668 lines
19 KiB
Markdown
668 lines
19 KiB
Markdown
# Temporal-Neural-Solver Integration Strategy
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## Executive Summary
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This document outlines the integration of the `temporal-neural-solver` crate into the Lean Agentic Learning System. Temporal-neural-solver combines neural network architectures with temporal logic solving to enable AI systems to reason about time-dependent constraints, temporal planning, and sequential decision-making with formal guarantees.
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## Research Background
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### Temporal Logic and Neural Networks
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**Temporal Logic** [1]: Formal system for reasoning about propositions qualified in terms of time.
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**Types of Temporal Logic**:
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1. **Linear Temporal Logic (LTL)** [2]:
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- `G φ` (Globally): φ holds at all future states
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- `F φ` (Finally): φ holds at some future state
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- `X φ` (Next): φ holds at next state
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- `φ U ψ`: φ holds until ψ becomes true
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2. **Computation Tree Logic (CTL)** [3]:
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- Branching-time logic
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- Multiple possible futures
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3. **Metric Temporal Logic (MTL)** [4]:
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- Time-bounded operators
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- Real-time constraints
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### Neural-Symbolic Integration
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**Key Approaches**:
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1. **Neural Theorem Proving** [5]: Use neural networks to guide logical proof search
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2. **Differentiable Logic** [6]: Make logical operations differentiable for gradient-based learning
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3. **Logic Tensor Networks** [7]: Embed logical knowledge in tensor space
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4. **Neural Module Networks** [8]: Compose neural modules for structured reasoning
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### Applications in AI
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**Temporal Planning**:
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- Robot navigation with safety constraints
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- Multi-step decision-making
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- Reinforcement learning with temporal specifications
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**Formal Verification**:
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- Prove neural network properties
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- Verify safety constraints
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- Generate certificates of correctness
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### References
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[1] Pnueli, A. (1977). "The temporal logic of programs." Proceedings of FOCS '77, 46-57.
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[2] Baier, C., & Katoen, J. P. (2008). "Principles of Model Checking." MIT Press.
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[3] Clarke, E. M., et al. (1999). "Model checking." MIT Press.
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[4] Koymans, R. (1990). "Specifying real-time properties with metric temporal logic." Real-Time Systems, 2(4), 255-299.
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[5] Rocktäschel, T., & Riedel, S. (2017). "End-to-end differentiable proving." NIPS 2017.
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[6] Sourek, G., et al. (2018). "Lifted relational neural networks." Journal of Artificial Intelligence Research, 62, 69-100.
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[7] Serafini, L., & Garcez, A. d'Avila. (2016). "Logic tensor networks." arXiv:1606.04422.
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[8] Andreas, J., et al. (2016). "Neural module networks." CVPR 2016.
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## Integration Architecture
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```
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┌─────────────────────────────────────────────────────────────┐
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│ Temporal-Neural-Solver Architecture │
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├─────────────────────────────────────────────────────────────┤
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│ │
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│ ┌──────────────────┐ ┌─────────────────┐ │
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│ │ Temporal Logic │ │ Neural │ │
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│ │ Specification │─────►│ Encoder │ │
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│ │ (LTL/MTL) │ │ │ │
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│ └──────────────────┘ └────────┬────────┘ │
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│ │ │
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│ ┌────────▼────────┐ │
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│ │ Differentiable │ │
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│ │ Reasoning │ │
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│ │ Engine │ │
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│ └────────┬────────┘ │
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│ │ │
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│ ┌──────────────────┐ ┌────────▼────────┐ │
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│ │ Constraint │◄─────│ Solution │ │
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│ │ Satisfaction │ │ Generator │ │
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│ └──────────────────┘ └────────┬────────┘ │
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│ │ │
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│ ┌────────▼────────┐ │
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│ │ Formal │ │
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│ │ Verification │ │
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│ └─────────────────┘ │
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│ │
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└─────────────────────────────────────────────────────────────┘
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```
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## Use Cases
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### 1. Safe Agent Planning
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**Problem**: Agent must satisfy temporal safety constraints (e.g., "never enter unsafe state").
