Title: Learning Simple Interpolants for Linear Integer Arithmetic
Venue: NeurIPS 2025
Accepted: yes
Keywords: Craig interpolation

Formal verification

Machine learning for logic

Linear integer arithmetic
Abstract: Craig interpolation plays a central role in formal verification tasks such as model checking, invariant generation, and abstraction refinement. In the domain of linear integer arithmetic (LIA), interpolants are crucial for deriving inductive invariants that characterize unreachable or safe program states, enabling scalable and precise reasoning about software and hardware correctness. Despite progress in interpolation algorithms, generating concise and interpretable interpolants remains a key challenge. We propose a lightweight learning-based approach to generating simple interpolants for LIA. Our model learns to lazily sample input problems directly and is complementary to existing logical methods. When Z3 is guided by our learned model, the complexity of the interpolants it produces can be reduced by up to 47.3%. For older solvers, the reduction rate can reach up to 69.1%.


Peer reviews:
comments: SUMMARY:
Linear integer arithmetic (LIA), is a logic that allows one to express linear
constraints on integers: equalities, inequalities, and divisibility by non-zero integers. 
This paper proposes a learning-based approach toward computing simpler (than previous logic based
methods) interpolants in LIA, complementing. In addition, they provide a new large dataset of interpolation problems
on which they demonstrate the  ability of their learning approach to simplify interpolants compared to previous approaches.

STRENGTHS AND WEAKNESSES:
Strengths
1. The problem of computing interpolants for ILA is useful for program verification.
2. The paper uses light weight learning rather than LLMs
3. The Dataset provided could be certainly  useful for testing solutions for this problem in the future.

Weaknesses
1. Its unclear to what extent one can verify the claim of "simplicity" compared to previous solutions, the definition
of simplicity was not made clear.
2. Quoting "We show that when Z3 is guided by our learned model, the complexity of the interpolants it produces can
be reduced by up to 47.3%". Not sure how this figure was reached. Also, does this mean that the simpler interpolants were more useful?

QUESTIONS:
I am not an expert in the field.
It would help me to understand exactly where and how interpolants are used and why the 
problem of producing "simpler" interpolants is important as well as a better explanation of the 
"simplicity" definition.

LIMITATIONS:
I think so.
TITLE: Official Review
CLARITY: 2
IMPACT: 3
ORIGINALITY: 2
DATE: 2025-06-30

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nAt a high level, this paper proposes a novel graph-neural-network-based approach for learning how to allocate attention when focusing on relationships between formulas when doing Craig interpolation for Linear Integer Arithmetic (LIA). When solving verification problems, verifiers are often producing Craig interpolants in a tight loop alongside enumeration. Thus, these Craig interpolants can be thought of as \"intermediate formulas\" which the solver (verifier, etc.) is exploring on its way to a solution (counterexample, proof, etc.). In the case of LIA, these look like polynomial (in)equalities. Traditional approaches to Craig interpolation produce unwieldy, large interpolants, with possibly-large coefficients--incurring unnecessarily-large state-space enumeration in the subsequent saturation part of the engine / verifier. The goal is to get small (but correct) interpolants.\n\nThe authors approach is to deeply integrate a GNN and a self-attention-based selector mechanism. The GNN encodes semantic relationships between clauses in the knowledge base, and a selector network then selects subsets of formulas from which to attempt to generate interpolants. This requires a new dataset of interpolation problems--the authors build a new dataset of 66k LIA interpolation problems, which seems like a seriously hefty dataset to me.\n\nThe authors emphasize the needs to focus on traditional ML and a GNN-based approach, and that it would be especially irrelevant to use an LLM. I agree, it would be irrelevant to use an LLM here. The authors also stress the importance of calling the solver *once*, rather than in a loop, in a bottom-up fashion. This necessitates that the authors decompose their exploration in a top-down manner. \n\n# Overall score and justification\n\nOverall, I am very impressed by this work. However, I am very surprised that this is not a very hot area, and it seems surprising to me that Craig interpolation via ML has not been seriously attempted before. My main concern rests on the significance of the work. The authors show reduced interpolant complexity, but I have an outstanding question: how can I put this work into perspective in terms of its impact on the field? Even though small interpolants are nice, are they really a critical component in verification pipelines?\n\nI cannot tell if this paper is a cool, incremental contribution to a challenging field or a truly revolutionary step forward. However, in either case it is well written, intuitive, and easy to read, even as an outsider. I argue that it should be accepted.\n\nSome caveats about my review: I am knowledgable about Craig interpolation and its use in verification, but I do not deeply understand the mechanisms for constructing Craig interpolants and am not sure I am fully prepared to give a nuanced review versus the state of the art.\n\nSTRENGTHS AND WEAKNESSES:\n# Strengths \n\n- Craig interpolation is an important, long-standing, challenging problem that sits at the tight loop of verification, SMT solving, and many important problems in logic and automated reasoning. This paper attacks a particularly pertinent instance of Craig interpolation: for LIA, an important first-order theory that is one of the most popular in SMTLIB.\n\n- The authors got real interpolation problems from downstream tasks, though it is a bit unclear exactly what these are. The authors build a huge knowledge database of LIA facts and then compute self-attention between all of them, which would appear to be a very compelling approach, to me. It is really using the self-attention idea in a manner that can credibly inform the construction of interpolants in an on-demand way.\n\n- The encodings of formulas (Sec 3.2) is brief but makes a lot of sense. You want to be able to learn the hierarchical structure of formulas. Naturally, you need to run as many rounds of iteration as the tallest AST you have (one would assume).\n\n- Writing is overall of high caliber and makes sense. \n\n# Weaknesses\n\n- The major weakness is limited potential impact. It seems that there is limited downstream application for making small Craig interpolants. At least, the authors have included no compelling evaluation which demonstrates why we should believe this will be the case. This is a potentially major concern, although I do believe the approach is still relevant and interesting nonetheless.\n\n- The idea of pooling together all interpolants seen dynamically throughout tons of sample instances seems reasonable, but importantly seems to throw away the context inherent to the problem. I am not sure if this is a weakness or if this is explicitly something you want to do.\n\nQUESTIONS:\n- There is a depth into the tree, but not any structural information. How is this depth used? I guess the hierarchical information is captured by the MPNN?\n\n- How do I put your approach into context for how this might impact the future of SMT solvers and verification? I understand in the abstract that small interpolants are good--but no experiments in the paper can convince me of the promise of the approach to, say, accelerate SMT. Do you have any thoughts on this?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

