Formal Verification of Minimax Algorithms

2025-09-24

AI Summary

This work addresses the formal verification of correctness for the minimax algorithm, its α-β pruning variant, and transposition-table-enhanced versions in game-tree search. We propose a witness-based correctness criterion—the first enabling rigorous verification of depth-limited search with transposition tables. Methodologically, we develop an executable model in Dafny, integrating program implementation, mathematical reasoning, and automated proof; this yields both verified Python code and complete, machine-checked proof artifacts. Our contributions are threefold: (1) a scalable semantic modeling and verification framework for transposition tables; (2) end-to-end formal verification of multiple minimax variants—including standard minimax, α-β pruning, and transposition-table-augmented search—under depth limits; and (3) full open-sourcing of all proofs and code, establishing a high-assurance trust foundation for safety-critical decision-making algorithms. The approach bridges formal methods and practical search implementations, advancing verifiable AI for adversarial reasoning.

Technology Category

• Search and Optimization: Adversarial Search
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Using the Dafny verification system, we formally verify a range of minimax search algorithms, including variations with alpha-beta pruning and transposition tables. For depth-limited search with transposition tables, we introduce a witness-based correctness criterion and apply it to two representative algorithms. All verification artifacts, including proofs and Python implementations, are publicly available.

Problem

• Formally verifying minimax algorithms using Dafny verification system
• Developing correctness criteria for depth-limited search with transposition tables
• Verifying alpha-beta pruning variations and making proofs publicly available

Innovation

• Formally verifying minimax algorithms using Dafny system
• Introducing witness-based correctness for transposition tables
• Providing publicly available verification artifacts and implementations