Adaptive Formal Verification Strategies

• Adaptive formal verification is a feedback-driven process that updates verification artifacts and proof strategies based on real-time outcomes.
• It integrates methods like counterexample-guided invariant generation and LLM-assisted repair to enhance verification completeness and efficiency.
• The approach applies to hardware, software, and circuit synthesis, adapting continuously to code changes, specifications, and resource constraints.

Searching arXiv for papers related to adaptive formal verification and AI-assisted formal methods. Adaptive formal verification denotes a class of verification workflows in which the verification process is modified in response to information obtained during verification itself, rather than remaining fixed after the initial problem encoding. Across contemporary usage, the term covers several distinct but related patterns: changing the proof environment after inductive-step failures by adding helper assertions (Kumar et al., 2024); reformulating natural-language requirements into executable verification artifacts in a way that adapts to differing requirement styles (Wang et al., 7 Jul 2025); reusing partial proof progress to recursively refine unsuccessful theorem-proving attempts (Kasibatla et al., 2024); dynamically selecting proof-refinement strategies based on the current proof state (Lu et al., 29 Oct 2025); adapting verification scope to code changes and regression artifacts in continuous integration workflows (Monteiro et al., 2019); and using formal-verification feedback as a control signal inside search, synthesis, or self-improvement loops (Ceska et al., 2020, Wilf et al., 20 Dec 2025). The unifying idea is not a single calculus, but a family of feedback-driven verification architectures in which proof obligations, specifications, invariants, context, search effort, or repair actions are updated as verification proceeds.

1. Conceptual scope and defining characteristics

The broadest common feature of adaptive formal verification is that verification outcomes are treated as structured signals for subsequent verification steps. In some settings, the adaptation occurs at the proof-object level: a failed induction step yields a counterexample, which is then used to generate new helper assertions or inductive invariants for the next run (Kumar et al., 2024). In others, the adaptation occurs at the requirements interface: natural-language requirements are formalized into verification assertions in a way that can absorb variation in wording, auxiliary variables, and non-standardized timing conditional expressions (Wang et al., 7 Jul 2025). Interactive theorem proving provides a third pattern, where failed proof attempts are not discarded but analyzed for reusable fragments, failing subgoals, or strategy-specific repair opportunities (Kasibatla et al., 2024, Lu et al., 29 Oct 2025).

This suggests a useful conceptual division between several adaptive loci. One locus is the specification layer, where the verifier adapts to heterogeneous or incomplete requirement descriptions by reformulating them into machine-checkable properties (Wang et al., 7 Jul 2025). A second is the proof-strengthening layer, where the system changes the current invariant, lemma set, or assumptions in response to failed proof obligations (Kumar et al., 2024). A third is the proof-search layer, where subsequent tactics, lemma discovery, or retrieval actions depend on the current proof state and error messages rather than a fixed heuristic schedule (Kasibatla et al., 2024, Lu et al., 29 Oct 2025). A fourth is the workflow layer, where verification effort is scoped, scheduled, or redirected according to code changes, regression tests, or prior partial results (Monteiro et al., 2019, Castaño et al., 2017). A fifth is the synthesis-and-learning layer, where formal verifiability or verification success becomes a feedback signal for optimization or curriculum design (Ceska et al., 2020, Wilf et al., 20 Dec 2025).

The literature represented here also distinguishes adaptive formal verification from fully autonomous verification. Several papers are explicit that their systems are heuristic and human-guided rather than end-to-end self-correcting. The induction-based RTL workflow in "Generative AI Augmented Induction-based Formal Verification" is described as “AI-augmented formal verification rather than a fundamentally new verification algorithm,” and its adaptation is “iterative and feedback-driven, but heuristic and largely human-guided rather than fully autonomous” (Kumar et al., 2024). "Supporting Software Formal Verification with LLMs: An Experimental Study" similarly emphasizes “human-machine collaboration” and notes that high-quality requirements documentation and human monitoring remain critical (Wang et al., 7 Jul 2025). By contrast, the Coq framework "Adaptive Proof Refinement with LLM-Guided Strategy Selection" formalizes refinement as a state-dependent control loop, though it still relies on prompted LLM reasoning rather than learned control policies (Lu et al., 29 Oct 2025).

