Mathematical Reasoning in Latent Space
We design and conduct a simple experiment to study whether neural networks can perform several steps of approximate reasoning in a fixed dimensional latent space. The set of rewrites (i.e. transformations) that can be successfully performed on a statement represents essential semantic features of the statement. We can compress this information by embedding the formula in a vector space, such that the vector associated with a statement can be used to predict whether a statement can be rewritten by other theorems. Predicting the embedding of a formula generated by some rewrite rule is naturally viewed as approximate reasoning in the latent space. In order to measure the effectiveness of this reasoning, we perform approximate deduction sequences in the latent space and use the resulting embedding to inform the semantic features of the corresponding formal statement (which is obtained by performing the corresponding rewrite sequence using real formulas). Our experiments show that graph neural networks can make non-trivial predictions about the rewrite-success of statements, even when they propagate predicted latent representations for several steps. Since our corpus of mathematical formulas includes a wide variety of mathematical disciplines, this experiment is a strong indicator for the feasibility of deduction in latent space in general.
= Summary Embeddings of mathematical theorems and rewrite rules are presented. An in-depth analysis of the resulting embeddings is presented, showing that a network can learn to *apply* embedded rewrite rules to embedded theorems, yielding results that are similar to the embedding of the rewritten theorem. [i.e., app*(emb(thm), emb(rule)) is near to emb(app(thm, rule))] This is an interesting property for the application of deep learning to automated theorem proving, though not directly a breakthrough result. = Strong/Weak Points + Simply a cute result, showing that proof search can remain in embedding space for a limited time horizon without having to switch back into the theorem prover environment. + Nicely designed experiments testing this (somewhat surprising) property empirically - Missed opportunity of better analysis of which theorem/rewrite rule properties are more likely to fail - Writing sometimes a bit overcomplicated (e.g., Sect. 4.5 could just be a figure of a commuting diagram and two sentences...) - Architecture choice unclear: Why are and separate networks. This is discussed on p4, but it*s unclear to me how keeping separate is benefitial for the analysis, and this is not picked up again explicitly again? = Recommendation Overall, this is a nice, somewhat surprising result. The writing and experiments could use some improvement, but I believe that the majority of the ICLR audience would enjoy seeing this result (even though it would have no impact on most people*s research) = Detailed Comments - page 4, Sect. 4.4: Architecture of would be nice (more than a linear layer?) - page 5, paragraph 3: *we from some* -> *we start from some* - p6par1: *much cheaper then computing* -> than - p6par6: *on formulas that with* -> no that - p6par7: *measure how rate* -> *measure the rate* - p8par1: *approximate embedding - is undefined and should probably be (this is also the case in the caption of Fig. 5), and should probably be included as well. However, I don*t understand the use of here. If Fig. 4 is following Fig. 3 in considering , then Fig. 4 should plot the performance of, e.g., (i.e., applied to approximate embedding of and (*true*) embedding of ). I believe that*s what *Pred (One Step)* expresses, but it would maybe be generally helpful to be more precise about the notation in Sect. 6.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Mathematical Reasoning in Latent Space
We design and conduct a simple experiment to study whether neural networks can perform several steps of approximate reasoning in a fixed dimensional latent space. The set of rewrites (i.e. transformations) that can be successfully performed on a statement represents essential semantic features of the statement. We can compress this information by embedding the formula in a vector space, such that the vector associated with a statement can be used to predict whether a statement can be rewritten by other theorems. Predicting the embedding of a formula generated by some rewrite rule is naturally viewed as approximate reasoning in the latent space. In order to measure the effectiveness of this reasoning, we perform approximate deduction sequences in the latent space and use the resulting embedding to inform the semantic features of the corresponding formal statement (which is obtained by performing the corresponding rewrite sequence using real formulas). Our experiments show that graph neural networks can make non-trivial predictions about the rewrite-success of statements, even when they propagate predicted latent representations for several steps. Since our corpus of mathematical formulas includes a wide variety of mathematical disciplines, this experiment is a strong indicator for the feasibility of deduction in latent space in general.
