🤖 AI Summary
This work demonstrates a fundamental limitation of unified formal verification methods within the standard Turing model when applied to nontrivial semantic invariants. By formalizing “acceptable verification schemes” as generator–verifier pairs through a model-theoretic lens, the study integrates Rice’s theorem with formal verification frameworks to prove that such schemes implicitly induce undecidable decision procedures. Crucially, this impossibility stems from the computational behavior inherent to the verification mechanism itself, rather than from unprovable complexity-theoretic assumptions. Leveraging computability theory, model theory, and Coq-based formalization, the authors construct an extended structural model capturing semantic–syntactic interactions and rigorously establish that properties related to P vs NP and cryptographic assumptions such as one-way functions cannot be certified by any such unified method. A complete Coq implementation accompanies the theoretical results.
📝 Abstract
This paper does not address the mathematical truth of P versus NP. Instead, it identifies a structural limitation of uniform proof-generation methods in the standard Turing model. The observation is model-theoretic: it concerns the interaction between semantic invariants and syntactic verification, not the provability of complexity statements. We formalise an admissible method as a generator-verifier pair that produces, for each program, a finite certificate establishing a semantic property. Admissibility forces the generator-verifier composition to behave uniformly with respect to the invariant being certified. In the standard model, such uniform semantic certification implicitly induces a decision procedure for the property. Rice's theorem shows that this implicit behaviour cannot be realised for non-trivial semantic invariants, revealing a structural constraint on formal certification. Understanding this requires a meta-computational perspective: the obstruction arises from the computational behaviour induced by certification, not from the complexity-theoretic status of the property. We apply this framework to two semantic invariants naturally associated with formal certification of P vs. NP and with cryptographic hardness assumptions (in particular, one-way functions). Both fall under the same limitation: no uniform admissible method can certify them in the standard model. A complete Coq formalisation is provided, capturing the extensional structure of admissible methods and the semantic-syntactic interaction underlying the result.