Title: The Logical Expressiveness of Temporal GNNs via Two-Dimensional Product Logics
Venue: NeurIPS 2025
Accepted: yes
Keywords: temporal graph neural networks

temporal logic

expressiveness
Abstract: In recent years, the expressive power of various neural architectures---including graph neural networks (GNNs), transformers, and recurrent neural networks---has been characterised using tools from logic and formal language theory. As the capabilities of basic architectures are becoming well understood, increasing attention is turning to models that combine multiple architectural paradigms. Among them particularly important, and challenging to analyse, are temporal extensions of GNNs, which integrate both spatial (graph-structure) and temporal (evolution over time) dimensions. In this paper, we initiate the study of logical characterisation of temporal GNNs by connecting them to two-dimensional product logics. We show that the expressive power of temporal GNNs depends on how graph and temporal components are combined. In particular, temporal GNNs that apply static GNNs recursively over time can capture all properties definable in the product logic of (past) propositional temporal logic PTL and the modal logic K. In contrast, architectures such as graph-and-time TGNNs and global TGNNs can only express restricted fragments of this  logic, where the interaction between temporal and spatial operators is syntactically constrained. These provide us with the first results on the logical expressiveness of temporal GNNs.


Peer reviews:
comments: SUMMARY:
This work proposes to use two-dimensional product logics to analyze the logical expressiveness of Temporal Graph Neural Networks (TGNNs).
The paper analyzes the expressiveness of three types of TGNNs: recursive, time-and-graph, and global TGNNs, using two types of logic: propositional temporal logic (PTL) and modal logic (K).
Theoretical results show that recursive TGNNs can express at least $PTL_{P,Y}\times K$, and that time-and-graph and global TGNNs can express fragments of $PTL_{P,Y}\times K$.

STRENGTHS AND WEAKNESSES:
Strengths: 
- The paper is the first to analyze the logical expressiveness of TGNNs using two-dimensional product logics.
- The paper has strong theoretical results, showing which TGNNs can express what type of logic.
- Overall, the paper is well-written and easy to follow.

Weaknesses: 
- The discussion on how the theoretical results could be utilized in practice is limited, despite the paper not including any experimental results.
- Reasoning behind the use of $PTL_{P,Y}$ is not stated clearly.

QUESTIONS:
- How could the theoretical results be utilized in practice? For example, could they be used to design more expressive TGNNs?
- Is there a reason for the use of truncated ReLU? I imagine those are not used in practice, and other activation functions such as sigmoid would be more common.
- Why do you only consider the operators $P$ and $Y$? Are there other past-time temporal logic operators that you could consider, such as ``always in the past''?
- More background details regarding some parts of the paper would be helpful, such as K^{\#} and time2vec.  While I understand they are given in the appendix, it would greatly help the readability of the paper if some details were included in the main text.