2. Feedback-driven proof strengthening and counterexample-guided refinement

One of the clearest adaptive patterns appears in induction-based hardware verification. The problem setting is standard safety proof for RTL designs using $k$-induction. The practical obstacle is that the inductive step may fail even when a property is true, because the step reasons over arbitrary states, including unreachable ones, whereas the base case uses initial-state constraints (Kumar et al., 2024). The resulting counterexamples are spurious with respect to reachable behavior. Traditional industrial practice addresses this by manually crafting strengthening lemmas or helper assertions that rule out the unreachable-state behavior. The adaptive contribution in (Kumar et al., 2024) is to place an LLM in that loop.

Two flows are described. In the proactive flow, the LLM receives a specification document and RTL, and generates candidate helper assertions before proof begins. In the reactive flow, formal verification is run first; if the inductive step fails, the counterexample or waveform plus RTL is provided to the LLM, which proposes helper assertions intended to prevent the same failure in the next iteration (Kumar et al., 2024). The generated helper assertions are then proven and, once proven, may be reused “as assumptions” to help prove more complex assertions. This is a layered proof methodology in which adaptation changes the hypothesis set and proof environment in response to prior failure artifacts.

The paper’s synchronized-counter example illustrates the mechanism. An assertion that “should obviously hold true” still fails under induction because of an unreachable-state trace described as “31st bit of count2 is not logic 1.” After giving the RTL and counterexample to the LLM, the system generated a helper assertion that then proved the original assertion faster (Kumar et al., 2024). Although the code and exact formulas are not reproduced, the narrative function is precise: the adaptive element is counterexample-guided invariant generation.

A closely related, but structurally richer, notion of adaptivity appears in knowledge-graph-centered SVA generation for hardware verification. "Knowledge Graphs, the Missing Link in Agentic AI-based Formal Verification" organizes the workflow around typed JSON IRs and a runtime knowledge graph that links SpecChunk, Requirement, Property, FormalResult, and CoverageMetrics, along with design hierarchy, signals, assumptions, syntax diagnostics, counterexamples, and coverage information (Viswambharan et al., 7 May 2026). The key adaptation loops are threefold: syntax repair guided by tool diagnostics, counterexample-guided correction using trace links, and coverage-directed property augmentation. Syntax failures trigger localized repair attempts; failing properties trigger VCD-guided analysis classifying the cause as RTL bug, over-specification, missing assumptions, or under-specification; coverage gaps trigger generation of new cover directives or assertions (Viswambharan et al., 7 May 2026). This yields a persistent, traceable refinement process that updates only property-granular downstream evidence rather than restarting from scratch.

The experimental evidence in (Viswambharan et al., 7 May 2026) is design-dependent but supports the basic adaptive mechanism. Syntax-repair overhead is low across seven benchmarks, with 0–3 fix attempts per design. CEX correction is highly effective on FIFO and less so on more invariant-heavy designs such as BST or TTC Counter. Coverage-directed augmentation substantially improves CSR, raising reachable RTL coverage from 33.8% to 84.6%, and final end-to-end formal coverage ranges in the visible table from 82.98% to 99.41% (Viswambharan et al., 7 May 2026). This suggests that adaptive closure mechanisms are strongest when failures are local and traceable, and weaker when correctness depends on deep temporal, arithmetic, or inductive reasoning.

Specification and testbench synthesis can also be adaptive without dynamically changing proof search. AutoSVA is a framework that generates formal testbenches for RTL module interactions from annotated interface structure, producing liveness and safety properties, auxiliary tracking logic, bindings, and tool scripts (Orenes-Vera et al., 2021). Its adaptation is structural rather than iterative: assertion polarity changes depending on whether a transaction is incoming or outgoing; richer attributes such as transid, data, ack, stable, and active trigger richer generated properties; and hierarchy-aware reuse turns assumptions into assertions in parent modules (Orenes-Vera et al., 2021). This supports a broader interpretation of adaptive formal verification in which the verification problem itself is synthesized and specialized to the interface contract rather than handwritten once and for all.

