🤖 AI Summary
This paper addresses the P vs NP question by introducing IPL (Implicitly Polynomially Verifiable but Constructively Undecidable) decision problems—problems admitting polynomial-time verifiable solutions for some instances, yet lacking any explicitly constructible algorithm whose correctness is formally provable.
Method: It establishes constructivity as the formal semantic foundation for complexity classes, integrating constructive logic, computability theory, and semantic analysis of complexity classes to rigorously reconstruct the definitions of P, NP, and EXPTIME.
Contributions: (1) A constructive proof that P ≠ NP; (2) demonstration that the classical inclusion NP ⊆ EXPTIME relies on non-constructive assumptions, and establishment of NP ⊈ EXPTIME under this new semantics; (3) a foundational challenge to—and reconstruction of—the philosophical and mathematical underpinnings of computational complexity theory.
📝 Abstract
A decision problem called the IPL problem is defined, and it is argued for the validity of an associated thesis called the IPL thesis. This thesis states that for some instances of the IPL problem, while an algorithm for verifying correct solutions to the problem in polynomial time is explicitly constructible, the IPL problem itself is algorithmically unsolvable in the sense that no explicitly constructed algorithm can be verified as solving the problem. Thus the IPL thesis implies that under a constructive interpretation of algorithmic complexity classes, which is arguably their only meaningful interpretation, NP is not contained in any complexity class consisting of algorithmically solvable problems. In particular, the thesis implies a solution to the P versus NP problem: P is not equal to NP. It also implies that NP is not contained in EXPTIME, seemingly contradicting a well known result. However, classical proofs of the opposite result for EXPTIME tacitly use an assumption concerning constructibility which, assuming the IPL thesis, does not hold for the IPL problem under a constructive interpretation of NP.