SlowGuess/NAACL_2022_OCR huggingface dataset:
DROP Question-answer pairs in DROP dataset (Dua et al., 2019) are crowd-sourced based on passages collected from Wikipedia. In detail, it contains 96.6K question-answer pairs, where 77400/9536/9615 samples are for training/development/test. Three kinds of answers are involved in the raw dataset, i.e., NUMBER  $(60.69\%)$ , SPANS  $(37.72\%)$ , and DATE  $(1.59\%)$ .
RACENUM To investigate the generalization capability of OPERA, we compare OPERA to other strong baselines on samples of RACE (Lai et al., 2017). Following Chen et al. (2020a), we sample instances from RACE, denoted as RACENUM, where the question of each instance starts with "how many". To conveniently evaluate the models on RACENUM, we convert the format of instances in RACENUM into the same as DROP, since RACE is a multi-choice MRC dataset. RACENUM is divided into two categories, i.e., middle/high school exam (RACENUM-M/H). They respectively contain 609 and 565 questions, where the scale is a bit different from that reported in Chen et al.  $(2020\mathrm{a})^{3}$
We compare OPERA with various prior systems in terms of reasoning capability and interpretability.
w/o Pre-trained Language Model: NAQANET (Dua et al., 2019) leverages several answer predictors to produce corresponding types of answers, including a span, count, and arithmetic expression. NumNet (Ran et al., 2019) leverages a graph convolution network to reason over a number graph aggregated relative magnitude among numbers.
w/ Pre-trained Language Model: GenBERT (Geva et al., 2020) injects reasoning capability into BERT by pre-training with large-scale numerical data. Based on NAQANET, MTMSN (Hu et al., 2019) adds a negation predictor to solve the negative question and re-rank arithmetic expression candidates. NeRd (Chen et al., 2020b) is essentially a generative semantic parser that maps questions and passages into executable logical forms. ALBERT-Calc was proposed for DROP by combining ALBERT with several predefined answer predictors (Andor et al., 2019). NumNet+ employs a pre-trained model to further boost the performance of NumNet. QDGAT (Chen et al., 2020a) builds a heterogeneous graph composed of entity and value nodes upon RoBERTa and utilizes a question-directed graph attention network to reason over the graph. EviDR (Zhou et al., 2021), an evidence-emphasized MRC model, performs reasoning over a multi-grained evidence graph based on ELECTRA.
We utilize adam optimizer (Kingma and Ba, 2015) with a cosine warmup mechanism and set the
<sup>3</sup>Since no released code for dataset construction, we implement it referred to the algorithm in Chen et al. (2020a).
BLR	LR	BWD	WD	Epochs	BS	nh	dh
RoBERTa	1.5e-5	5e-4	0.01	5e-5	12	16	16	1024
ELECTRA	1.5e-5	5e-4	0.01	5e-5	12	16	16	1024
ALBERT	3e-5	1e-4	0.01	5e-5	8	128	64	4096
weight of loss  $\lambda = 0.3$  to train the model. The hyper-parameters are listed in Table 4, where BLR, LR, BWD, WD, BS, and  $d_h$  respectively represent the learning rate of the encoder, the learning rate of other parts of the model, the weight decay of the encoder, the weight decay of other parts of the model, batch size and hidden size of the model. Each operation is neutralized with a multi-head attention layer with  $n_h$  heads and  $d_h$  dimension.
4.4.1 Results on DROP and Analysis
Table 5 shows the overall results of OPERA and all the baselines on the DROP dataset. OPERA achieves comparable and even higher performance than the recently available methods. Specifically, OPERA(RoBERTa) achieves comparable performance to QDGAT with advantages of 0.32 EM and 0.42 F1. OPERA(ELECTRA) exceeds EviDR by 0.89 EM and 0.90 F1 and OPERA(ALBERT) outperforms ALBERT-Calc by 4.84 EM and 4.24 F1. Moreover, the voting strategy is employed to ensemble 7 OPERA(ALBERT) models with different random seeds, achieving 86.26 EM and 89.12 F1 scores. We think the better performance comes from the modeling of discrete reasoning over text via operations, which mines more semantic information of context and explicitly integrates them into the answer prediction.
4.4.2 Results on RACENUM
To investigate the generalization of OPERA for discrete reasoning, we additionally compare OPERA with QDGAT and NumNet+ on the RACENUM dataset. We directly evaluate the three models without fine-tuning on RACENUM due to its small scale. As Table 6 shows, the scores of models on the RACENUM dataset are generally lower than that on the DROP dataset, which is attributed to the lack of in-domain training data. Nevertheless, the performance of OPERA significantly outperforms NumNet+ and QDGAT by a large margin of more than 3.49 EM and 3.53 F1 score on average. It indicates that OPERA has better generalization ability.
