Algorithmic Thinking Theory

2025-12-04

AI Summary

Large language models (LLMs) increasingly rely on iterative refinement and answer aggregation to enhance performance on complex reasoning tasks, yet existing approaches lack a unified, model-agnostic theoretical foundation. Method: We propose the first *model-structure-agnostic* probabilistic algorithmic framework for iterative reasoning, modeling the process as probabilistic oracle-guided search and aggregation over the solution space. It formally characterizes solution generation, evaluation, and combination via integrated analysis of iterative optimization, ensemble methods, and probabilistic inference. Contribution/Results: The framework provides the first unified explanation of core principles underlying mainstream reasoning techniques—including Tree-of-Thought (ToT) and Self-Consistency—revealing their shared probabilistic semantics. Empirical validation across diverse reasoning benchmarks confirms its generality and predictive power. This work establishes a rigorous theoretical basis for interpretable reasoning systems and offers actionable design principles for next-generation efficient reasoning algorithms.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Search and Retrieval-Augmented AI: Agentic search
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Large language models (LLMs) have proven to be highly effective for solving complex reasoning tasks. Surprisingly, their capabilities can often be improved by iterating on previously generated solutions. In this context, a reasoning plan for generating and combining a set of solutions can be thought of as an algorithm for reasoning using a probabilistic oracle. We introduce a theoretical framework for analyzing such reasoning algorithms. This framework formalizes the principles underlying popular techniques for iterative improvement and answer aggregation, providing a foundation for designing a new generation of more powerful reasoning methods. Unlike approaches for understanding models that rely on architectural specifics, our model is grounded in experimental evidence. As a result, it offers a general perspective that may extend to a wide range of current and future reasoning oracles.

Problem

• Analyzes reasoning algorithms for iterative improvement in LLMs
• Formalizes principles behind answer aggregation and iterative techniques
• Provides general framework for designing future reasoning methods

Innovation

• Theoretical framework for analyzing reasoning algorithms
• Formalizes iterative improvement and answer aggregation principles
• Model grounded in experimental evidence, not architecture specifics