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**Solution**: Use temporal logic to specify constraints, neural solver to find satisfying plans.
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**Implementation**:
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```rust
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// Specify temporal constraint
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let safety_constraint = ltl!(
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G(not(unsafe_state)) & F(goal_state)
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);
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// Create solver
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let solver = TemporalNeuralSolver::new();
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// Find plan that satisfies constraint
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let plan = solver.solve_with_constraint(
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initial_state,
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safety_constraint,
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max_steps,
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)?;
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// Verify plan
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assert!(solver.verify_plan(&plan, &safety_constraint));
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```
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### 2. Temporal Reward Shaping
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**Problem**: Shape rewards to encourage temporally extended behavior.
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**Solution**: Encode temporal patterns as rewards using neural-symbolic integration.
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**Implementation**:
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```rust
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// Define temporal pattern: "eventually reach A, then reach B"
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let pattern = mtl!(
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F[0..100](at_location_A) & F[100..200](at_location_B)
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);
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// Create reward shaper
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let shaper = TemporalRewardShaper::from_specification(pattern);
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// Use in RL training
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let shaped_reward = base_reward + shaper.compute_bonus(&trajectory);
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```
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### 3. Multi-Agent Coordination
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**Problem**: Coordinate multiple agents with temporal constraints on interactions.
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**Solution**: Solve multi-agent temporal logic specifications.
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**Implementation**:
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```rust
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// Specify coordination constraint
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let coordination = ctl!(
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AG(agent1.action == pickup => AF(agent2.action == deliver))
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);
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// Solve for coordinated policy
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let policies = solver.solve_multi_agent(
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&agents,
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&coordination,
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)?;
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```
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## Technical Specifications
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### API Design
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```rust
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pub struct TemporalNeuralSolver {
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encoder: TemporalEncoder,
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reasoning_engine: DifferentiableReasoner,
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verifier: FormalVerifier,
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config: SolverConfig,
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}
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pub enum TemporalFormula {
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LTL(LTLFormula),
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CTL(CTLFormula),
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MTL(MTLFormula),
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}
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pub struct LTLFormula {
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// G φ, F φ, X φ, φ U ψ
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operator: LTLOperator,
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operands: Vec<Box<LTLFormula>>,
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}
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pub struct Solution {
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pub trajectory: Vec<State>,
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pub actions: Vec<Action>,
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pub satisfaction_proof: Proof,
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pub confidence: f64,
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}
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impl TemporalNeuralSolver {
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pub fn new(config: SolverConfig) -> Self;
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pub fn solve_with_constraint(
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&self,
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initial_state: State,
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constraint: TemporalFormula,
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horizon: usize,
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) -> Result<Solution, SolverError>;
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pub fn verify_plan(
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&self,
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plan: &Plan,
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constraint: &TemporalFormula,
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) -> bool;
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pub fn synthesize_controller(
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&self,
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specification: TemporalFormula,
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) -> Controller;
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pub fn compute_robustness(
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&self,
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trajectory: &Trajectory,
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formula: &MTLFormula,
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) -> f64;
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}
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// Macros for convenient specification
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macro_rules! ltl {
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(G($e:expr)) => { LTLFormula::globally($e) };
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(F($e:expr)) => { LTLFormula::eventually($e) };
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(X($e:expr)) => { LTLFormula::next($e) };
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($e1:expr & $e2:expr) => { LTLFormula::and($e1, $e2) };
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}
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```
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### Performance Requirements
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| Operation | Target | Rationale |
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|-----------|--------|-----------|
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| Formula encoding | <10ms | Real-time planning |
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| Solution search | <500ms | Interactive response |
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| Verification | <100ms | Quick validation |
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| Robustness calc | <50ms | Online monitoring |
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## Integration Points
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### 1. Constrained Agent Planning
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**Location**: `src/lean_agentic/agent.rs`
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**Enhancement**:
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```rust
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impl AgenticLoop {
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pub async fn plan_with_temporal_constraints(
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&self,
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context: &Context,
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constraints: Vec<TemporalFormula>,
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) -> Result<Plan, Error> {
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let solver = TemporalNeuralSolver::new(config);
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// Combine constraints
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let combined = constraints.into_iter()
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.fold(TemporalFormula::true_(), |acc, c| {
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TemporalFormula::and(acc, c)
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});
