REVIEW 3 major objections 5 minor 14 references
Towards LLM-based Generation of Human-Readable Proofs in Polynomial Formal Verification
T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read The paper argues that large language models can generate human-readable induction proof sketches for BDD-based polynomial formal verification, but that a formal reasoning engine must validate and repair them.
desk verdict A short, honest vision paper on using LLMs to draft induction proofs for BDD-based verification, but the key 'easy link' to formal engines is asserted, not shown, and the abstract oversells what is demonstrated. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The argument turns on reduced ordered binary decision diagrams (BDDs) — the canonical graph representation of Boolean functions whose node count is the resource measure in PFV — and on the induction step over blocks of variables (pairs for $f_2$, triples for $f_3$, single levels for symmetric functions). The LLM's contribution is the narrative version of these size-bound arguments; the load-bearing mechanism is the proposed link to graph-based proof-pattern techniques, which would repair and formally check the draft. The author identifies that link explicitly but only asserts it.
What would settle it
Benchmark the proposed pipeline on a BDD family with a known linear size bound that is absent from LLM training data: have the LLM draft an induction sketch, then require the graph-based proof engine to turn that sketch into a formally checked proof without hand-written repair, and count the repair steps. A quicker arithmetic check also cuts: for $f_1 = x_1x_2$ with $x_1 < x_2$, the reduced BDD has 2 internal nodes, not the 3 quoted in the paper's transcript, so the draft's numerical details are already wrong where the author says the LLM 'works well'.
Extended reading notes
Core claim
On its own terms, the paper's discovery is that LLMs can already produce the induction skeletons that human-readable BDD proofs need, while remaining unreliable in their details. For $f_2 = x_1x_2 + x_3x_4 + \cdots + x_{2n-1}x_{2n}$, for its three-variable analogue $f_3$, and for totally symmetric functions, the LLM gives the correct shape of the argument — inductive hypothesis, adding one block of variables, and a linear or quadratic node count. When asked about multiplication, it correctly denies that BDD size is polynomially bounded, yet it states that this holds only under 'most' variable orderings, when the known result says all orderings, and it even suggests an ordering that it claims can make multiplication polynomial in practice. The author takes these examples to show feasibility of LLM-assisted proof generation, with the formal proof engine as the validator, not as evidence that the LLM itself proves anything.
Load-bearing premise
The working assumption is that an LLM's rough proof outline can be connected to the existing graph-based proof-generation tool 'easily' — the paper states this in Section 2 but does not demonstrate it on a completed example, and if the connection is not straightforward the proposal has no demonstrated core.
Editorial extensions
If this is right
- If the LLM-plus-engine pairing works, PFV results gain a human-readable proof front-end, so designers can see why an adder or multiplier is correct rather than trusting a black-box resource bound.
- The same induction principles could be applied to modern verification data structures beyond BDDs, since the paper explicitly extends its examples to that direction.
- Retrieval-augmented generation could feed previously proved proof fragments back into the LLM, letting it compose longer proofs while the formal engine tracks correctness.
- Chain-of-thought prompting and error feedback from the formal tool could improve the drafts, but the paper insists the final correctness check must remain with the reasoning engine.
- Every LLM-generated proof sketch, even one with the right conclusion, must pass an independent formal check; the multiplier example shows the supporting statements can be false.
Reading between the lines
- A stronger test of the paper's thesis would use BDD families that are not already well represented in LLM training data; the paper's examples are classic textbook material and may therefore flatter the tool.
- If the 'easy link' to graph-based proof generation turns out to need a hand-written adapter per proof pattern, the practical value shifts from proof discovery to proof explanation: the LLM translates machine-found proofs into readable prose rather than finding the proof steps itself.
- The false quantification in the multiplier answer ('most' instead of 'all') is a testable instance of a general pattern: LLM confidence and correctness can diverge, and follow-up prompting corrected it here — whether such correction is reliably repeatable is an open empirical question.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper is a short invited position paper that proposes using large language models (LLMs), specifically ChatGPT-4o, to generate human-readable proof sketches for polynomial formal verification (PFV), with the understanding that correctness must subsequently be checked by formal reasoning engines. The body reports four illustrative Q&A interactions: an induction-proof outline for binary decision diagrams, an induction proof for the function f2 = x1x2 + x3x4 + ... + x_{2n-1}x_{2n}, a variant f3 with three-variable monomials, and a discussion of BDD sizes for totally symmetric functions and for multipliers. The author acknowledges that LLM outputs can contain errors and argues that the generated sketches can be linked to the graph-based proof generation techniques of references [5,6]. The conclusion, however, states that linking formal tools with LLMs is a promising direction for future work, which is somewhat in tension with the abstract's phrasing that the examples show how LLMs can interact with proof engines.