3. Adaptive proof search and refinement in interactive theorem proving

Interactive theorem proving supplies a more explicit formulation of adaptation as proof-state-driven control. Cobblestone, developed for Coq, changes what counts as useful progress during proof search. Instead of treating an unsuccessful proof attempt as a total failure, it decomposes the generated script into a recursive result structure with a prefix, a Boolean prefix_succeeds?, and nested subproof_results (Kasibatla et al., 2024). Fail-safe execution uses admit tactically to preserve proof structure and localize exactly which proof fragments succeed and which fail. Partial successes then become recursive subproblems, possibly patched by CoqHammer or solved by recursive calls to Cobblestone (Kasibatla et al., 2024).

This is adaptive in three senses identified directly in the paper: it adapts to internal proof feedback, adapts proof-search granularity from theorem-level to subgoal-level and deeper, and adapts to external assistance such as relevant lemmas or decompositions supplied by oracles (Kasibatla et al., 2024). The quantitative effect is large. On CoqGym100, Cobblestone proves 48% of theorems, compared with 30% for CoqHammer, 22% for ChainOfThought, and 17% for Proverbot9001; on Wigderson100, it proves 38%, compared with 27%, 17%, and 10%, respectively (Kasibatla et al., 2024). Relative to all prior tools combined, Cobblestone adds 34.2% on CoqGym100 and 21.9% on Wigderson100. Oracle-assisted variants reach up to 58% on CoqGym100 and 55% when all Cobblestone variants are unioned, supporting the claim that partial structure from humans or other tools can be amplified by adaptive proof recombination (Kasibatla et al., 2024).

Adapt generalizes this idea from fragment reuse to explicit strategy selection. It models proof refinement in Coq as an iterative loop over the theorem statement $t$, global context $g$, a decision maker $\mathit{DM}$, the current proof state, and a working context (Lu et al., 29 Oct 2025). The available actions are Lemma Discovery, Context Enrichment, and Regeneration. After each failed proof attempt, the LLM-based decision maker receives the theorem statement, definitions, stuck proof state, erroneous tactic, error message, partial proof, current working context, and a similar proof. It then outputs one of three strategy choices:

• Lemma Discovery; [name_1, name_2 ...]
• Context Enrichment; [keyword_1, keyword_2 ...]
• Regeneration (Lu et al., 29 Oct 2025)

The core significance is that adaptation is state-dependent. Different failure modes require different responses: missing intermediate lemmas, insufficient context, or a simply poor proof direction. Experimentally, Adapt outperforms four baselines on both CoqDev and CoqStoq, reaching 651 solved theorems (37.85%) on CoqDev and 484 (41.33%) on CoqStoq, above PALM, Rango, Self-Refine, and Self-Refine+RAG (Lu et al., 29 Oct 2025). Ablations show that removing lemma discovery or context enrichment degrades performance, with lemma discovery “especially critical.” The LLM-based decision maker also outperforms random, rule-based, and feature-based DNN selection in overall solves (Lu et al., 29 Oct 2025). This is one of the most explicit instances of adaptive formal verification in the strong algorithmic sense: proof automation is treated as dynamic control over proof states rather than repeated one-shot generation.

4. Requirement adaptation, natural-language interfaces, and socio-technical feedback

A different branch of adaptive formal verification focuses not on proof search but on the bottleneck of requirement formalization. In software verification, tools such as CBMC and ESBMC can already reason directly over code, but still require users to provide suitable formal properties. "Supporting Software Formal Verification with LLMs: An Experimental Study" addresses this gap with SpecVerify, a framework that takes natural-language requirements and C code, asks an LLM to produce a semi-formal requirement specification, then generates C assertions for ESBMC (Wang et al., 7 Jul 2025).

The adaptive contribution is concentrated at the front end. The framework adapts to different requirement styles because the LLM accepts unconstrained natural language rather than demanding FRETish or fixed temporal templates. It can encode historical dependencies, interactions between multiple variables, non-standardized timing conditional expressions, and non-functional requirements by generating auxiliary variables such as previous-state or event-counter storage (Wang et al., 7 Jul 2025). It also supports iterative discrepancy analysis: when verification results differ from CoCoSim or SLDV, the authors replay counterexamples as test cases, inspect runtime behavior, and determine whether the disagreement came from the assertion, the verifier, the model, or the benchmark (Wang et al., 7 Jul 2025). Humans can review the intermediate formalized specification, which the paper treats as an intended adaptation point for human-machine collaboration.

