SlowGuess/NAACL_2022_OCR huggingface dataset:
We compare with the SOTA entailment tree generation method EntialmentWriter (Dalvi et al., 2021), which directly generates the linearized trees (e.g.,  $s_2 + s_5 \to i_1$ ): eruptions block sunlight;  $s_4 + i_1 \to H$ ) given  $H + QA + S$  with an end-to-end encoder-decoder framework. We also follow the "Iterative" ProofWriter (Tafjord et al., 2021), which is one of the SOTA proof generation methods for logical reasoning, to extend EntialmentWriter to EntialmentWriter-Iter. EntialmentWriter-Iter iteratively generates a part of the linearized tree in one forward process (e.g.,  $s_2 + s_5 \to i_1$ ): eruptions block sunlight;) and concatenates all parts to make the final tree. It completes the step selection and entailment reasoning in a seq2seq model and does not provide the reasoning types of steps.
Modules. We implement the entailment modules on top of T5-large (Raffel et al., 2020) with the following two implementations. (1) Separated. We implement each module separately. We have six models in total, corresponding to the three reasoning types of deductive and abductive versions. (2)
Prefixed. We implement all modules with a single model. To specify which reasoning type the model should perform, we follow Raffel et al. (2020) to add a type-specific prefix (e.g., "deductive substitution:") to the input before feeding it to the model. To evaluate the modules, we annotate the types of 275 steps in the dev split. We train the modules with a batch size of 20 for 100 epochs.
Controller. The controller is implemented with albert-xxlarge-v2 (Lan et al., 2019). We train two individual controllers for Task1 and Task2. For Task3, we reuse the Task2 model without additional training. The controllers are trained with a batch size of 10 for 1,000 epochs. The margins  $m_{step}$ ,  $m_{fact}$ , and  $m_{state}$  are tuned on the development split and all set to 0.1.
Algorithm. For Task1, we iterate until all facts in  $S$  are used. For Task2, we use a fact score threshold of 0.001 to filter distractors and a maximum reasoning depth of 5. We select the top  $10\%$  steps for each state and set the beam size to 10. All hyper-parameters are selected using the dev split (Appendix C). For Task3, we follow Dalvi et al.
<sup>1</sup>The threshold was picked using 300 manually labeled pairs (Dalvi et al., 2021).
2We repair a bug in the official evaluation code, which makes the Intermediate AllCorrect  $= 1$  if the precision  $= 1$  (rather than if  $\mathrm{F}1 = 1$ ), which leads to an overestimation on the Intermediate AllCorrect.
Task	Method	\(n_{para}\)	Leaves	Steps	Intermediates	Overall
F1	AllCorrect	F1	AllCorrect	F1	AllCorrect	F1	AllCorrect
Task1(no-distractor)	EntialmentWriter (T5-11B)\(^\dagger\)	11.00	99.0	89.4	51.5	38.2	71.2	52.9†	35.6
EntialmentWriter (T5-large)	0.77	98.4	84.1	50.0	38.5	67.0	35.9	34.4
EntialmentWriter-Iter (T5-large)	0.77	99.8	97.6	51.6	38.5	68.3	36.5	35.0
METGEN-separated (Ours)	0.22+6×0.77	100.0	100.0	57.9	42.1	71.3	39.2	37.0
METGEN-prefixed (Ours)	0.22+0.77	100.0	100.0	57.7	41.9	70.8	39.2	36.5
Task2(distractor)	EntialmentWriter (T5-11B)\(^\dagger\)	11.00	89.1	48.8	41.4	27.7	66.2	53.2†	25.6
EntialmentWriter (T5-large)	0.77	83.2	35.0	39.5	24.7	62.2	28.2	23.2
EntialmentWriter-Iter (T5-large)	0.77	85.2	40.9	38.9	26.8	63.5	29.1	25.0
METGEN-separated (Ours)	0.22+6×0.77	83.7	48.6	41.7	30.4	62.7	32.7	28.0
METGEN-prefixed (Ours)	0.22+0.77	82.7	46.1	41.3	29.6	61.4	32.4	27.7
Task3(full-corpus)	EntialmentWriter (T5-11B)\(^\dagger\)	11.00	39.7	3.8	7.8	2.9	36.4	13.2†	2.9
EntialmentWriter (T5-large)	0.77	30.9	1.2	4.4	1.2	28.8	5.6	1.2
EntialmentWriter-Iter (T5-large)	0.77	32.4	1.8	4.4	1.5	29.7	6.5	1.5
METGEN-separated (Ours)	0.22+6×0.77	34.8	8.7	9.8	8.6	36.7	20.4	8.6
METGEN-prefixed (Ours)	0.22+0.77	34.8	8.7	9.8	8.6	36.6	20.4	8.6
Table 3: Automatic evaluation results on the EntailmentBank test split.  $\dagger$  indicates results from the published paper2.  $n_{para}$  denotes the number of model parameters (B).