Significance. If the proposed pipeline were realized, it could help automate the production of human-readable proofs in polynomial formal verification, thereby complementing existing resource-bounded verification methods. The paper's strengths are its transparency—raw LLM transcripts are included, and one substantial error in the multiplier example is explicitly acknowledged—and its clear separation of the LLM-generated sketch from the required formal validation. However, the demonstrated contribution is limited: no reasoning engine is actually invoked, none of the transcripts is validated, and the acknowledged false claim in the multiplier discussion is left uncorrected. The paper should be read as a proposal and a call for future work rather than as a demonstration of the claimed LLM-to-proof-engine interaction.
major comments (3)
- [Section 2] The central enabling assumption is asserted rather than demonstrated. The sentence 'But this can easily be linked to the graph-based techniques from [5,6]' appears after the examples, yet no such link is exhibited anywhere in the paper: the appendix contains only raw LLM answers, no proof-engine output, and no pattern-based proof generator is applied to any of the four examples. Moreover, Section 3 states that 'a linking of formal tools with the LLM is a promising direction for future work,' which directly contradicts the abstract's claim that the examples 'show how LLMs can interact with proof engines.' This contradiction is load-bearing because the paper's proposal rests on the feasibility of that link. The authors should either add a concrete demonstration for at least one example (e.g., taking the f2 sketch and turning it into a pattern-based proof in the style of [5,6]) or substantially temper the abstract and title claims.
- [Section 2] The multiplier example is not handled correctly even at the level of the sketch. The LLM answer claims that the variable ordering a0, a1, ..., a_{n-1}, b0, b1, ..., b_{n-1} can 'significantly reduce' BDD size and 'sometimes mak[e] it polynomial in practice,' which is false under the known result cited as [7] that multiplication has exponential BDD size for every variable ordering. The paper itself notes that the last sentence is not correct, but it leaves the erroneous transcript without a corrected formal statement or an illustration of how a proof engine would reject and repair it. This is precisely the hallucination scenario the paper says must be caught by formal tools, and the absence of any such correction or validation demonstrates that the proposed pipeline is not yet operational.
- [Appendix, Q2 and Q3] The node counts in the 'successful' induction examples are inaccurate. In Q2, the BDD for f1 = x1 AND x2 under the ordering x1 < x2 has two internal decision nodes, not three, and the claimed inductive invariant 'size ≤ 3k + 1' is inconsistent with a base of three nodes for k=1. In Q3, the statement that the base case for x1x2x3 has three internal nodes is correct, but the inductive step is said to add '4 nodes (including the OR node),' which is not justified for a reduced ordered BDD, where no separate OR gate appears as a node. Since these two examples are the paper's positive evidence that 'LLMs work well' for induction, the unchecked counts undermine the claimed reliability of the LLM-generated proof sketches.
minor comments (5)
- [Abstract] The phrase 'Examples are given that show how LLMs can interact with proof engines' overstates what is presented; the examples show LLM conversations only, and no interaction with a proof engine is demonstrated.
- [Section 1] The sentence 'so far, its lacks automation' contains a typo; it should read 'so far, it lacks automation.'
- [Section 2] The sentence 'Most variablesis not correct, since it can be proven that it holds for all' has a grammatical and clarity problem; it should read something like 'The LLM's phrase "most variable orderings" is not correct, since it holds for all variable orderings.'
- [Section 2] The statement that for small numbers of variables 'the program might terminate within the given resources' is vague; the authors should specify what resource bounds are intended and how this relates to the formal result on exponential blow-up.
- [Section 3] The future-work item on RAG is plausible but would benefit from a concrete indication of which PFV proof patterns could be stored and retrieved; as written, it is only a general suggestion.
Circularity Check
No circularity: the LLM proof-generation examples reproduce standard BDD induction arguments and no fitted parameter or self-referential derivation is present.
full rationale
The paper's examples are raw LLM transcripts reproducing textbook BDD facts (Bryant 1986), and the claims about induction and symmetric functions stand on standard external results rather than on the paper's own constructions. The frequently cited prior PFV work [5,6] is invoked to suggest that formal proof generation can be linked to graph-based techniques, but that linkage is asserted rather than demonstrated; the paper explicitly states that formal-tool linking is future work. This is a missing-support or feasibility concern, not a circularity. There is no fitted parameter renamed as a prediction, no definition of one quantity in terms of another, and no self-citation used as an unverified load-bearing theorem. The paper's acknowledged LLM errors are correctness risks, not circular steps. Thus the circularity score is 0.
Assumptions & free parameters
assumptions (3)
- standard math Standard mathematical induction is a valid proof technique.
- domain assumption Known BDD complexity results from Bryant's work (linear for f2, O(n^2) for symmetric functions, exponential for multipliers) are correct.
- ad hoc to paper LLM outputs can be mechanically linked to the graph-based proof generation techniques of refs [5,6] with manageable effort.