user
<think>Okay, to do a decent peer review, I should start with summarizing what the work is about. This includes novelty, contributions, clarity, and impact. I might include suggestions to issues that I find critical to the usefulness of the work. Starting with novelty, there are different types of novelty. The first aspect of novelty is scope and scale. Is the work covering a broader perspective than previous works? The second aspect is methodology. Is the proposed methodology novel compared to previous methods? Do they provide any significant advantage over previous methodologies? The third aspect is results. Are the results different compared previous work? Does this lead to any novel theories or an extension of current theories? Next is clarity, I will be reading the work in depth and analyze if any sentences can be potentially ambiguous or misleading. If any sentences are ambiguous, I will state them and give a suggestion on how to rephrase such sentence or how to make more clear to the audience. Lastly, impact is important. I will attempt to predict how such work will impact the scientific community. This might include the applications of such work and how the work might transform the way we study a certain topic. Finally, I will might only mention issues that are critical. This is to keep the review as concise and helpful as possible.</think>



= Summary Embeddings of mathematical theorems and rewrite rules are presented. An in-depth analysis of the resulting embeddings is presented, showing that a network can learn to *apply* embedded rewrite rules to embedded theorems, yielding results that are similar to the embedding of the rewritten theorem. [i.e., app*(emb(thm), emb(rule)) is near to emb(app(thm, rule))] This is an interesting property for the application of deep learning to automated theorem proving, though not directly a breakthrough result. = Strong/Weak Points + Simply a cute result, showing that proof search can remain in embedding space for a limited time horizon without having to switch back into the theorem prover environment. + Nicely designed experiments testing this (somewhat surprising) property empirically - Missed opportunity of better analysis of which theorem/rewrite rule properties are more likely to fail - Writing sometimes a bit overcomplicated (e.g., Sect. 4.5 could just be a figure of a commuting diagram and two sentences...) - Architecture choice unclear: Why are and separate networks. This is discussed on p4, but it*s unclear to me how keeping separate is benefitial for the analysis, and this is not picked up again explicitly again? = Recommendation Overall, this is a nice, somewhat surprising result. The writing and experiments could use some improvement, but I believe that the majority of the ICLR audience would enjoy seeing this result (even though it would have no impact on most people*s research) = Detailed Comments - page 4, Sect. 4.4: Architecture of would be nice (more than a linear layer?) - page 5, paragraph 3: *we from some* -> *we start from some* - p6par1: *much cheaper then computing* -> than - p6par6: *on formulas that with* -> no that - p6par7: *measure how rate* -> *measure the rate* - p8par1: *approximate embedding - is undefined and should probably be (this is also the case in the caption of Fig. 5), and should probably be included as well. However, I don*t understand the use of here. If Fig. 4 is following Fig. 3 in considering , then Fig. 4 should plot the performance of, e.g., (i.e., applied to approximate embedding of and (*true*) embedding of ). I believe that*s what *Pred (One Step)* expresses, but it would maybe be generally helpful to be more precise about the notation in Sect. 6.
assistant
= Summary Embeddings of mathematical theorems and rewrite rules are presented. An in-depth analysis of the resulting embeddings is presented, showing that a network can learn to *apply* embedded rewrite rules to embedded theorems, yielding results that are similar to the embedding of the rewritten theorem. [i.e., app*(emb(thm), emb(rule)) is near to emb(app(thm, rule))] This is an interesting property for the application of deep learning to automated theorem proving, though not directly a breakthrough result. = Strong/Weak Points + Simply a cute result, showing that proof search can remain in embedding space for a limited time horizon without having to switch back into the theorem prover environment. + Nicely designed experiments testing this (somewhat surprising) property empirically - Missed opportunity of better analysis of which theorem/rewrite rule properties are more likely to fail - Writing sometimes a bit overcomplicated (e.g., Sect. 4.5 could just be a figure of a commuting diagram and two sentences...) - Architecture choice unclear: Why are and separate networks. This is discussed on p4, but it*s unclear to me how keeping separate is benefitial for the analysis, and this is not picked up again explicitly again? = Recommendation Overall, this is a nice, somewhat surprising result. The writing and experiments could use some improvement, but I believe that the majority of the ICLR audience would enjoy seeing this result (even though it would have no impact on most people*s research) = Detailed Comments - page 4, Sect. 4.4: Architecture of would be nice (more than a linear layer?) - page 5, paragraph 3: *we from some* -> *we start from some* - p6par1: *much cheaper then computing* -> than - p6par6: *on formulas that with* -> no that - p6par7: *measure how rate* -> *measure the rate* - p8par1: *approximate embedding - is undefined and should probably be (this is also the case in the caption of Fig. 5), and should probably be included as well. However, I don*t understand the use of here. If Fig. 4 is following Fig. 3 in considering , then Fig. 4 should plot the performance of, e.g., (i.e., applied to approximate embedding of and (*true*) embedding of ). I believe that*s what *Pred (One Step)* expresses, but it would maybe be generally helpful to be more precise about the notation in Sect. 6.