candidates, unverified]
↓
┌──────────────────────┐
│ Initial Population │
│ (seed from LLM) │
└──────────────────────┘
↓
┌──────────────────────┐
│ CEGIS + CEGAR + │
│ Genetic Evolution │
│ (intense analysis) │
└──────────────────────┘
↓
┌──────────────────────┐
│ Strongly Typed │
│ Verified Patch │
│ (with proof) │
└──────────────────────┘
Implementation¶
class LLMGuidedSynthesis:
"""Use LLM candidates as seeds for formal synthesis."""
def __init__(
self,
llm_client: LLMClient,
cegis: CEGISEngine,
cegar: CEGAREngine,
genetic: GeneticSynthesizer
):
self.llm = llm_client
self.cegis = cegis
self.cegar = cegar
self.genetic = genetic
async def synthesize_patch(
self,
context: SynthesisContext,
k: int = 10
) -> Optional[StronglyTypedPatch]:
"""
Full pipeline: LLM → genetic → CEGIS → strongly typed.
"""
# Step 1: Get top-k candidates from LLM
llm_candidates = await self._get_llm_candidates(context, k)
# Step 2: Initialize genetic population with LLM candidates
population = self.genetic.initialize_from_candidates(llm_candidates)
# Step 3: Extract specification from context
spec = self._extract_specification(context)
# Step 4: CEGIS loop with genetic refinement
max_epochs = 50
counterexamples = []
for epoch in range(max_epochs):
# Evolve population
population = self.genetic.evolve_epoch(
population, counterexamples, spec
)
# Get best candidate
best = self.genetic.select_best(population)
# Verify with CEGAR
result = self.cegar.verify(best.code, spec)
if result.verified:
return self._finalize_patch(best, result)
# Add counterexample for next epoch
counterexamples.append(result.counterexample)
# Log progress
self._log_epoch(epoch, best.fitness, len(counterexamples))
return None # Synthesis failed
async def _get_llm_candidates(
self,
context: SynthesisContext,
k: int
) -> List[CodePatch]:
"""Query LLM for k candidate patches."""
prompt = self._build_synthesis_prompt(context)
# Request multiple completions
responses = await self.llm.complete(
prompt=prompt,
n=k,
temperature=0.8, # Higher temp for diversity
max_tokens=2000
)
# Parse code from responses
candidates = []
for response in responses:
code = self._extract_code(response)
if code and self._syntactically_valid(code):
candidates.append(CodePatch(code=code, source='llm'))
return candidates
Strongly Typed Patch Output¶
The Final Artifact¶
@dataclass
class StronglyTypedPatch:
"""A verified, formally typed patch with proof certificate."""
# The code itself
code: str
diff: str # Unified diff format
# Type information
type_signature: TypeSignature
generic_parameters: List[TypeVar]
type_constraints: List[TypeConstraint]
# Formal verification
preconditions: List[Predicate]
postconditions: List[Predicate]
invariants: List[Predicate]
proof_certificate: ProofCertificate
# Provenance
synthesis_method: Literal['cegis', 'cegar', 'genetic', 'hybrid']
llm_seed_id: Optional[str]
counterexamples_resolved: int
epochs_required: int
verification_time_ms: int
# Confidence metrics
entropy_at_synthesis: float
fitness_score: float
abstraction_level_verified: int
def to_pr_description(self) -> str:
"""Generate PR description with verification details."""
return f"""
## Strongly Typed Patch
### Type Signature
{self.type_signature}
### Verification
- **Method**: {self.synthesis_method}
- **Counterexamples resolved**: {self.counterexamples_resolved}
- **Abstraction level**: {self.abstraction_level_verified}
- **Proof certificate**: {self.proof_certificate.hash[:16]}...
### Preconditions
{self._format_predicates(self.preconditions)}
### Postconditions
{self._format_predicates(self.postconditions)}
### Confidence
- Entropy at synthesis: {self.entropy_at_synthesis:.3f}
- Fitness score: {self.fitness_score:.3f}
"""
The Fundamental Insight¶
“Only by using calculation-intense and entropy-aware methods can we hope to steer the models we are using into actual structured output of the strongly typed variety.”
This captures the essential asymmetry:
LLM Outputs |
Formal Synthesis |
|---|---|
O(1) generation |
O(2^n) worst case |
Probabilistic |
Deterministic |
Plausible |
Provable |
Weakly typed |
Strongly typed |
No guarantees |
Formal guarantees |
The synthesis framework bridges this gap: use LLMs for cheap candidate generation, then invest computational resources to achieve formal guarantees. The genetic algorithm maintains population diversity (entropy-aware) while CEGIS/CEGAR provide the verification backbone.
This is the only path to trustworthy AI-generated code.