Method	Task1	Task2
Automatic	Manual	Automatic	Manual
EntialmentWriter (T5-large)	35	46	21	26
EntialmentWriter-Iter (T5-large)	35	47	25	35
METGEN-prefixed (Ours)	36	53	27	39
Table 4: Entailment tree evaluation results on 100 uniformly sampled questions from the test split. We report the proportion (\%) of the predicted trees that are rated as valid, following automatic and manual evaluation.
(2021) to retrieve 25 sentences from the corpus  $C$  using the  $H$  as the query. We use the same retrieval results as EntailmentWriter for a fair comparison. Model checkpoints are selected using the dev split. More implementation details can be found in the Appendix.
6 Result Analysis
As shown in Table 3, our methods outperform all baseline methods on the strictest metric Overall AllCorrect for all three tasks. Notice that the trees generated by our methods only contain 2-premise steps, which would lead to a 0 Overall AllCorrect score on  $26\%$  of test samples whose annotations contain  $n$ -premise  $(n > 2)$  steps. Even so, our METGEN-separated still obtains an absolute improvement of  $1.4\% / 2.4\% / 5.7\%$  on Task1/2/3 in comparison to the strongest baseline. With only  $9.0\%$  of the model parameters, METGEN-prefixed can outperform the EntialmentWriter (T5-11B) by absolute  $0.9\% / 2.1\% / 5.7\%$  on Task1/2/3. In the case of using a comparable amount of model parameters, METGEN-prefixed also outperforms the EntialmentWriter-Iter (T5-large) by a large margin. For Task3, we note that all methods perform
Figure 4: Manual evaluation results of 100 single-step entailments uniformly sampled from the predicted trees of Task2 test spilt. EW denotes EntailmentWriter.
poorly. The main reason is that the retrieved facts may not contain all the required facts  $S_{gold}$  (68% of the cases). We note that METGEN underperforms the baselines on some metrics, probably due to the inaccuracy of the tree alignment algorithm in the automatic evaluation (Appendix G).
6.2 Manual Evaluation
As analysed by Dalvi et al. (2021), the automated metrics might misjudge some valid trees and thus underestimate the performance. To make a more accurate comparison, we perform the manual evaluation. We compare three methods with a comparable amount of model parameters, EntialmentWriter (T5-large), EntialmentWriter-Iter (T5-large), and METGEN-prefixed. For each step and tree, we invite three students as experts to evaluate the validity. The inter-annotator agreement (Cohen's kappa statistic) is 0.85/0.76 for the step/tree, indicating the substantial agreement between annotators.
Validity of Full Entailment Trees. As shown in Table 4, under the manual evaluation, METGEN outperforms the baselines with large margins.
Validity of Individual Entailment Steps. We review the validity of the single-step entailment and annotate each step with one of the four categories:
- Valid: The step conclusion can be inferred from
Implemen-tation	Models	Reasoning Type	Training Data	Overall AllCorrect	Single-step Accuracy
(a)	Sep	6×T5-large	✓	S+E	28.0	81.0
(b)	Sep	6×BART-large	✓	S+E	26.2	77.0
(c)	Sep	6×T5-base	✓	S+E	27.3	78.0
(d)	Sep	6×T5-large	✓	E	27.8	79.5
(e)	Sep	6×T5-large	✓	S	23.5	43.6
(f)	Pre	1×T5-large	✓	S+E	27.7	78.4
(g)	Pre	1×T5-large	✓	E	27.4	78.1
(h)	Pre	1×T5-large	×	E	25.9	76.0
Table 5: Ablation results on entailment modules. Sep/Pre indicates seperated/prefixed. S/E denotes the synthesis/EntailmentBank step training data.
Task	Method	Leaves	Steps	Intermediate	Overall
Task1	controller	100.0	42.1	39.2	37.0
w/o abduction	100.0	41.4	38.4	36.2
heuristic	100.0	31.2	31.2	28.8
Task2	controller	48.6	30.4	32.7	28.0
w/o abduction	44.5	28.3	31.6	27.0
heuristic	3.2	3.2	12.1	3.2
Table 6: Ablation results on the reasoning controller. We report the AllCorrect scores on the test split.
the premises and does not trivially repeat them.
-Unsupported: The conclusion is in conflict with, irrelevant with, or not followed from the premises.
- Repeat premises: The conclusion trivially repeats one or more of the premises.
- Missing premises: The conclusion uses knowledge unstated in the premises. The step would be correct if one additional premise from  $S$  is added.
