The Quest For Decidability: Seeking Resolution On Long-Standing Open Problems

Seeking Decidability: The P vs. NP Problem

The P vs. NP problem is a central open question in the mathematical fields of computational complexity theory and algorithmic analysis. It asks whether all problems with solutions that can be quickly verified by a computer can also be quickly solved by a computer. More formally, the P class contains decision problems that can be solved in polynomial time by a deterministic Turing machine, while the NP class contains problems with solutions that can be verified in polynomial time.

The entities involved in defining the P vs. NP problem include the complexity classes P and NP, decision problems, deterministic Turing machines, polynomial time complexity, and verifiability of solutions. The key predicates assert that P is a set of problems solvable in polynomial time, NP is a set verifiable in polynomial time, and the open question asks whether P = NP or P ≠ NP.

Defining the P vs. NP Problem

To fully define the P vs. NP problem requires formalizing the complexity classes P and NP, the concepts of algorithms and computation, and what constitutes efficient solvability versus efficient verifiability. The class P consists of decision problems that can be solved by a deterministic Turing machine in polynomial time, with respect to the size of the input. The class NP consists of problems with the property that given any proposed solution, its correctness can be verified by a deterministic Turing machine in polynomial time.

Key attributes include deterministic computation, polynomial time complexity as a definition of algorithmic efficiency, and the asymmetry between the difficulty of blindly finding a solution versus checking a given solution. While verifying solutions can be done by systematically checking them against the problem parameters, finding solutions often requires more guesswork and creativity.

Implications of Resolving P ≠ NP

Resolving that P does not equal NP would have sweeping implications both theoretically and for practical computing applications. It would eliminate the possibility of generic efficient algorithms for hard problems like integer factoring, graph isomorphism, and the traveling salesman problem. This would cement our understanding of computational complexity classes and the inherent difficulty of broad categories of problems.

Practical applications impacted include cryptography, machine learning, economics, quantum physics simulation, and more. Nearly all current cryptography relies on the difficulty of problems believed to be in NP but not in P, so a proof of P ≠ NP could validate most encryption schemes. But it would also spur research into alternative cryptography based on different hard problems not ruled out by the solution.

Approaches to Proving P ≠ NP

Many approaches to resolving the P vs. NP problems have been considered without success so far. These include diagonalization, proving super-polynomial lower bounds on specific problems, introducing complexity measures beyond time, reducibility and oracles, algebraic and geometric techniques, and logic-based proofs.

Attributes of possible proof approaches include technical tools like diagonalization and reducibility, complexity measuring techniques, the relating of complexity classes via oracles, exploiting special structures like algebraic geometry, and even indirect logical contradiction and paradoxes. But all attempts so far have fallen short, and each failure provides more data points to guide and motivate future work.

Candidate Hard Problems for NP

Good candidate problems to shed light on proving P ≠ NP require a careful balance of apparent intrinsic hardness, wide connections to other problems, and some special structure that could enable a breakthrough proof. The most studied NP problems include the Boolean satisfiability problem, integer factoring, graph isomorphism, traveling salesman problems, and graph coloring.

Attributes prioritized include difficulty of exhaustive search, parity between best known algorithms and brute force, problem simplicity helping isolate hardness, versatile reduction targets enabling indirect attacks, and mathematical qualities like symmetry or topology harboring leverage for proofs.

Exploring Alternatives to Provable Resolution

Given the sustained elusiveness of a definitive answer, some researchers advocate focusing more effort on developing unconditional hardness assumptions that bypass resolving P vs. NP either way. This would involve identifying “natural problems” that are assumed practically intractable regardless, and from these assumptions deriving security for cryptographic protocols and rigorous approximations for optimization problems.

Key qualities of such alternative natural problems are practical use cases like cryptography, empirical hardness against all known algorithms, connections to fundamental mathematics giving confidence in longevity, and versatility enabling multiple applications.

The Future of Research on the P vs. NP Problem

Despite over 60 years of intense study, the P vs. NP problem remains widely open with relatively little consensus among experts whether P = NP or P ≠ NP is more likely. Continued research is motivated by the problem’s simplicity, approachability from many angles, and the paradigm shift its solution could spur across computer science and mathematics.

Ongoing work focuses on complexity measure variants tailored for possible proofs, restrictions of problems to special structures with additional leverage, empirical investigations of concrete hard problems, and surprising connections between complexity, logic, philosophy, and physics hinting at potential breakthrough reasoning.