A Formal Framework for Naturally Specifying and Verifying Sequential Algorithms

2025-04-28

AI Summary

Formal verification of sequential algorithms faces two key bottlenecks: (1) specification languages struggle to naturally express stateful programs with complex, flexible control flow; and (2) Hoare-logic-based verification fails to reflect the intuitive, hierarchical structure of human-written proofs. To address these challenges, we propose a novel Coq-based formal framework. Our approach introduces the *state-relation monad*, a new semantic modeling mechanism that enables concise, compositional representation of both program state and control flow. We further design a two-phase proof methodology that cleanly separates high-level logical reasoning from low-level mechanical verification, thereby faithfully mirroring natural proof structure. We have fully verified the KMP string-matching algorithm and formally developed a generic depth-first search (DFS) algorithm within this framework. Experimental evaluation demonstrates significant improvements in specification readability, verification modularity, and proof maintainability.

Technology Category

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

Application Category

• 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
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

Current approaches for formal verification of algorithms face important limitations. For specification, they cannot express algorithms naturally and concisely, especially for algorithms with states and flexible control flow. For verification, formal proof based on Hoare logic cannot reflect the logical structure of natural proof. To address these challenges, we introduce a formal framework for naturally specifying and verifying sequential algorithms in Coq. We use the state relation monad to integrate Coq's expressive type system with the flexible control flow of imperative languages. It supports nondeterministic operations and customizable program states, enabling specifying algorithms at an appropriate level of abstraction. For verification, we build a Hoare logic for the monad and propose a novel two-stage proof approach that separates natural logical reasoning from mechanical composition. It reflects the logical structure of natural proof, enhancing modularity and readability. We evaluate the framework by formalizing the Depth-First Search (DFS) algorithm and verifying the Knuth-Morris-Pratt (KMP) algorithm.

Problem

• Limitations in natural specification of sequential algorithms
• Inadequate reflection of logical structure in formal proofs
• Challenges in verifying algorithms with flexible control flow

Innovation

• State relation monad integrates Coq's type system
• Two-stage proof approach enhances modularity
• Supports nondeterministic operations and customizable states