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// Solve
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let solution = solver.solve_with_constraint(
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context.current_state(),
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combined,
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self.config.max_planning_depth,
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)?;
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// Convert to plan
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Ok(Plan::from_solution(solution))
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}
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}
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```
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### 2. Safe Learning with Temporal Specifications
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**Location**: `src/lean_agentic/learning.rs`
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**Enhancement**:
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```rust
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pub struct SafeStreamLearner {
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learner: StreamLearner,
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solver: TemporalNeuralSolver,
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safety_specs: Vec<TemporalFormula>,
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}
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impl SafeStreamLearner {
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pub async fn learn_safely(
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&mut self,
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experience: Experience,
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) -> Result<(), Error> {
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// Propose update
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let proposed_update = self.learner.compute_update(&experience);
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// Verify safety
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if !self.verify_safety(&proposed_update) {
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return Err(Error::SafetyViolation);
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}
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// Apply update
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self.learner.apply_update(proposed_update)?;
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Ok(())
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}
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fn verify_safety(&self, update: &Update) -> bool {
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let predicted_trajectory = self.simulate_with_update(update);
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self.safety_specs.iter().all(|spec| {
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self.solver.verify_plan(&predicted_trajectory, spec)
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})
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}
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}
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```
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### 3. Temporal Knowledge Reasoning
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**Location**: `src/lean_agentic/knowledge.rs`
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**Enhancement**:
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```rust
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impl KnowledgeGraph {
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pub fn query_temporal(
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&self,
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query: TemporalQuery,
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) -> Vec<TemporalFact> {
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let solver = TemporalNeuralSolver::new(config);
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// Encode query as temporal formula
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let formula = query.to_temporal_formula();
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// Find satisfying facts
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self.temporal_facts.iter()
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.filter(|fact| solver.check_satisfaction(fact, &formula))
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.cloned()
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.collect()
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}
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pub fn infer_temporal_relations(
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&mut self,
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solver: &TemporalNeuralSolver,
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) {
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// Infer new temporal facts from existing knowledge
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for fact1 in &self.temporal_facts {
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for fact2 in &self.temporal_facts {
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if let Some(inferred) = solver.infer_relation(fact1, fact2) {
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self.add_temporal_fact(inferred);
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}
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}
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}
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}
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}
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```
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## Implementation Phases
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### Phase 1: Core Solver (Week 1-2)
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- [ ] Implement LTL parser and encoder
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- [ ] Create neural encoding network
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- [ ] Build differentiable reasoning engine
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- [ ] Add basic SAT solver
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- [ ] Write unit tests
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### Phase 2: Temporal Operators (Week 3)
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- [ ] Implement all LTL operators (G, F, X, U)
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- [ ] Add MTL time-bounded operators
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- [ ] Create CTL branching operators
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- [ ] Implement robustness semantics
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- [ ] Write integration tests
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### Phase 3: Neural Integration (Week 4)
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- [ ] Create logic tensor networks
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- [ ] Implement neural module networks
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- [ ] Add gradient-based optimization
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- [ ] Integrate with agent planning
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- [ ] Benchmark performance
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### Phase 4: Verification (Week 5)
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- [ ] Implement model checking
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- [ ] Add counterexample generation
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- [ ] Create certificate generation
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- [ ] Add safety verification
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- [ ] Write documentation
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## Benchmarking Strategy
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### Benchmark Suite
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```rust
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#[bench]
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fn bench_ltl_encoding(b: &mut Bencher) {
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let formula = ltl!(G(safe) & F(goal));
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let solver = TemporalNeuralSolver::new(config);
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b.iter(|| {
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solver.encode_formula(&formula)
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});
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}
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#[bench]
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fn bench_planning_with_constraints(b: &mut Bencher) {
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let solver = TemporalNeuralSolver::new(config);
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let constraint = ltl!(G(not(unsafe)) & F(goal));
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b.iter(|| {
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solver.solve_with_constraint(
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initial_state.clone(),
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constraint.clone(),
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100,
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)
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});
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}
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#[bench]
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fn bench_robustness_calculation(b: &mut Bencher) {
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let solver = TemporalNeuralSolver::new(config);
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let trajectory = generate_trajectory(100);
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let formula = mtl!(F[0..50](goal));
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b.iter(|| {
