Formal Derivation and Verification of Critical Path Algorithm for Directed Acyclic Graph

Zhen You, Xinwu Yi, Jinyun Xue, Hongwen Hu, Jiewen Huang, Zhuo Cheng

Abstract: Graph structure is widely used in network design, path planning, relational processing, electronic circuit design, and power grid tide management. However, the complexity and variety of relationships between different data objects create difficulties in the derivation of graph algorithms, the correctness of algorithms cannot be easily guaranteed in some complex problems. In this paper, we formally derive the loop invariant of critical path by using the new definition and new strategies of loop invariant in PAR method. Furthermore, the Apla abstraction algorithm program is designed, and the executable code is generated by the PAR platform. Finally, the correctness of the algorithm is proved by using the Dijkstra’s weakest precondition method. The critical path is a typical dynamic programming problem. The recursive relation of the critical path can be automatically detected when the loop invariant is derived by using the PAR method, and the reliability of the Apla algorithm program is ensured by using the formal verification technique. The formal derivation and verification of the critical path algorithm in this paper can be extended to solve other dynamic programming type problems.