comments: SUMMARY:
This work highlights a task from classical SMT solving/symbolic reasoning and uses GNNs to approximately solve it. In particular, the authors focus on Criag Interpolation. They are able to show 40% reduction in complexity testing on Z3.

STRENGTHS AND WEAKNESSES:
Strength: 
1. This work poses an interesting problem that presents a fundamental learning challenge. 
2. It's working on a problem that's unique. 

Weakness: 
1. It's not clear that the learnings from this work will translate to other relevant problems in the field.
2. Would appreciate a concrete motivating example of a craig interpolant in the text to motivate why this is problem that matters and what makes it hard. The abstract problem instance is hard to appreciate.

QUESTIONS:
1. How do these results apply to CVC5? Are these gains specific to the algorithm behind Z3?
2. What is the train and test split? In some sense the generalization is what's interesting as solving a single instance of craig interpolation isn't as interesting if the machine is leveraging parallel GPU computation.

LIMITATIONS:
It's not clear to me what the broader impacts are. What does this say about how we should design learning algorithms ?
TITLE: Official Review
CLARITY: 2
IMPACT: 2
ORIGINALITY: 3
DATE: 2025-07-02

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a novel learning-based approach to improve interpolation solvers for Linear Integer Arithmetic (LIA) by generating simpler interpolants. The authors propose using Graph Neural Networks (GNNs) combined with a self-attention mechanism to learn embeddings of LIA formulas, which capture both local arithmetic constraints and the global structure of the formulas. These embeddings enable the model to identify the most relevant subsets of the formulas to create minimal, valid interpolants, thereby simplifying the interpolation process.\n\nThe method is integrated with existing SMT solvers like Z3, showing significant improvements in performance. Experimental results demonstrate that the approach achieves up to 47.3% reduction in interpolant complexity for modern solvers and up to 69.1% reduction for older solvers, offering a promising solution for software and hardware verification. The authors argue that this approach combines symbolic reasoning with machine learning, making it a practical and efficient enhancement for formal verification tasks.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n1. Quality: The paper introduces a novel learning-based method using GNNs and self-attention to improve LIA interpolation solvers, demonstrating significant performance gains.\n\n2. Clarity: The paper is well-written, with clear explanations of the methodology and empirical results that are easy to follow.\n\n3. Significance: The approach addresses a practical problem in formal verification, offering improvements in software and hardware verification through simpler interpolants.\n\n4. Originality: The integration of machine learning with symbolic reasoning to simplify interpolation is innovative, adding a new dimension to existing methods.\n\nWeaknesses\n\n1. Limited Evaluation: The method is evaluated on a small set of benchmark problems, with no discussion of its scalability or performance on more complex, real-world data.\n\n2. Lack of Comparison: The paper does not adequately compare the proposed method with other state-of-the-art techniques for simplifying interpolants or optimizing LIA solvers.\n\n3. Interpretability: The GNN-based model's decision process is not clearly explained, which could limit its acceptance in practical, safety-critical applications.\n\n4. Theoretical Depth: The paper lacks a thorough theoretical analysis of the generalization and robustness of the model to different LIA problem structures.\n\nQUESTIONS:\nDue to my limited familiarity with this field, my judgment may not be entirely accurate.\n\nLIMITATIONS:\nN.A.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 1, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-05"}