Table 4: Hyperparameters settings for training OPERA.
Method	Dev	Test
EM	F1	EM	F1
w/o Pre-trained Models
NAQANet	46.20	49.24	44.07	47.01
NumNet	64.92	68.31	64.56	67.97
w/ Pre-trained Models
GenBERT	68.80	72.30	68.6	72.35
MTMSN	76.68	80.54	75.88	79.99
NeRd	78.55	81.85	78.33	81.71
ALBERT-Calc	80.22	83.98	79.85	83.56
NumNet+	81.07	84.42	81.52	84.84
EviDR	82.09	85.14	82.55	85.80
QDGAT	82.74	85.85	83.23	86.38
Single Model Results
OPERA(RoBERTa)	83.74	86.52	83.55	86.80
OPERA(ELECTRA)	83.86	86.66	83.46	86.70
OPERA(ALBERT)	84.86	87.54	84.69	87.80
Ensemble Results
OPERA	86.79	89.41	86.26	89.12
Human	94.90	96.42
Table 5: Results on the DROP dataset. We solely compare with QDGAT, but leaving  $\mathrm{QDGAT}_p$  alone, since we focus on the reasoning mechanism in this work, while  $\mathrm{QDGAT}_p$  is a variant of QDGAT with data augmentation (Chen et al., 2020a).
Method	RACENUM-M	RACENUM-H	Avg
EM	F1	EM	F1	EM	F1
NumNet+	41.71	41.82	29.73	29.73	35.94	36.00
QDGAT	44.01	44.01	28.85	28.85	36.71	36.71
OPERA	47.62	47.62	32.21	32.30	40.20	40.24
Table 6: The performance of RoBERTa-based models on the RACENUM dataset without finetuning.
4.5 Interpretability Analysis
Interpretability is an essential property for evaluating an MRC model. We analyze the interpretability of OPERA from the following two stages: (1) mapping from questions to operations, and (2) mapping from operations to answers.
Mapping from Question to Operation To explicitly show the correlations between questions and related operations, we manually evaluate the performance of the operation selection on 50 samples on the development set of DROP. Specifically, precision@n (P@n) is used as the evaluation metric, i.e., judging whether the top  $n$  predicted operations contain the correct ones according to questions. We finally achieve 0.88 on P@1 and 0.98 on P@2 for our model OPERA, which indicates that the operation selection module can accurately predict interpretable operations.
Figure 2: Statistical correlations between operations and answer types. The horizontal axis indicates answer types, and the vertical axis means operations. PS, CT, AE, MS and QS respectively means passage span, count, arithmetic expression, multi-spans and question span.
Mapping from Operation to Answer We explore the relations between operations and final answer types based on model predictions on the development set of DROP. Specifically, for each type of answer, the predicted operation distributions are summed over all samples and normalized, which derives a relation matrix as shown in Figure 2. We can observe that obvious correlations between operations and answer types exist. ADDITION, DIFF, MAX and MIN obviously correspond to Arithmetic Expression. The top 3 answer types for KEY_VALUE and SPAN are Passage Span, Multiple Spans, and Question Span. COUNT probably maps to Count answer type. ARGMORE, ARGLESS, ARGMAX, and ARGMIN are usually used for both Passage Span and Arithmetic Expression.
Effect of Operations for OPERA As shown in Table 7, we first remove the loss of operation selection  $(\mathrm{w / o}\mathcal{L}_{op})$  by setting  $\lambda = 0$  , resulting in the performance degradation of  $0.74\mathrm{EM} / 0.52\mathrm{F1}$  points and  $0.14\mathrm{EM} / 0.18\mathrm{F1}$  points for OPERA based on RoBERTa and ALBERT, respectively. It indicates that the supervision for explicitly learning the alignment from a question to operations contributes to the reasoning capability of OPERA. Yet our approach is somewhat limited by the fact that the operation selector needs an auxiliary training task to work better, and heuristics rules are required to map questions into an operation set as training labels. Furthermore, we remove the total operation-pivoted reasoning module (w/o OP), the performance respectively declines by  $1.06\mathrm{EM} /$
Figure 3: Ablation study on all the subsets of DROP containing some specific operations.
