🤖 AI Summary
This work addresses the long-standing absence of explicit deterministic polynomial-time solvers for concrete NP-complete problems by presenting the first complete deterministic Turing machine implementation for specific problems such as SAT and Subset-Sum. Building upon an NP-verifier simulation framework, the approach extends verification mechanisms to deterministic FNP solving—without increasing the polynomial time complexity—through techniques including dynamic computation graphs, feasible graph construction, and verification-path traversal. A fully functional simulator is implemented in Python, and experimental results demonstrate that the system strictly adheres to polynomial time bounds while effectively generating valid witnesses for satisfiable instances. The source code is publicly released to ensure transparency and reproducibility of the results.
📝 Abstract
While prior work established a verifier-based polynomial time framework for NP, explicit deterministic machines for concrete NP-complete problems have remained elusive. In this paper, we construct fully specified deterministic Turing Machines (DTMs) for SAT and Subset-Sum within a polynomial-time NP verifier simulation framework. We show that both machines operate in polynomial time and, for satisfiable instances, deterministically generate valid witnesses, thereby extending the framework to deterministic FNP computation without increasing the degree of polynomial complexity. Furthermore, we provide a complete implementation of the framework, including the dynamic computation graph, feasible-graph construction, verification walks, and Turing-machine simulation via edge extensions. The implementation behaves in accordance with the predicted polynomial-time bounds. To ensure transparency and reproducibility, the complete Python implementation and source code are made available in a public online repository.