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

Abstract: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 $\Sigma_k^{\mathrm{P}}\subseteq\Sigma_k^{\mathrm{RV}}\subseteq\Sigma_{k+1}^{\mathrm{P}}$; 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 from core-versus-support labels that can change under refactoring. For recursive self-improvement, uniformly bounded self-modification under a common sound interpreter and fixed verification protocol remains within the same verification class. A separate conditional-error budget controls false selection across adaptively chosen candidates. A quota-enforced XOR-synthesis family separates unbounded ratios of search success from changes in the accepted languages; exact and probabilistic audits check the resulting evidence requirements. The framework ties self-improvement claims to obligations on correctness, admissible evidence, verification resources, and selection error.

26 pages, 5 figures. Includes proofs and reproducibility artifacts

Artificial Intelligence (cs.AI); Computational Complexity (cs.CC)

arXiv:2610.10611 [cs.AI]

(or arXiv:2610.10611v1 [cs.AI] for this version)

https://doi.org/10.48550/arXiv.2610.10611 Focus to learn more arXiv-issued DOI via DataCite

code/exact_audit/audit.json

code/exact_audit/fused_qbf_audit.py

code/exact_audit/patch_agent_audit.py

code/exact_audit/patch_agent_declaration.json

code/exact_audit/test_fused_qbf_audit.py

code/exact_audit/test_patch_agent_audit.py

formalization/StagedAscentFused.lean

integrated/audit.py

integrated/audit_results.json

integrated/figures/amplification.dat

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integrated/figures/false_acceptance.dat

integrated/figures/generate_data.py

integrated/figures/xor_success.dat

integrated/test_xor_envelope.py

integrated/xor_envelope.py

integrated/xor_envelope_results.json

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