P vs. NP and the Difficulty of Computation: A ruliological approach
6 days ago
- #computational complexity
- #Turing machines
- #theoretical computer science
- The article explores empirical approaches to theoretical computer science, particularly focusing on the P vs. NP question.
- It discusses the use of Turing machines to empirically study computational complexity by enumerating possible programs and observing their behavior.
- The author highlights the concept of computational irreducibility, where certain computations cannot be shortcut and must be run step-by-step.
- The article details experiments with small Turing machines (1-state, 2-state, and 3-state) to understand their computational capabilities and limitations.
- It examines the runtime distributions and space complexity of these Turing machines, providing insights into their behavior.
- The author introduces the idea of multiway (nondeterministic) Turing machines and compares their computational power to deterministic ones.
- The article discusses the 'everything machine,' a theoretical construct that includes all possible rules, and its implications for understanding computation.
- It reflects on the challenges and insights gained from empirical studies of Turing machines, including the ubiquity of computational irreducibility.
- The author shares personal notes on the evolution of his work on ruliology and its connections to theoretical computer science.
- The article concludes by considering the implications of these findings for the P vs. NP question and the broader field of theoretical computer science.