Below is a conceptual, skeletal implementation of LLM‑Ω with all extensions. It is not runnable as a full system but illustrates the architecture, components, and interactions. Real implementation would require thousands of lines for each module (transformers, proof assistant integration, etc.).
# llm_omega.py
# Conceptual blueprint for LLM-Ω: a mathematics-only language model
# with pattern recognition, core math engine, theorem proving, discovery,
# multi-modal input, RL from feedback, and self-play.
import torch
import torch.nn as nn
import torch.nn.functional as F
from typing import Any, Dict, List, Optional, Tuple
import numpy as np
import sympy as sp
from dataclasses import dataclass
# ----------------------------------------------------------------------
# 1. Pattern Recognition Engine (PRE)
# ----------------------------------------------------------------------
class PatternRecognitionEngine(nn.Module):
"""Multi-modal encoder: text, images, diagrams -> math object."""
def __init__(self, hidden_dim=512):
super().__init__()
self.text_encoder = nn.TransformerEncoder(...) # placeholder
self.image_encoder = nn.Sequential(...) # CNN
self.diagram_encoder = DiagramEncoder() # see below
self.fusion = nn.Linear(hidden_dim*3, hidden_dim)
def forward(self, text, image, diagram) -> "MathObject":
# Encode each modality
t_emb = self.text_encoder(text) if text else torch.zeros(1,512)
i_emb = self.image_encoder(image) if image else torch.zeros(1,512)
d_emb = self.diagram_encoder(diagram) if diagram else torch.zeros(1,512)
fused = self.fusion(torch.cat([t_emb, i_emb, d_emb], dim=-1))
# Convert to a math expression tree (simplified as a string for now)
return MathObject.from_embedding(fused)
class DiagramEncoder(nn.Module):
"""CNN + GNN to parse commutative diagrams, geometric figures."""
def __init__(self):
super().__init__()
self.cnn = nn.Conv2d(...)
self.gnn = nn.GraphConv(...)
def forward(self, diagram_image):
# Extract nodes and edges -> adjacency matrix
return self.gnn(self.cnn(diagram_image))
# ----------------------------------------------------------------------
# 2. Core Mathematics Engine (CME)
# ----------------------------------------------------------------------
class CoreMathEngine(nn.Module):
"""Transformer over math expression trees."""
def __init__(self, vocab_size=10000, hidden_dim=512, num_layers=6):
super().__init__()
self.embed = nn.Embedding(vocab_size, hidden_dim)
self.tree_transformer = TreeTransformer(hidden_dim, num_layers)
self.output_proj = nn.Linear(hidden_dim, vocab_size)
def forward(self, math_tree: "MathTree") -> "MathTree":
# Tree is serialized as a sequence with special tokens (e.g., parentheses)
seq = math_tree.to_sequence()
emb = self.embed(seq)
out = self.tree_transformer(emb) # custom attention that respects tree structure
logits = self.output_proj(out)
# Decode back to tree
return MathTree.from_logits(logits)
class TreeTransformer(nn.Module):
"""Transformer with tree-structured attention (placeholder)."""
def __init__(self, hidden_dim, num_layers):
super().__init__()
self.layers = nn.ModuleList([nn.TransformerEncoderLayer(hidden_dim, 8) for _ in range(num_layers)])
def forward(self, x):
for layer in self.layers:
x = layer(x)
return x
# ----------------------------------------------------------------------
# 3. Mathematical Objects
# ----------------------------------------------------------------------
class MathObject:
"""Abstract base for all mathematical structures."""
pass
class MathTree(MathObject):
"""Expression tree (e.g., ( + 2 ( * 3 x ) )."""
def __init__(self, root):
self.root = root
def to_sequence(self):
# Convert tree to a flat sequence (preorder) with special tokens
return [1,2,3] # dummy
@classmethod
def from_logits(cls, logits):
# Dummy: argmax, then build tree
return cls(None)
# ----------------------------------------------------------------------
# 4. Interactive Theorem Proving (ITP) Module
# ----------------------------------------------------------------------
class ITPModule:
"""Communicates with Lean/Coq proof assistant."""
def __init__(self, proof_assistant="lean"):
self.prover = None # would launch a process
def check_proof(self, theorem_stmt: str, proof_script: str) -> Tuple[bool, str]:
"""Returns (success, error_message)."""
# In reality: write to a file, call lean, parse output.
# Placeholder: always succeed for dummy.
return True, ""
def step_feedback(self, partial_proof: str) -> List[str]:
"""Return remaining subgoals as mathematical objects."""
# Parse proof assistant output into a list of goal expressions.
return []
# ----------------------------------------------------------------------
# 5. Mathematical Discovery Module
# ----------------------------------------------------------------------
class DiscoveryModule:
def __init__(self, cme: CoreMathEngine, itp: ITPModule, novelty_estimator):
self.cme = cme
self.itp = itp
self.novelty = novelty_estimator # e.g., a neural net trained on known theorems
def generate_conjecture(self) -> MathObject:
# Sample from CME's distribution conditioned on a "conjecture" prefix
dummy_tree = MathTree(None)
output_tree = self.cme(dummy_tree) # in reality, use generation loop
return output_tree
def is_novel(self, conjecture: MathObject) -> float:
return self.novelty(conjecture) # high score = novel
def discover(self):
conjecture = self.generate_conjecture()
if self.is_novel(conjecture) > 0.7:
# Try to prove using ITP
success, err = self.itp.check_proof(str(conjecture), "") # need proof script
if success:
# Add to knowledge base
return conjecture
return None
# ----------------------------------------------------------------------
# 6. Multi-Modal Diagram Encoder (part of PRE already)
# ----------------------------------------------------------------------
# Already included in PatternRecognitionEngine via DiagramEncoder.
# ----------------------------------------------------------------------
# 7. Reinforcement Learning from Mathematical Feedback (RLMF)
# ----------------------------------------------------------------------
class RLMFEnvironment:
"""Environment that uses a Computer Algebra System (CAS) for reward."""
def __init__(self):
self.cas = sp # sympy
def step(self, expression: str) -> Tuple[float, bool]:
"""Evaluate correctness of a mathematical transformation."""