Empirically, SpecVerify is evaluated on 58 requirements from nine Lockheed Martin cyber-physical systems. Claude 3.5 Sonnet plus ESBMC forms 58/58 properties and verifies 27/58, for 46.5% verification accuracy, matching CoCoSim’s 46.5% but with 0 false positives and 2 false negatives, compared with CoCoSim’s 2 false positives and 6 false negatives (Wang et al., 7 Jul 2025). ChatGPT+ESBMC is much weaker, with 35/58 formed, 15/58 verified, 8 false positives, and 23 assertion errors. Logic-equivalence analysis over 58 cases reports 46 logic-equivalent properties (79.31%), with smaller fractions due to LLM misunderstanding, missing assumptions, sequence reversal, or over-verification (Wang et al., 7 Jul 2025). The strongest case study is a floating-point median-selection bug that CoCoSim missed because it reasoned over a rationalized abstraction, while ESBMC, operating on code-level floating-point semantics, found the concrete falsifiable behavior (Wang et al., 7 Jul 2025).

This suggests that adaptive verification at the requirements interface is not merely about convenience. It changes what properties become expressible and how reliably they can be checked against implementation semantics. At the same time, the paper is clear that adaptation is not yet requirement repair in the strong sense: when documentation is wrong or underspecified, the LLM may preserve that defect rather than repair it (Wang et al., 7 Jul 2025).

A parallel hardware-oriented line appears in "Using LLMs to Facilitate Formal Verification of RTL." There, GPT4 is iteratively taught to generate SystemVerilog Assertions from RTL using a growing rule set that captures SVA syntax and temporal semantics, including the distinction between $|->$ and $|=>$, correct use of \$past, internal signal binding, and reduction-operator meaning (Orenes-Vera et al., 2023). The feedback loop is concrete: generate assertions, run FPV, inspect syntax failures, wrong timing, wrong semantics, and internal-signal binding issues, then refine prompt rules and retry. On a FIFO benchmark, this required 23 iterations and about 8 hours of engineering to reach correct syntax, full proof, and full coverage (Orenes-Vera et al., 2023). On larger CVA6 modules, multiple GPT4 assertion batches improved statement and toggle coverage substantially over AutoSVA liveness properties alone, and one generated assertion exposed a real PTW bug that had eluded prior evaluation (Orenes-Vera et al., 2023). Here, the adaptive element lies in prompt refinement driven by formal feedback, rather than in adapting the proof engine itself.

5. Continuous, resource-aware, and verifiability-driven adaptation

Not all adaptive formal verification is LLM-mediated. Some work adapts verification scope or effort directly in response to change history or search dynamics.

Continuous Formal Verification (CFV) integrates software model checking into CI and DevOps by using code modifications and regression tests to decide what to reverify (Monteiro et al., 2019). Its core workflow is change-aware and resource-aware. It identifies modified functions from SCM history, tries a fast AST equivalence check and then a slower formal equivalence check to determine whether the old and new versions are semantically the same, and only if equivalence fails or times out does it generate generalized test harnesses for regression-test-guided bounded model checking (Monteiro et al., 2019). Rather than verifying the whole system after every commit, it adapts verification scope to code changes and to developer artifacts already present in CI. The paper explicitly states that this verification “should run fast (e.g., in less than 5 minutes)” (Monteiro et al., 2019). This is adaptive formal verification at the workflow-orchestration level: the verification problem is narrowed and reformulated according to diff-time relevance.

Verification Coverage addresses a related issue from another angle: incomplete verification runs often return only unknown, with no measure of progress (Castaño et al., 2017). The paper defines verification coverage for ART-based model checking using feasible terminating executions and an Assumption Automaton summarizing explored behavior. A statement is covered if it appears on some feasible terminating execution whose prefix remains within the explored region, formally through the existence of $t \in \mathcal{T}$ and a prefix $\pi$ satisfying $isPrefix(\pi,t)$, $s \in \pi$, and $t$0 (Castaño et al., 2017). The resulting statement-coverage ratio

$t$1

acts as an intermediate progress signal for incomplete verification (Castaño et al., 2017). The paper also proposes a heuristic based on the least fixed point

$t$2

to guide subsequent exploration toward assumption-automaton states that can potentially cover more CFA locations (Castaño et al., 2017). This is adaptive in the sense that information from an interrupted run is used to steer the next run.