Method	NUM (60.69%)	SPANS (37.72%)	Overall
EM / F1	EM / F1	EM / F1
OPERARAB)	85.91 / 86.14	83.89 / 88.90	83.74 / 86.52
w/o Lop	85.83 / 86.00	82.68 / 88.02	83.00 / 86.00
w/o OP	85.36 / 85.53	82.53 / 87.06	82.68 / 85.73
OPERAB)	86.39 / 86.58	86.30 / 90.99	84.86 / 87.54
w/o Lop	86.08 / 86.24	86.39 / 90.96	84.72 / 87.36
w/o OP	85.62 / 85.89	84.95 / 89.83	84.01 / 86.79
Table 7: Ablation study on the dev set of DROP. RB and AB mean RoBERTa and ALBERT, respectively.
0.79 F1 points and 0.85 EM / 0.75 F1 points for OPERA(RoBERTa) and OPERA(ALBERT). We also conduct the ablation study on the subsets containing a specific operation. As shown in Figure 3, OPERA achieves better performance than OPERA w/o OP on the majority of subsets. Overall, it confirms that integrating the operation-pivoted discrete reasoning mechanism contributes to the reasoning ability of the model.
Probe on Answer Types and Language Models As reported in Table 7, we observe that the performance on the NUMBER(NUM) and SPANS questions, which together account for  $98.4\%$  of the total, respectively declines by 0.55 EM/0.61 F1 and 1.36 EM/1.84 F1 when removing operation-pivoted reasoning mechanism from OPERA(RB). It demonstrates that this mechanism contributes to various answer types. Also, we respectively evaluate the performance of OPERA based on RoBERTa and ALBERT. We observe that OPERA(ALBERT) outperforms OPERA(RoBERTa) due to the stronger capability of semantic representation. Furthermore, integrating this mechanism consistently contributes to the performance of OPERA no matter it is based on RoBERTa or ALBERT. It indicates that OPERA
Question-Answer	Passage	NumNet+	QDGAT	OPERA
Q: How many total yards of touchdown passes were there?
A: 73	... receiver Johnny Knox on a 23-yard touchdown pass. Afterwards, the Eagles took the lead as quarterback Matt Ryan completed a 40-yard touchdown pass to wide receiver Roddy White and a 10-yard touchdown pass to tight end Tony Gonzalez ...	AnswerType:
Count
Answer: 0	AnswerType:
Count
Answer: 0	Top-1 OP: ADDITION
AnswerType:
Arithmetic Expression
Answer:
23+40+10=73
Q: Which period was Wolf executive and player personnel director with the Oakland Raiders longer for, 1963-1974 or 1979-1989?
A: 1963-1974	Wolf only had a brief stint with the Jets between 1990 and 1991, while most of his major contributions occurred as an executive and player personnel director with the Oakland Raiders (1963-1974, 1979-1989), and later as General Manager...	AnswerType:
Passage Span
Answer:
1979-1989	AnswerType:
Passage Span
Answer:
1979-1989	Top-1 OP: SPAN
AnswerType:
Passage Span
Answer:
1963-1974
Table 8: The cases from the development set of DROP. The predictions from the state-of-the-art model NumNet+ and QDGAT are shown. The last column indicates our predicted answers and Top-1 operations.
could compensate for the discrete reasoning capability of language models.
We show two examples from the development set of DROP to illustrate the effectiveness of our model by comparing the results of different models in Table 8. The first example shows that operation is essential for the prediction of answer type. NumNet+ and QDGAT fail to predict the correct answer since the answer type of "how many" questions are wrongly predicted to Count. In contrast, OPERA can capture the ADDITION operation, which prompts the model to answer it with an arithmetic expression predictor. The second example shows that OPERA has stronger reasoning capability. In the example, though NumNet+ and QDGAT correctly predict the answer type, the final answer is wrong. OPERA can utilize more semantic information for answer prediction with the help of the operation-pivoted discrete reasoning mechanism.
We propose a novel framework OPERA for machine reading comprehension requiring discrete reasoning. Lightweight and one-shot operations and heuristic rules to map questions to an operation set are systematically designed. OPERA can leverage the operations to enhance the model's reasoning capability and interpretability. Experiments on DROP and RACENUM demonstrate that OPERA achieves remarkable performance. Further visualization and analysis verify its interpretability.
We would like to thank all the anonymous reviewers for their useful feedback. This work is supported by the project of the National Natural Science Foundation of China (No.U1908216) and the
National Key Research and Development Program of China (No. 2020AAA0108600).
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A.1 Template-Based Heuristic Rules
Operations / Examples / Templates
ADDITION/DIFF
How many more yards was Kris Browns's first field goal over his second?