Cite this review
Pith. "Pith review of Towards LLM-based Generation of Human-Readable Proofs in Polynomial Formal Verification." pith.science (2026). https://pith.science/paper/TFPA5RCU
@misc{pith2026250523311,
author = {Pith},
title = {Pith review of: Towards LLM-based Generation of Human-Readable Proofs in Polynomial Formal Verification},
year = {2026},
howpublished = {\url{https://pith.science/paper/TFPA5RCU}},
note = {Machine review of arXiv:2505.23311}
}
read the original abstract
Verification is one of the central tasks in circuit and system design. While simulation and emulation are widely used, complete correctness can only be ensured based on formal proof techniques. But these approaches often have very high run time and memory requirements. Recently, Polynomial Formal Verification (PFV) has been introduced showing that for many instances of practical relevance upper bounds on needed resources can be given. But proofs have to be provided that are human-readable. Here, we study how modern approaches from Artificial Intelligence (AI) based on Large Language Models (LLMs) can be used to generate proofs that later on can be validated based on reasoning engines. Examples are given that show how LLMs can interact with proof engines, and directions for future work are outlined.
Reference graph
Works this paper leans on
-
[7]
Bryant, R,: On the complexity of VLSI implementations and graph representations of Boolean functions with application to integer multiplication, IEEE Transactions on Computers, vol. 40, no. 2, pp. 205-213 (1991)
work page 1991
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[1]
Kluwer Academic Publishers, 2004
Drechsler, R.: Advanced Formal Verification. Kluwer Academic Publishers, 2004
work page 2004
-
[2]
Drechsler, R.: Formal System Verification: State-of the-Art and Future Trends. Springer, 2017
work page 2017
-
[3]
Drechsler, R.: PolyAdd: Polynomial Formal Verification of Adder Circuits. 24th IEEE International Symposium on Design and Diagnostics of Electronic Circuits and Systems (2021)
work page 2021
-
[4]
International Conference on Computer Aided Design (2022)
Drechsler, R., Mahzoon, A.: Polynomial Formal Verification: Ensuring Correctness under Resource Constraints. International Conference on Computer Aided Design (2022)
work page 2022
-
[5]
International Conference on Formal Methods and Models for System Design (2023)
Drechsler, R., Schnieber, M.: Next-Generation Automatic Human-Readable Proofs Enabling Polynomial Formal Verification. International Conference on Formal Methods and Models for System Design (2023)
work page 2023
-
[6]
Philosophical Transactions of the Royal Society A (2025)
Schnieber, M., Drechsler, R.: Automated polynomial formal verification using generalized binary decision diagram patterns. Philosophical Transactions of the Royal Society A (2025)
work page 2025
-
[8]
IEEE Asian Test Symposium (2024)
Qayyum, K., Ahmadi-Pour, S., Jha, C., Hassan, M., Drechsler, R.: LLMs for Hardware Verification: Frameworks, Techniques, and Future Directions. IEEE Asian Test Symposium (2024)
work page 2024
Show all 14 references
-
[9]
C-35, no
Bryant, R.: Graph-Based Algorithms for Boolean Function Manipulation, IEEE Transactions on Computers, vol. C-35, no. 8, pp. 677-691 (1986)
1986
-
[10]
36th International Conference on Neural Information Pro- cessing Systems (2022)
Wei, J., Wang, X., Schuurmans, D., Bosma, M., Ichter, B., Xia, F., Chi, E., Le, Q., Zhou, D.: Chain-of-thought prompting elicits reasoning in large language models. 36th International Conference on Neural Information Pro- cessing Systems (2022)
2022
-
[11]
IEEE 30th International Require- ments Engineering Conference Workshops (2022)
Fey, G., Fränzle, M., Drechsler, R.: Self-Explanation in Systems of Systems. IEEE 30th International Require- ments Engineering Conference Workshops (2022)
2022
-
[12]
37th International Conference on Neural Information Processing Systems (2023)
Yao, S., Yu, D., Zhao, J., Shafran, I., Griffiths, T., Cao, Y., Narasimhan, K.: Tree of thoughts: deliberate problem solving with large language models. 37th International Conference on Neural Information Processing Systems (2023)
2023
-
[13]
Conference of the Nations of the Americas Chapter of the Association for Computational Linguistics: Human Language Technologies (2025)
Liu, T., Xu, W., Huang, W., Zeng, Y., Wang, J., Wang, X., Yang, H., Li, J.: Logic-of-Thought: Injecting Logic into Contexts for Full Reasoning in Large Language Models. Conference of the Nations of the Americas Chapter of the Association for Computational Linguistics: Human La...
2025
-
[14]
Drechsler, R., Plump, C., Schnieber, M.: The Future is Hybrid: Next Generation Data Structures for Formal Verification, Asian Test Symposium (2024) 4 R. Drechsler 4 Appendix Q1: How can I perform an induction proof for binary decision diagrams? A: To perform an induction proof...
2024
Reviewed August 7, 2026 · model on record in the stance chip above.
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