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 3
DATE: 2025-06-30

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents an expressivity investigations for a temporal extension of Graph Neural Networks (GNN) hereinafter referred to as TGNNs. Instead of simply dealing with the topology of a graph as in GNNs, the model adds a temporal dimension considering a sequence of graphs marked by a timestamp. The paper recall the definition of GNN and then presents three different ways of extending their semantics to the temporal dimension given rise to the three classes of Recursive TGNNs, Time-and-graph TGNNs and Global TGNNs. The different way for including the temporal information (basically the way the embeddings on previous graphs are used to fix the embedding in the following ones) affects the expressiveness of corresponding classes. Following an idea  already successfully adopted in the literature for GNNs the expressiveness is compared by using a logical approach: which properties (expressed by a suitable logic) can be recognized by the considered class. In particular the logic exploited in the paper is a suitable product combination of a standard Past Linear Time Logic (for the temporal dimension) and the K logic (but also a fragment of CTL would suit as well) for the spatial dimension. The main result is that the logic is subsumed by Recursive TGNNs but not by Time-and-graph TGNNs and Global TGNNs. In addition, for by Time-and-graph TGNNs and Global TGNNs, the paper propose fragments of the product logic which are respectively subsumed.\n\nSTRENGTHS AND WEAKNESSES:\nStrength.  From the theoretical viewpoint, the paper focus on a relevant and interesting topic extending the expressiveness investigation on the temporal aspects of Timed GNN. The investigation provides an initial picture on the topic. Some aspects in the adopted technique are original (e.g. the adoption of the product logic). The technical machinery is appropriate and seems to be sound. The paper is clearly written even if some technical definitions needed to understand the paper are confined to the appendix.\nWeakeness. Some results seems to immediately derive from corresponding results of the untimed case (e.g. Theorem 2 concerning presburger quantification). Quoting from the abstract: “These results  yields the first logical characterizations of temporal GNNs”.  Usually to have a “characterization” one has to prove that two formalisms are equivalent in some sense, so that claim should be removed. The correct claim is that stated by the title. If there had been a technical characterization the results would have assumed a much greater theoretical significance. The authors should explain more convincingly the usefulness of the temporal dimension.\n\nQUESTIONS:\nPage 3. Definition of Temporal Graph. The sequence of graphs in a temporal graph shares the same set of nodes. No constraint is required on the set of edges admitting different structures over the same set of nodes. Authors should spend some word on this point.\nThe logic K used in combination with PLTL allows only to express a modality for neighborhood. What would be the effects if a modality for reachability were considered (something like EF in CTL)?\nThe weakness of Global TGNNs seems to be the fact that they are not able to focus on specific instants or intervals on the past. Is there any significant class able to interact in more selective way with the past (not only the immediately preceding instant)?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the authors study the expressiveness of temporal graph\nneural networks (TGNNs).\n\nA GNN is defined by a sequence of aggregation and combination\nfunctions that are applied to a graph G = (V, E, c), where c is a\nfunction that maps each vertex in V to a vector in R^k. A temporal\ngraph is defined as (G_1, t_1), (G_2, t_2), ..., (G_n, t_n), where\neach t_i is a timestamp and t_1 < t_2 < ... < t_n. Then, a TGNN is an\nextension of a GNN that takes as input a temporal graph (G_1, t_1),\n(G_2, t_2), ..., (G_n, t_n), processes each graph G_i as a GNN, but\nassigns to each node u in (G_{i+1}, t_{i+1}) a label that is defined\nas the concatenation of the label of u in G_{i+1} with the resulting\nlabel from the execution on G_i.\n\nThe paper considers different classes of TGNNs, with the class of\nrecursive TGNNs being the most expressive. To study the expressiveness\nof these classes, the paper considers different product logics, the\nmost expressive being defined as the product of path temporal logic\nPTL_{P,Y}, with operators \"yesterday\" (Y) and \"sometime in the past\"\n(P), with the modal logic K. Then, the authors prove that each formula\nin PTL_{P,Y} × K can be expressed by a recursive TGNN. Moreover, the\nauthors consider less expressive TGNNs and identify fragments of\nPTL_{P,Y} × K that are and are not contained in them. As a corollary\nof these results, they provide separations in terms of expressive\npower between the different classes of TGNNs considered in the paper.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nS1 Understanding the expressive power of a TGNN is a fundamental\nproblem. In particular, understanding what cannot be expressed by a\nclass of TGNNs helps to understand not only the limitations of such an\narchitecture but also to identify what needs to be added to the\narchitecture to increase its expressive power.\n\nS2 The paper provides some non-trivial results about the\nexpressiveness of some classes of TGNNs, which help in understanding\nthe expressiveness of these classes.\n\nWeaknesses:\n\nW1 The inexpressibility results presented in the paper are\ninteresting, as they help us understand what a class of TGNNs cannot\nexpress. However, the authors do not make an effort to explain such\ninexpressibility results in a way that the reader can clearly\nunderstand the limitations of a class of TGNNs.\n\nW2 The paper does not provide convincing evidence that product logics\nare the right formalism to study the expressive power of TGNNs. In\nparticular, they do not clearly explain why the expressiveness of\nTGNNs cannot be studied in terms of temporal extensions of first-order\nlogic.\n\nW3 The paper does not identify any logic that exactly captures the\nexpressive power of any of the classes of TGNNs studied. Notice that\nthis has been done for (non-temporal) GNNs.\n\nQUESTIONS:\nQ1 What are the complexities of the translations presented in the\npaper? This information is important, for example, to determine\nwhether a TGNN is exponentially more succinct than a product logic.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates the expressive power of Temporal Graph Neural Networks (TGNNs) using two-dimensional product logics, which combine temporal and modal logics. The authors characterize the logical expressiveness of three classes of TGNNs—recursive, time-and-graph, and global TGNNs—by connecting them to fragments of these logics. They show that recursive TGNNs can express all properties definable in the product logic, while time-and-graph and global TGNNs are limited to specific fragments due to constraints on how temporal and spatial operators interact. These results provide foundational insights into the capabilities of different TGNN architectures and establish relative expressiveness relationships between them.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n\n1. The paper presents a novel and rigorous logical characterization of temporal graph neural networks (TGNNs) using two-dimensional product logics, effectively bridging a gap in the literature.\n2. The theoretical results are well-supported by detailed proofs and clearly defined concepts, enhancing the paper’s credibility and rigor.\n3. The work provides practical insights into the expressive power of different TGNN architectures, which could inform model selection for specific tasks.\n\n## Weaknesses\n\n1. The proposed method can derive logical interpretations for TGNNs but does not provide edge-specific or global weights (e.g., confidence scores), limiting interpretability at a granular level.\n2. The paper lacks empirical validation, which could have strengthened the practical applicability of its theoretical findings.mpirical validation, which could have strengthened the practical relevance of the theoretical findings.\n3. The technical complexity of the proofs and logical frameworks might limit accessibility for readers without a strong background in formal logics.\n\nQUESTIONS:\n1. How can the proposed method be extended to provide edge-specific or global confidence scores to enhance interpretability?\n2. What is the time complexity (or computational cost) of extracting logical rules from a TGNN?\n\nLIMITATIONS:\nThe discussion on limitations, while present, could be more detailed, particularly regarding the assumptions and their impact on real-world applications.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}