Reliably knowing when a language model is correct is almost as important as being correct. We introduce prover-verifier deliberation (PVD), an inference-time protocol grounded in interactive proof theory, as a mechanism for selective prediction: the protocol produces both an answer and a structured confidence verdict, allowing a system to report high-confidence answers while abstaining on uncertain cases. In each dialogue, a prover defends a candidate answer through checkable sub-claims while a verifier issues targeted challenges and returns \textsc{Accept}, \textsc{Challenge}, or \textsc{Reject}. Because frozen language models are imperfect provers and verifiers operating over a noisy channel, formal soundness and completeness guarantees do not transfer; instead, we characterize the protocol empirically through its coverage-precision behavior. Our main experiment uses Claude Sonnet 4.6 as prover and Claude Haiku 4.5 as verifier on GPQA Diamond. Questions accepted with no answer revision, which we call Accept + No Change (ANC), are reported as the high-confidence subset; we evaluate this subset by its precision and coverage. ANC separates reliable from unreliable answers, yielding a ∼30pp HC-Prec gap over the non-ANC complement. Robustness experiments with GPT and Gemini pairings show that high HC-Prec can transfer across model families, while verifier strictness and domain competence largely determine the size of the selection gap. On Humanity's Last Exam, weaker prover-verifier pairings can collapse or invert the ANC signal, illustrating a practical failure mode when the verifier operates outside its effective region. Comparisons with self-consistency, universal self-consistency, multi-agent debate, and Reflexion suggest that prover-verifier deliberation supplies a distinct argument-defensibility signal for selective prediction.
When an LLM supplies an argument that a user could not readily construct, how can the user decide whether to accept its claim? Inspired by interactive proofs, we model human-LLM deliberation as an interaction between a prover with unrestricted internal search and a resource-bounded human verifier. The verifier requests and checks supporting details without access to the LLM's internal state. Passed checks accumulate evidence toward an acceptance threshold. We prove anytime-valid soundness against adaptive provers: the probability of ever accepting a false claim is at most a chosen error level, provided the task supplies bounds on false passes and human checking errors that remain valid after every relevant history. A finite-horizon completeness bound additionally requires bounds on the adequacy of honest responses and sufficient diagnostic progress. Further checks can strengthen the evidence for acceptance, but each requires another adequate response and reliable human effort. Whether this tradeoff permits certification depends on the verifier's effort budget, cognitive load, expertise, and fatigue. We identify conditions under which the supplied bounds certify a specified sequence of local checks but not a specified global check under the same resource budgets.
LLM-as-Judge systems are widely deployed for automated evaluation, yet practitioners lack reliable methods to know when a judge's verdict should be trusted. Token log-probabilities, the standard post-hoc confidence signal, are unavailable for many commercial LLMs and, even when accessible, saturate above 0.999 with structured JSON output. We introduce VERDI (VERification-Decomposed Inference), a method that extracts confidence from the reasoning trace a structured judge already produces, with no additional inference calls. VERDI decomposes each verification-style evaluation into sub-checks and derives three structural signals: Step-Verdict Alignment, Claim-Level Margin, and Evidence Grounding Score. We combine them with Platt-scaled logistic regression. On three public benchmarks, VERDI achieves AUROC 0.72-0.91 on GPT-4.1-mini and 0.66-0.80 on GPT-5.4-mini. On Qwen3.5-4B/9B/27B, where answer-token logprobs are anti-calibrated (higher confidence on errors, AUROC 0.32-0.49), VERDI achieves 0.56-0.70. We additionally validate on a production system with eight rubrics (AUROC 0.73-0.88 on factual rubrics), demonstrate cross-model transfer (AUROC 0.66-0.69), and show that a 33M-parameter NLI (Natural Language Inference) model provides a scalable alternative to regex extraction.
In a long conversation, an LLM can produce a plausible continuation that rests on premises the conversation has already abandoned. No runtime check ties its output to what the conversation has established, a gap that context-manipulation attacks on deployed agents exploit. We close this gap with a runtime verifier: an LLM Interpreter classifies each utterance into one of eight epistemic operations, and a symbolic engine applies them to a dependency map that records what every claim rests on and whether it still stands. Whether a continuation is grounded reduces to a walk over the map, linear in its size, with no LLM call. Retraction propagates through the same map with a conflict-free guarantee, flagging exactly the conclusions that lose support. On ReviseQA for belief revision and MemoryAgentBench's fact-consolidation split, two third-party benchmarks where earlier premises are superseded, the verifier leads a budget-matched retrieval baseline across five QA models and lifts MemoryAgentBench single-hop accuracy from 0.46--0.95 to 0.93--0.98. With the verifier, even the 7B model overtakes unaided GPT-4o. These runs feed the engine the benchmarks' own structured updates. When a GPT-4o Interpreter extracts every update from raw text instead, accuracy is statistically unchanged. Per-query cost is flat in conversation length, prompts staying near 0.8k tokens where full context reaches 114k and retraction queries under a microsecond at 2000 turns.