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solver.compute_robustness(&trajectory, &formula)
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});
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}
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```
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### Validation Tests
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```rust
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#[test]
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fn test_safety_verification() {
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let solver = TemporalNeuralSolver::new(config);
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// Constraint: never enter unsafe state
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let safety = ltl!(G(not(unsafe_state)));
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// Safe plan
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let safe_plan = generate_safe_plan();
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assert!(solver.verify_plan(&safe_plan, &safety));
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// Unsafe plan
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let unsafe_plan = generate_unsafe_plan();
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assert!(!solver.verify_plan(&unsafe_plan, &safety));
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}
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#[test]
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fn test_liveness_verification() {
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let solver = TemporalNeuralSolver::new(config);
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// Constraint: eventually reach goal
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let liveness = ltl!(F(goal_state));
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let plan_with_goal = generate_plan_reaching_goal();
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assert!(solver.verify_plan(&plan_with_goal, &liveness));
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let plan_without_goal = generate_plan_not_reaching_goal();
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assert!(!solver.verify_plan(&plan_without_goal, &liveness));
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}
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```
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## Neural Network Architecture
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### Temporal Encoder
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```rust
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pub struct TemporalEncoder {
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embedding: nn::Embedding,
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lstm: nn::LSTM,
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attention: nn::MultiheadAttention,
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output: nn::Linear,
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}
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impl TemporalEncoder {
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pub fn encode(&self, formula: &TemporalFormula) -> Tensor {
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// Convert formula to sequence
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let sequence = formula.to_sequence();
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// Embed
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let embedded = self.embedding.forward(&sequence);
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// LSTM encoding
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let (encoded, _) = self.lstm.forward(&embedded);
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// Attention over temporal operators
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let attended = self.attention.forward(&encoded, &encoded, &encoded);
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// Final encoding
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self.output.forward(&attended)
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}
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}
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```
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### Differentiable Reasoner
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```rust
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pub struct DifferentiableReasoner {
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rule_network: nn::Sequential,
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memory: nn::Parameter,
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iterations: usize,
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}
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impl DifferentiableReasoner {
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pub fn reason(&self, encoded_formula: &Tensor, state: &State) -> Tensor {
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let mut reasoning_state = self.memory.clone();
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for _ in 0..self.iterations {
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// Apply reasoning rules
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reasoning_state = self.rule_network.forward(&vec![
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reasoning_state.clone(),
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encoded_formula.clone(),
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state.to_tensor(),
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]);
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}
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reasoning_state
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}
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}
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```
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## Success Criteria
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- [ ] Formula encoding < 10ms
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- [ ] Planning with constraints < 500ms
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- [ ] Verification < 100ms
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- [ ] 95%+ accuracy on benchmark temporal logic problems
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- [ ] Successfully verify safety properties
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- [ ] Generate correct counterexamples
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- [ ] Full test coverage (>90%)
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## Safety Guarantees
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### Soundness
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```rust
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/// Guarantee: If solver.verify_plan returns true,
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/// the plan ACTUALLY satisfies the specification
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#[test]
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fn test_soundness() {
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let solver = TemporalNeuralSolver::new(config);
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for _ in 0..1000 {
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let spec = generate_random_specification();
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let plan = generate_random_plan();
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if solver.verify_plan(&plan, &spec) {
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// Formally check with external model checker
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assert!(model_check(&plan, &spec));
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}
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}
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}
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```
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### Completeness
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```rust
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/// Best-effort: If a solution exists, try to find it
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/// (May not always succeed due to search complexity)
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#[test]
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fn test_completeness_on_simple_problems() {
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let solver = TemporalNeuralSolver::new(config);
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// For simple, known-solvable problems
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let simple_specs = generate_simple_specifications();
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for spec in simple_specs {
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let solution = solver.solve_with_constraint(
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initial_state,
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spec.clone(),
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100,
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);
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// Should find solution for simple problems
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assert!(solution.is_ok());
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}
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}
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```
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## Future Enhancements
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1. **Probabilistic Temporal Logic**: Handle uncertainty
|
||
2. **Multi-Objective Optimization**: Balance multiple temporal constraints
|
||
3. **Continuous-Time Logic**: Support hybrid systems
|
||
4. **Quantitative Verification**: Compute satisfaction probabilities
|
||
5. **Adaptive Complexity**: Adjust solver based on problem difficulty
|
||
|
||
## References
|
||
|
||
[1] Pnueli (1977). The temporal logic of programs.