As shown in Figure 4, METGEN achieves considerable improvement in the validity of steps compared to the baseline methods. We note that  $17\%$  of the steps of EntialmentWriter belong to missing premises. METGEN constrains the reasoning types of steps and uses the premise-related and context-independent entailment modules to perform every single step. This can reduce the cases of missing premises (from  $17\%$  to  $2\%$ ) and improve the validity of the conclusions (from  $38\%$  to  $70\%$ ).
Entailment Modules Analysis. Table 5 reports the ablation results on modules. We report the Overall AllCorrect on test spilt and the single-step entailment accuracy on the labeled dev steps, and can make the following observations. (1) Separated vs. Prefixed. We can see that METGEN-prefix achieved slightly worse performance than METGEN-separated ((a) vs. (f) and (d) vs. (g)). This is mainly because separate modules could better learn different types of reasoning. However, in our final system, we still choose to use METGEN-prefix
Figure 5: Results on different ratios (0.01, 0.05, 0.10, 0.20, 0.50, 1.00) of EntailmentBank training data.
due to the consideration of model size. (2) Clarifying Reasoning Types. We train a module to infer without distinguishing or assigning specific reasoning types. We find that the performance drops from  $27.4\%$  to  $25.9\%$  ((g) vs. (h)), suggesting that clarifying the reasoning types of the entailment steps is crucial for generating entailment trees. (3) Training Data. Comparing (a) and (d), we find that training with the synthesis data could improve the accuracy. Without tuning on EntailmentBank (setting (e)), the modules might not adapt to the science domain and obtain low step accuracy. However, the well-trained controller would verify and filter the error conclusions, thus our method can still achieve  $23.5\%$  on Overall AllCorrect. (4) Generative Model. A stronger generative model, which achieves higher single-step accuracy, could achieve higher tree generation performance (comparing (a), (b) and (c)), indicating that our method can be further improved with stronger entailment modules.
Controller and Algorithm Analysis. (1) Is the reasoning controller necessary? To answer this question, we design a heuristic generation algorithm without the controller (Appendix D). It uses the BLEURT scores as heuristic information to guide the reasoning. As shown in Table 6, the heuristic method achieves observable lower performance. The controller could aid in eliminating the error steps and states, so as to find the valid trees efficiently and accurately. Without the controller, we find it difficult to find effective heuristic information. (2) Effect of Abductive Steps. The generation performance drops when abductive steps are not used. This suggests that abductive steps, as a way of backward searching, could help improve the quality of generated trees.
6.4 Data-scarce Setting
Figure 5 reports the results in the data-scarce setting. Our method is more data-efficient. With only  $1\%$  of the EntailmentBank training data, our
Method	eQASC	eOBQA
P@1	NDCG	P@1	NDCG
EntialmentWriter (T5-large)	52.48	73.14	69.07	89.05
EntialmentWriter-Iter (T5-large)	52.56	73.28	72.15	90.19
METGEN-prefix (Ours)	55.81	74.19	74.89	90.50
Table 7: Cross-dataset results on the eQASC and eOBQA test split.
method obtains  $14.7\%$  on Task2 Overall AllCorrect, in comparison to  $10.0\%$  of the strongest baseline. When the data is scarce, the advantage of training our modules with synthetic data becomes more significant. It can help alleviate the overfitting on few EntailmentBank sentences.
6.5 Cross-dataset Setting
To test the generalization capability of our method, we conduct cross-dataset experiments on datasets eQASC and eOBQA (Jhamtani and Clark, 2020), which collect one-step entailment trees for questions from QASC (Khot et al., 2020) and OpenBookQA (Mihaylov et al., 2018), respectively. Given  $H$  and  $S$ , their task requires selecting the valid one-step trees (e.g.,  $s_1 + s_2 \rightarrow H$ ) from a candidate set. We apply the Task2 models (without fine-tuning on eQASC or eOBQA) to select from the candidate trees (Appendix E). Following Jhamtani and Clark (2020), we evaluate models with the P@1 and NDCG metrics. Questions with no valid tree are filtered. As shown in Table 7, our method achieves better generalization performance. Instead of training a seq2seq model with a single generation loss, our method explicitly models the step and state selection ability (equation (1) and (3)) and guides the controller with specific losses to rank the correct ones ahead of incorrect ones. Such a manner could aid in alleviating the overfitting on training data and improve the generality.
We propose METGEN, a module-based framework to generate the entailment trees for explaining answers. METGEN reasons with single-step entailment modules and the reasoning controller. Experiments on EntailmentBank benchmark show METGEN can generate valid trees with reliable steps and achieve SOTA performance.
We appreciate the anonymous reviewers for their insightful comments. We would like to
thank Zhihong Shao for the helpful discussions. This work is supported by the National Key Research and Development Program of China (No. 2018AAA0100701), and the Guoqiang Institute of Tsinghua University with Grant No. 2020GQG0005.
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