O-Forge Unleashing the Power of LLMs and CAS for Proving Asymptotic Inequalities

Asymptotic analysis plays a crucial role in mathematics and computer science, especially when comparing the growth rates of functions. Proving inequalities involving asymptotic behavior often requires both creative insight and rigorous symbolic verification. The recent framework called O-Forge combines the strengths of large language models (LLMs) with computer algebra systems (CAS) to tackle this challenge effectively. This post explores how O-Forge works, why it matters and how it advances the state of asymptotic inequality verification.

O-Forge framework integrating LLM and CAS for asymptotic analysis

The Challenge of Proving Asymptotic Inequalities

At its core, an asymptotic inequality compares two functions \( f(x) \) and \( g(x) \) for sufficiently large \( x \). A typical statement looks like this:

\[

f(x) \leq C \cdot g(x) \quad \text{for all } x \geq x_0

\]

Here, \( C \) and \( x_0 \) are constants that exist to make the inequality true. This can be rewritten as a quantified real inequality:

\[

\exists C > 0, \exists x_0, \forall x \geq x_0 : f(x) - C g(x) \leq 0

\]

This kind of statement is challenging because it involves quantifiers and inequalities over real numbers, which are generally hard to prove symbolically. Computer algebra systems can handle parts of this problem when the functions belong to supported classes, using techniques like quantifier elimination and decision procedures. However, the complexity often grows quickly, especially when the domain is large or the functions have complicated behavior.

How O-Forge Combines LLMs and CAS

The key innovation in O-Forge is to combine the creativity of LLMs with the symbolic rigor of CAS in a feedback loop. Large language models excel at proposing proof ideas and decompositions of the problem, while CAS tools provide formal verification of each proposed piece.

Domain Decomposition

O-Forge breaks down the infinite domain \([x_0, \infty)\) into a finite union of intervals:

\[

[x_0, \infty) = \bigcup_{i=1}^m I_i

\]

On each interval \( I_i \), the CAS attempts to prove:

\[

\forall x \in I_i : f(x) - C g(x) \leq 0

\]

The LLM suggests a "good" partition based on mathematical features like turning points, monotonicity changes, or singularities. This decomposition makes the problem tractable for the CAS, which can then certify each piece axiomatically.

The Feedback Loop

The process iterates as follows:

• The LLM proposes a partition of the domain.

• The CAS verifies the inequality on each piece.

• If verification fails on some intervals, the LLM refines the partition or adjusts constants.

• This loop continues until the entire domain is certified or a counterexample is found.

  
  

This collaboration leverages the best of both worlds: human-like intuition from LLMs and the precision of symbolic computation.

Practical Example of the Verification Loop

Here is a simplified pseudocode illustrating the core verification loop used in O-Forge. While O-Forge uses powerful CAS like Mathematica, this example uses SymPy as a stand-in:

```python

def verify_piecewise(f, g, C, pieces):

# pieces: list of intervals, e.g. [(x0, 1), (1, 10), (10, oo)]

for (a, b) in pieces:

if not CAS_prove_forall(f(x) - C*g(x) <= 0, domain=(x >= a) & (x <= b)):

return False

return True

x0 = 1

C = propose_constant()

pieces = propose_partition()

result = verify_piecewise(f, g, C, pieces)

```

In this setup:

• `propose_constant()` is a function where the LLM suggests a suitable constant \( C \).

• `propose_partition()` is the LLM's proposal for domain decomposition.

• `CAS_prove_forall()` is the CAS's symbolic verification routine.

  
  

The loop can be extended to refine \( C \) and the partition until verification succeeds.

Why O-Forge Matters

Traditional methods for proving asymptotic inequalities often rely on manual insight and tedious symbolic manipulation. O-Forge automates much of this process by:

• Reducing human effort in identifying critical points and intervals.

• Ensuring symbolic correctness through CAS verification.

• Handling complex functions that are difficult to analyze by hand.

• Providing a scalable framework that can be adapted to various classes of functions.

  
  

This approach opens new possibilities for research in algorithm analysis, complexity theory and mathematical proofs where asymptotic behavior is central.

Potential Applications and Impact

The combination of LLMs and CAS in O-Forge can impact several areas:

• Algorithm analysis: Automatically proving runtime bounds and complexity inequalities.

• Mathematical research: Assisting in conjecture verification involving asymptotic relations.

• Education: Providing tools for students to explore and verify inequalities interactively.

• Symbolic computation: Enhancing CAS capabilities with AI-driven heuristics.

  
  

By bridging creative reasoning and formal verification, O-Forge represents a step forward in automated mathematical reasoning.

Summary and Next Steps

O-Forge demonstrates how large language models and computer algebra systems can work together to prove asymptotic inequalities effectively. The framework’s domain decomposition strategy and feedback loop enable tackling complex quantified inequalities with symbolic rigor.