Psi-Turing Machines: Bounded Introspection for Complexity Barriers and Oracle Separations
For complexity theorists, this offers a new model to study relativization and barriers, but the results are theoretical and incremental relative to existing oracle separations.
The paper introduces Psi-Turing Machines, a model with bounded introspection, and proves an oracle separation P^Ψ ≠ NP^Ψ along with a strict depth hierarchy, providing a framework for complexity barriers.
We introduce Psi-Turing Machines (Psi-TM): classical Turing machines equipped with a constant-depth introspection interface $ ι$ and an explicit per-step information budget $ B(d,n)=c\,d\log_2 n $. With the interface frozen, we develop an information-theoretic lower-bound toolkit: Budget counting, $ Ψ$-Fooling, and $ Ψ$-Fano, with worked examples $ L_k $ and $ L_k^{\mathrm{phase}} $. We prove an oracle-relative separation $ P^Ψ \neq NP^Ψ $ and a strict depth hierarchy, reinforced by an Anti-Simulation Hook that rules out polynomial emulation of $ ι_k $ using many calls to $ ι_{k-1} $ under the budget regime. We also present two independent platforms (Psi-decision trees and interface-constrained circuits IC-AC$^{0}$/IC-NC$^{1}$) and bridges that transfer bounds among machine, tree, and circuit with explicit poly/log losses. The model preserves classical computational power outside $ ι$ yet enables precise oracle-aware statements about barriers (relativization; partial/conditional progress on natural proofs and proof complexity). The aim is a standardized minimal introspection interface with clearly accounted information budgets.