# DB-CRAWL-AGENT
Agentic framework that translates natural language queries into analytic features, decomposes them into SQL tasks, executes them across databases, and leverages graph-based memoization for reuse.

Table of Contents
1.	Overview
2.  Mathematical Foundations of Graph-Memoized Reasoning


# Overview

db-crawl is an agentic data-analysis framework that lets you query structured data warehouses using natural language.
It decomposes human intent into features, tasks, and SQL plans, then executes them across databases such as PostgreSQL, Snowflake, and AWS RDS.

User Query → Feature Orchestrator → Task Decomposer → SQL Executor → Insight Generator

Key Goals:
	•	LLM-driven feature reasoning
	•	Automated SQL generation
	•	Reuse via graph-memoization
	•	Multi-database compatibility

# Mathematical Foundations of Graph-Memoized Reasoning

The decision-graph memoization layer in **db-crawl** is not merely a system optimization — it is grounded in mathematical optimization principles that define how agentic reasoning can be made efficient, consistent, and auditable.

### Optimization over Graph States

Each reasoning workflow can be represented as a property graph  
\( G = (V, E) \),  
where nodes represent intents, decompositions, or execution steps, and edges capture information dependencies.

The system seeks an **optimal reasoning graph** that minimizes total computational cost while maintaining internal consistency:

\[
\min_G \; \mathcal{L}(G)
   = \text{Cost}(G) + \lambda \, \text{Inconsistency}(G)
\]


- **Cost(G)** represents resource usage: number of LLM calls, latency, or plan complexity.  
- **Inconsistency(G)** captures logical or causal divergence across equivalent reasoning paths.  
- **λ** is a Lagrange-like hyperparameter balancing efficiency and reliability.

This optimization lives in a **hybrid space** — continuous (edge weights, confidence values) and discrete (graph structure).  
Practical solvers combine:
- **Gradient-based methods** for differentiable parameters (confidence, utility scores).  
- **Combinatorial search or relaxations** (e.g., Gumbel-Softmax, simulated annealing) for discrete structure updates.

### Constrained Optimization for Causal Consistency

Causal consistency introduces structural constraints on permissible graph edges.  
For a directed acyclic reasoning graph (DAG),  
\[
A_{ij} = 0 \quad \text{if no causal link } X_j \to X_i
\]

The optimization thus becomes **constrained**:
\[
\min_G \; \mathcal{L}(G) \quad \text{s.t. } h(G) = 0
\]
where \( h(G) \) encodes acyclicity and dependency rules.

This enables **identifiability** — ensuring that updates or agent inferences remain explainable and reproducible under schema changes.  
Constraint handling follows classical **Lagrangian** and **KKT** formulations, allowing causal rules to act as soft or hard regularizers.

### Multi-Objective Optimization for Ethical Reasoning

Beyond efficiency, trustworthy systems must optimize across **multiple conflicting objectives**:

\[
\min_G \; [ f_1(G), f_2(G), f_3(G) ]
\]
where typical objectives are:
- \( f_1 \): computational efficiency,  
- \( f_2 \): fairness or resource balance,  
- \( f_3 \): interpretability or transparency.

The result is a **Pareto frontier** of reasoning graphs, each offering a different trade-off between utility and ethics.  
Selecting among them involves *multi-objective optimization* techniques (weighted sums, evolutionary methods, or Pareto ranking).

### Summary

Mathematically, the **memoization layer** transforms reasoning reuse into an optimization problem defined over structured decision graphs.  
It blends **continuous optimization** (for model parameters and scores) with **discrete optimization** (for structural choices and cache retrieval), while maintaining causal and ethical constraints.  
This formulation ensures that every reused workflow is not just efficient, but **formally auditable, stable, and ethically grounded**.