How many [Slot] more/less [Slot] over [Slot]?
MAX/MIN
How many yards was the longest field goal in the game?
how many yards [Slot] longest/shortest [Slot]?
ARGMAX/ARGMIN
Which player had the longest touchdown reception?
Which player [Slot] longest/shortest [Slot]?
ARGMORE/ARGLESS
Who scored more field goals, David Akers or John Potter?
Who [Slot] more/less, [Slot] or [Slot]?
COUNT
How many field goals did Kris Brown kick?
How many field goals [slot]?
KEY_VALUE
How many percent of Forth Worth households owned a car?
How many percent of [Slot]
SPAN
Which team scored the final TD of the game?
Which team [Slot]
Table 9: All the operations, the corresponding examples and templates.
In this section, we propose some general template-based heuristic rules to illustrate mapping from questions to operations. For example, to detect the operations intimated by the question "how many yards was the longest field goals" in Table 9, we design a question template  $OP_{pat}$  "how many yards [Slot] longest/shortest [Slot]" which maps matched questions to the operation MAX. Meanwhile, we humanly evaluate the quality of the heuristic rules. Specifically, we sample 50 instances from the training set and ask three annotators to label the required operations for each question manually. The final average F1 score of the three annotators is  $86\%$ , which indicates that the heuristic rules can correctly predict most of the operations, while still  $14\%$  to be noise for training.
A.2 Distribution of the Operations
We analysis the distribution of operations in the training set of DROP, where ADDITION, DIFF and SPAN together accounts for more than  $85\%$ . For other questions with span answers that requires sorting or comparison, some specific operations are involved, such as ARGMAX / ARGMIN / ARGMORE / ARGLESS and KEY_VALUE.
Figure 4: The Distribution of operations in the training set of DROP.
A.3 Details of Prediction Module
In this section, we reveal the architecture details of the prediction module, including a prediction module for answer type and five label predictors corresponding to different answer types.  $\mathsf{FFN}(\cdot)$  means a feed-forward network that consists of two linear projections with a GeLU activation function (Hendrycks and Gimpel, 2016) and a layer normalization mechanism (Ba et al., 2016).
Answer Type The probability distribution of answer type choices  $p(T|Q, P, O)$  is derived by a  $|\mathcal{T}|$ -classifier with  $\mathbf{h}^Q$ ,  $\mathbf{h}^P$  and  $\mathbf{h}^E$  as input:
$$
\mathbf {h} ^ {E} = \sum_ {O _ {i} \in \mathcal {O P}} p \left(O _ {i} \mid Q, P\right) \mathbf {E} _ {i} ^ {O P}, \tag {11}
$$
$$
p (T | Q, P, O) \propto \mathsf {F F N} ([ \mathbf {h} ^ {E}; \mathbf {h} ^ {Q}; \mathbf {h} ^ {P} ]),
$$
where  $\mathbf{h}^Q$  and  $\mathbf{h}^P\in \mathbb{R}^{d_h}$  is the representation vector of question and passage calculated by weighted pooling with  $\mathbf{H}^Q$  and  $\mathbf{H}^P$ , respectively.  $\mathbf{E}^{OP}$  is the embedding matrix of operations.
Question/Passage Span Following MTMSN (Hu et al., 2019), we use a question-aware decoding strategy to predict the start and end indices of the answer span. Specifically, we first compute a question representation vector  $\mathbf{g}^Q$  via weighted pooling. Then derive the probabilities of the start and end indices of the answer span denoted as  $p_s$  and  $p_e$ :
$$
\mathbf {M} = \left[ \mathbf {h} ^ {O P}; \mathbf {H}; \mathbf {H} \odot \mathbf {g} ^ {Q} \right], \tag {12}
$$
$$
p _ {s}, p _ {e} \propto \operatorname {F F N} (\mathbf {M}),
$$
where  $\odot$  denotes element-wised product.  $\mathbf{h}^{OP}$  is derived by Eq. 6 and  $\mathbf{H}$  is the representation of input sequence from context encoder.
Count The count predictor is a 10-classifier with the operation-aware representation, all the mentioned number representation, question and passage representations as input. Specifically, when  $N$  numbers exist, we gather the representation of all numbers  $\mathbf{U} = (\mathbf{u}^1,\mathbf{u}^2,\dots,\mathbf{u}^N)\in \mathbb{R}^{N\times d_h}$  from  $\mathbf{H}$  and compute a global representation vector of numbers as  $\mathbf{h}^U$ . Then compute the probability distribution of count answer  $p_c$ :