Verification and Self-Improvement in Agentic AI: Foundations and Limits

Chien-Ping Lu

Agentic AI systems can improve by searching longer, receiving additional support, or modifying how they propose and verify outputs. A performance score does not distinguish these mechanisms. We compare these changes through bounded verification with hidden terminal randomness. A stage specifies admissible transcripts, polynomial bounds, an alternating verification protocol, and a terminal checker. Its native reach uses default support; its closure frontier permits all support already admitted by the interface. Under a uniform pointwise probability gap and task-relative soundness, these are well-defined languages. We prove that independent majority amplification preserves both languages, whereas existential acceptance over random tapes can admit incorrect outputs. Exact verification is the zero-randomness case, with placement and completeness results. The randomized-verifier classes satisfy $Σ_k^{\mathrm{P}}\subseteqΣ_k^{\mathrm{RV}}\subseteqΣ_{k+1}^{\mathrm{P}}$; strict enlargement and depth separation require explicit complexity assumptions, while $\mathrm{BPP}=\mathrm{P}$ yields exact companions with the same frontiers. Representation analysis separates invariant acceptance…