A still more direct resource adaptation appears in approximate arithmetic circuit synthesis. "Adaptive Verifiability-Driven Strategy for Evolutionary Approximation of Arithmetic Circuits" puts formal verification inside an evolutionary search loop and treats verifiability itself as a control signal (Ceska et al., 2020). The target problem is to minimize circuit size subject to a normalized worst-case absolute error bound: $t$3 with goal

$t$4

A candidate circuit is considered infeasible not only when it violates the error bound, but also when the SAT-based proof of compliance cannot be completed within the current resource limit $t$5, in which case $t$6 (Ceska et al., 2020). The adaptive controller changes $t$7 online according to recent improvement counts. If improvements are frequent, the limit is decreased to increase throughput; if improvements dry up, the limit is increased to allow harder proofs. The update logic includes

$t$8

or

$t$9

depending on thresholded improvement counts, clipped between minLimit and maxLimit (Ceska et al., 2020). The best adaptive strategy, ada4, achieves an average versatility score of 106.3 versus 114.1 for the best fixed-limit strategy, with Friedman test $g$0 and $g$1 (Ceska et al., 2020). This is a strong example of adaptive formal verification in which proof cost is treated as an online optimization signal.

6. Closed-loop verification in learning, generation, and broader systems

Adaptive formal verification also appears in systems where the verifier is not just validating a fixed artifact, but steering learning or synthesis.

"Propose, Solve, Verify: Self-Play Through Formal Verification" places formal verification inside a self-play loop for verified code generation in Verus (Wilf et al., 20 Dec 2025). At iteration $g$2, a solver $g$3 attempts multiple solutions per specification, the verifier checks each one, and only verified solutions are retained in the training set: $g$4 The solver is then updated by minimizing

$g$5

The proposer uses empirical verifier pass rates

$g$6

to classify problems as Easy, Medium, [Hard](https://www.emergentmind.com/topics/livecodebench-hard), or Impossible according to thresholds $g$7 and $g$8, and then generate new problems at target difficulties (Wilf et al., 20 Dec 2025). Thus verification results change both what the system learns from and what tasks it generates next. On MBPP in test-time training, PSV-Verus reaches 36.78% pass@1 versus 3.83% for iterative RFT and 6.48% for AlphaVerus; on Dafny2Verus it reaches 65.63% versus 34.46% and 24.06%; and ablations show that removing formal verification or difficulty awareness sharply reduces performance (Wilf et al., 20 Dec 2025). The verifier here acts as a sound curriculum signal inside a closed self-improvement loop.

ProofWright applies a related pattern to CUDA kernels. It combines an LLM-based VerCors agent for memory and thread safety with a Rocq-based semantic equivalence framework for selected element-wise kernels (Chatterjee et al., 15 Nov 2025). The adaptive aspect comes from annotation synthesis, proof repair, and a dynamic Annotation Guide distilled from prior examples and verification episodes. The agent classifies kernel families, introduces helper indexing functions, selects permission strategies, and iteratively revises annotations based on VerCors error messages (Chatterjee et al., 15 Nov 2025). On KernelBench L1, ProofWright verifies safety properties for 74% of generated kernels, with 14% full semantic equivalence and about 3 minutes per kernel (Chatterjee et al., 15 Nov 2025). Ablations show that without the evolved annotation guide, only 13 kernels verify; without knowledge base and guide, zero kernels verify (Chatterjee et al., 15 Nov 2025). This indicates that adaptive proof construction, not just deductive verification, is the scalability enabler.

Veri-Sure shows a similar feedback architecture for RTL code generation. It builds a contract IR from natural-language specifications, lint-checks it, derives sequential and combinational obligations

$g$9

then uses trace-driven temporal analysis, assertion-based checking, Boolean equivalence, and static dependency slicing to guide localized patching (Liu et al., 27 Jan 2026). Earliest divergence is defined as

$\mathit{DM}$0

and combinational equivalence uses a miter

$\mathit{DM}$1

with goal $\mathit{DM}$2 (Liu et al., 27 Jan 2026). Static slicing restricts edits to a bounded suspect-block set derived from read/write dependencies, and patches are accepted only if they improve the pair $\mathit{DM}$3 by delaying first failure or reducing mismatches at the same $\mathit{DM}$4 (Liu et al., 27 Jan 2026). On VerilogEval-v2-EXT, Veri-Sure with GPT-5.2 reaches 93.30% overall functional Pass@1 and 85.07% on Hard tasks, far above standalone GPT-5.2 and above ablations without formal verification or without tracing, slicing, and patching (Liu et al., 27 Jan 2026). This is adaptive formal verification used as a repair controller.