|
||
[2] Baier & Katoen (2008). Principles of Model Checking.
|
||
[3] Clarke et al. (1999). Model checking.
|
||
[4] Koymans (1990). Metric temporal logic.
|
||
[5] Rocktäschel & Riedel (2017). End-to-end differentiable proving.
|
||
[6] Sourek et al. (2018). Lifted relational neural networks.
|
||
[7] Serafini & Garcez (2016). Logic tensor networks.
|
||
[8] Andreas et al. (2016). Neural module networks.
|
||
|
||
## Appendix A: LTL Semantics
|
||
|
||
### Syntax
|
||
|
||
```
|
||
φ ::= p | ¬φ | φ₁ ∧ φ₂ | X φ | φ₁ U φ₂
|
||
```
|
||
|
||
### Derived Operators
|
||
|
||
```
|
||
F φ ≡ true U φ (Eventually)
|
||
G φ ≡ ¬F¬φ (Globally)
|
||
φ₁ R φ₂ ≡ ¬(¬φ₁ U ¬φ₂) (Release)
|
||
```
|
||
|
||
### Semantics
|
||
|
||
```
|
||
σ, i ⊨ p iff p ∈ σ(i)
|
||
σ, i ⊨ ¬φ iff σ, i ⊭ φ
|
||
σ, i ⊨ φ₁∧φ₂ iff σ, i ⊨ φ₁ and σ, i ⊨ φ₂
|
||
σ, i ⊨ X φ iff σ, i+1 ⊨ φ
|
||
σ, i ⊨ φ₁Uφ₂ iff ∃j≥i: σ,j ⊨ φ₂ and ∀i≤k<j: σ,k ⊨ φ₁
|
||
```
|
||
|
||
## Appendix B: Example Usage
|
||
|
||
```rust
|
||
use midstream::temporal_neural_solver::*;
|
||
|
||
// Create solver
|
||
let solver = TemporalNeuralSolver::new(config);
|
||
|
||
// Define safety specification
|
||
let safety = ltl!(
|
||
G(not(collision)) & // Never collide
|
||
F(goal_reached) & // Eventually reach goal
|
||
G(battery > 20) // Always maintain battery
|
||
);
|
||
|
||
// Solve for plan
|
||
let solution = solver.solve_with_constraint(
|
||
robot.current_state(),
|
||
safety,
|
||
horizon = 100,
|
||
)?;
|
||
|
||
// Verify solution
|
||
assert!(solver.verify_plan(&solution, &safety));
|
||
|
||
// Execute plan
|
||
for action in solution.actions {
|
||
robot.execute(action);
|
||
}
|
||
|
||
// Monitor robustness online
|
||
let robustness = solver.compute_robustness(
|
||
&robot.trajectory(),
|
||
&safety,
|
||
);
|
||
|
||
if robustness < 0.0 {
|
||
eprintln!("Warning: Safety specification violated!");
|
||
}
|
||
```
|