# Example: given expression "2+2", expected "4"
# This is a placeholder. Real implementation would have ground truth.
try:
result = sp.sympify(expression)
# Compare with expected (needs state)
return 1.0, True
except:
return -1.0, False
class RLMFAgent:
def __init__(self, cme: CoreMathEngine):
self.cme = cme
def act(self, state_math: MathObject) -> MathObject:
return self.cme(state_math)
def update(self, reward, old_action, new_state):
# Use PPO or REINFORCE
pass
# ----------------------------------------------------------------------
# 8. Self-Play and Curriculum Learning
# ----------------------------------------------------------------------
class SelfPlayOrchestrator:
def __init__(self, generator: CoreMathEngine, solver: CoreMathEngine, verifier: ITPModule):
self.generator = generator
self.solver = solver
self.verifier = verifier
def generate_problem(self) -> MathObject:
# Sample from generator
return self.generator(MathTree(None))
def solve_problem(self, problem: MathObject) -> MathObject:
return self.solver(problem)
def verify_solution(self, problem: MathObject, solution: MathObject) -> bool:
# Use ITP to check that solution solves the problem
# For an equation: solution substituted into problem should hold.
# Dummy implementation.
return True
def self_play_step(self):
problem = self.generate_problem()
solution = self.solve_problem(problem)
if self.verify_solution(problem, solution):
# Reward both generator and solver
return True
return False
# ----------------------------------------------------------------------
# 9. Main LLM-Ω Model (Integrates all)
# ----------------------------------------------------------------------
class LLM_Omega:
def __init__(self):
self.pre = PatternRecognitionEngine()
self.cme = CoreMathEngine()
self.itp = ITPModule()
self.discovery = DiscoveryModule(self.cme, self.itp, None)
self.rl_agent = RLMFAgent(self.cme)
self.selfplay = SelfPlayOrchestrator(self.cme, self.cme, self.itp)
self.knowledge_base = [] # list of proved theorems
def forward(self, raw_input: Dict[str, Any]) -> MathObject:
"""Main inference: raw_input may contain text, image, diagram."""
math_obj = self.pre(raw_input.get("text"), raw_input.get("image"), raw_input.get("diagram"))
result = self.cme(math_obj)
return result
def train_step(self, batch: List[Tuple[MathObject, MathObject]]):
"""Supervised training on pairs (input, target)."""
# Standard next-token prediction on math trees
for inp, target in batch:
output = self.cme(inp)
loss = F.cross_entropy(output, target) # simplified
loss.backward()
def rl_train_step(self, state: MathObject, target: Optional[MathObject] = None):
"""Use RLMF environment to get reward."""
action = self.rl_agent.act(state)
reward, done = RLMFEnvironment().step(str(action))
self.rl_agent.update(reward, action, state)
def self_play_epoch(self, steps=100):
for _ in range(steps):
self.selfplay.self_play_step()
def discover_theorems(self):
new_theorem = self.discovery.discover()
if new_theorem:
self.knowledge_base.append(new_theorem)
print(f"Discovered: {new_theorem}")
# ----------------------------------------------------------------------
# 10. Example Usage
# ----------------------------------------------------------------------
if __name__ == "__main__":
model = LLM_Omega()
# Train supervised on math expressions
# model.train_step(batch)
# Run self-play
model.self_play_epoch(10)
# Discover new theorems
model.discover_theorems()
# Inference on a text query (converted by PRE)
result = model.forward({"text": "derivative of x^2"})
print(result) # should output something like "2*x"Explanation of the code structure:
PatternRecognitionEngine: Multi-modal encoder; outputs aMathObject(here simplified as an expression tree).CoreMathEngine: Transformer that operates on expression trees; uses a customTreeTransformer(placeholder for tree-structured attention).ITPModule: Interface to external proof assistant; methods to check proofs and get subgoals.DiscoveryModule: Generates conjectures via CME, filters by novelty, attempts proof via ITP.DiagramEncoder: Part of PRE; uses CNN + GNN to parse diagrams into math.RLMFEnvironmentandRLMFAgent: Use a CAS (SymPy) to evaluate correctness of algebraic transformations; update model with policy gradient.SelfPlayOrchestrator: Generator-solver-verifier loop; both generator and solver share the same CME (or can be separate copies).LLM_Omega: Main class integrating all components; provides training and inference methods.
This code is a blueprint – real implementation would require:
- A proper tokenizer and tree‑structured attention for math expressions.
- Integration with Lean/Coq (calling external processes, parsing errors).
- Training on large corpora (e.g., arXiv, Mathlib).
- GPU‑accelerated training loops.
But it captures the essential design of LLM‑Ω.