sc-tir-mathematical-reasoning

SC-TIR: Self-Consistency with Tool-Integrated Reasoning

Problem

Large language models struggle with mathematical reasoning due to:

1. Calculation errors: Text-only reasoning often makes arithmetic mistakes
2. Lack of precision: Cannot reliably compute complex expressions
3. Single-path failure: One reasoning trace may hit a dead end
4. No verification: Cannot check intermediate results

Symptoms:

• Model gives correct reasoning but wrong final answer
• Long calculations contain arithmetic errors
• Inconsistent answers across multiple runs
• Competition scores plateau around 30-40%

Context / Trigger Conditions

Use SC-TIR when:

• Working on mathematical reasoning tasks (AIMO, AIME, MATH benchmarks)
• LLM outputs need precise calculation (algebra, calculus, combinatorics)
• Single-pass reasoning is insufficient or unreliable
• Need state-of-the-art performance on math competitions
• Problem: "My model gets the reasoning right but the answer wrong"

Competition context:

• AIMO (AI Mathematical Olympiad) - 0-99999 integer answers
• MATH benchmark - High school competition problems
• AIME - American Invitational Mathematics Examination

Solution

SC-TIR Algorithm Overview

SC-TIR combines two powerful techniques:

1. Self-Consistency: Sample multiple reasoning paths and vote
2. Tool-Integrated Reasoning: Model generates Python code for precise calculation

Core parameters:

• N (width): Number of reasoning traces to generate (typical: 4-48)
• M (depth): Number of iterations per trace (typical: 1-4)

Algorithm flow:

Input: Mathematical problem P
Output: Final answer A

1. Initialize N candidates with problem text
2. For each candidate i in 1..N:
   For iteration j in 1..M:
     a) Model generates Python code to solve current step
     b) Execute code, capture output
     c) Append output to context
     d) If no code generated → restart or prune
   End For
3. Extract all \boxed{answer} from candidates
4. Filter invalid answers (non-numeric, negative)
5. Apply modulo 1000 (if required by competition)
6. Majority vote: select most common answer
7. Return A