Predictor-Impossibility Theorem and Applications

Abstract

We introduce a hierarchy consisting of stage machines, stage domains, and stage languages generated by semantic operators. The central result is a Predictor Impossibility Theorem (PITT), which shows that no effective predictor family can uniformly determine all stage languages of our hierarchy. The proof makes use of a pseudo-complement construction to obtain a language that yields a contradiction with every language in P. We then define an aggregate language MIS and establish a formal Slice Theorem connecting aggregate inputs to individual stage languages. This provides a rigorous Bridge Theorem from polynomial-time decidability of MIS to the existence of an effective predictor family. By utilizing succinct representations, the aggregate language is shown to be undecidable in deterministic polynomial time. Under the aggregate growth condition defining valid aggregate objects, MIS is shown to belong to NP. Combining these two results yields our main theorem: MIS in NP setminus P. The paper is organized so that PITT stands independently as a theoretic result, while the complexity-theoretic consequences are derived from the aggregate-language framework. The method does not relativize, algebrize, or naturalize.