7. Boundaries, misconceptions, and unresolved questions

A recurring misconception is that adaptive formal verification always means a new proof calculus or adaptive solver. Much of the recent literature shows the opposite. Several influential systems leave the underlying formal engine untouched and instead adapt the artifacts supplied to it: helper invariants, assertions, requirement formalizations, proof contexts, or scoping decisions (Kumar et al., 2024, Wang et al., 7 Jul 2025, Monteiro et al., 2019). The contribution is architectural and workflow-level rather than foundational.

Another misconception is that LLM integration removes the need for human oversight. The literature consistently rejects that interpretation. The RTL induction work explicitly warns about hallucinations and recommends “human-in-the-loop AI” (Kumar et al., 2024). SpecVerify states that high-quality documentation and human monitoring remain critical because models misinterpret specifications (Wang et al., 7 Jul 2025). The SVA-from-RTL work finds that full coverage does not imply correct assertions, and that proved assertions may still encode the wrong intent if not audited (Orenes-Vera et al., 2023). Even in more agentic systems such as ProofWright and Veri-Sure, adaptation is constrained by hand-designed contract schemas, annotation guides, bounded repair scopes, or manual validation of lowering steps (Chatterjee et al., 15 Nov 2025, Liu et al., 27 Jan 2026).

A further misconception is that adaptivity automatically yields stronger soundness guarantees. In fact, adaptivity often improves completeness, throughput, or applicability more than soundness. Soundness generally remains anchored in the underlying checker: Coq’s kernel in Cobblestone (Kasibatla et al., 2024), Coq in Adapt (Lu et al., 29 Oct 2025), ESBMC in SpecVerify (Wang et al., 7 Jul 2025), Verus in PSV (Wilf et al., 20 Dec 2025), VerCors and Rocq in ProofWright (Chatterjee et al., 15 Nov 2025), or SAT/SMT proofs in circuit synthesis and Boolean equivalence (Ceska et al., 2020, Liu et al., 27 Jan 2026). When adaptive layers introduce approximations, such as sigmoid-based binarization for object-detection matching, the paper is explicit that properties relying on the approximation become incomplete (Elboher et al., 2024).

Several unresolved questions emerge repeatedly. One is automatic validation and minimization of generated artifacts: how should helper assertions, retrieved lemmas, or generated properties be ranked, minimized, and integrated soundly (Kumar et al., 2024, Lu et al., 29 Oct 2025)? Another is benchmarking and metrics: what measures best capture throughput gains, engineer-effort reduction, proof-success improvement, or partial progress (Kumar et al., 2024, Castaño et al., 2017)? A third is broader proof-state semantics: Cobblestone notes that its internal-progress model works best when proofs decompose into reusable subgoals, and struggles with long linear proofs (Kasibatla et al., 2024). A fourth is specification reliability: context-aware or natural-language front ends inherit ambiguity, omissions, or wrong assumptions from source documents unless explicit repair mechanisms exist (Wang et al., 7 Jul 2025, Viswambharan et al., 7 May 2026). A fifth is scalability of exact formal components: BMC remains limited by bounds and solver timeouts (Wang et al., 7 Jul 2025), and object-detection verification remains bottlenecked by IoU encoding, matching, and postprocessing semantics (Elboher et al., 2024).

Adaptive formal verification therefore remains a heterogeneous research area rather than a closed theory. Its most stable core is the shift from static, one-shot verification to verification as an iterative, feedback-driven process in which formal results actively reshape subsequent verification effort. The literature surveyed here shows that this shift can occur at many levels—requirements, assertions, invariants, proof strategies, workflows, synthesis loops, or resource budgets—and that its success depends on preserving the soundness of the formal backend while making the front-end and orchestration layers responsive to evidence generated during verification itself (Kumar et al., 2024, Wang et al., 7 Jul 2025, Kasibatla et al., 2024, Lu et al., 29 Oct 2025, Monteiro et al., 2019, Ceska et al., 2020, Wilf et al., 20 Dec 2025).