Despite some similarities that have been pointed out in the literature, the parallelism between automated planning and natural language processing has not been fully analysed yet. However, the success of Transformer-based models and, more generally, deep learning techniques for NLP, could open interesting research lines also for automated planning. Therefore, in this work, we investigate whether these impressive results could be transferred to planning. In particular, we study how a BERT model trained on plans computed for three well-known planning domains is able to understand how a domain works, its actions and how they are related to each other. In order to do that, we designed a variation of the typical masked language modeling task which is used for the training of BERT, and two additional experiments into which, given a sequence of consecutive actions, the model has to predict what the agent did previously (Previous Action Prediction) and what it is going to do next (Next Action Prediction).
In 2007, the work in [
In this work, we focus on one of the most recent deep learning architectures for NLP: BERT
[
Inspired by the parallelism pointed out in [
BERT (Bidirectional Encoder Representations from Transformer) [
At first, each token is associated to an index. Next, an embedding layer calculates a vector of real numbers for each token, as in initial representation of its meaning. Considering a sequence of tokens of length , each token ∈ is represented by a vector ∈ R. We denote as ∈ R× the overall matrix of token embeddings.
After calculating this representation, these vectors are processed by several encoding layers. Each layer applies in parallel multiple self-attention mechanisms, called heads, into which the relation between each possible pair of tokens is computed. This mechanism produces a matrix , ∈ R× , into which is the number of the encoding layer and is the head number. For each token ∈ , the vector ∈ , contains the attention weights that represent how much is related to the other tokens in . In order to calculate these weights, in each head the input representation of the token sequence ∈ R× is projected into three new representations called key (), query () and value ( ) with three weight matrices , and : = × , = × , = × Then, the self-attention weights are calculated using the softmax function on the scaled dotproduct between and . Finally, this mechanism computes a new vector representation of the input tokens () by multiplying the attention weights for .
︂( × ⊺ )︂ = √ , = × Given that in each encoding layer there are several heads (typically 8, 12, 16 or 24), in order to create a single representation provided by the multi-head attention mechanism the result of each head is concatenated and then passed to a feed-forward layer.
As described in [
In the original model, designed for NLP tasks, the training was performed using 16GB of
documents for a total of more than 3 billions of words [
In order to perform PLM, we approach the planning domains as a NLP task. In fact, our BERT model is trained using a slight variation of the typical language modeling task that we call Planning Language Modeling. In this technique, given a sequence of contiguous actions ¯ composed of actions , with ∈ [1, ], we consider each action as a separated token, like words in a sentence. First, we divide the action sequence into separated tokens, using the WordPiece tokenizer [23]. Next, we substitute a certain percentage of actions with the special token [MASK] and give this incomplete sequence as input. The BERT model has the task of predicting the missing actions from the overall context of the action sequence. If the model is capable of performing this operation, we claim that it has the ability to understand how the planning domain works, the impact of the actions, when they are performed by an agent, and which efects they have. In order to perform the planning language modeling, the BERT model needs processing a large quantity of these training instances, progressively adjusting the training weights of the model with the backpropagation algorithm until it reaches a satisfying level of accuracy.
Our model architecture is very similar to the one proposed in [
(a) (b)
Our choice is to develop a multi-domain model, considering Logistics, Blocksworld and Satellite. With a single model, following the current trend of multi-language models, we want to verify whether BERT could understand diferent domains at the same time. Next, we aim to study the training process for a model which was already trained on a diferent domain. The model was first trained on Logistics. It took 40 epochs to understand the domain and reach a good accuracy. Then we trained the model on Blocksworld and Satellite and it took less than 10 epochs to achieve good performance. Therefore, we verified that adding another domain to a pre-trained model does not require another, full training procedure. Instead, the model can easily adapt to the new domain as in fine-tuning operations, requiring only a few epochs to achieve a good accuracy on the new domain.
In this section, we present a simple example of how we structure our planning language modeling task. For simplicity’s sake, let us assume that our BERT works only for this domain. We use a very simple plan instance in the well-known Blocksworld domain. In this domain, the agent has the goal of building one or more stacks of blocks, and only one block may be moved at a time. The agent can perform four types of actions: Pick-Up a block from the table, Put-Down a block on the table, both these operations take only one argument, Stack which puts a block on top of another one and Unstack which removes a block that is on another one, both have two arguments. We assume that our plans involve at most 22 blocks, therefore we have 22 possible Pick-Up action, 22 Put-Down actions, 22 * 21 Stack actions and 22 * 21 Unstack actions, for a total number of 968 actions.
Let’s consider the plan formed by the following 6 actions: (Unstack Block_R Block_G), (PutDown Block_R), (Unstack Block_M Block_O), (Stack Block_M Block_R), (Unstack Block_G Block_L), (Stack Block_G Block_M) with ids corresponding to indices 59, 462, 121, 703, 309 and 240. Next, we mask 1 action (17%) chosen randomly. Assuming that the special token [MASK] has index 0, the input of our BERT model will be the following sequence: [59, 462, 121, 0, 309, 240].
This sequence is then processed by the embedding layer and by all the encoding layers, producing a final representation of each element as a vector of 768 real numbers. Given that the fourth element of the sequence is masked, its embedding vector is passed to a feed-forward ƒlayer composed of 968 neurons (one for each possible action of the domain) with the softmax activation function. Given that the original action had index 703, we want in output a vector formed by all zeroes except for a 1 at index 703. Therefore, the result produced by the feedforward layer is compared with the desired output. This procedure is repeated for every training instance, evaluating the overall error made by the architecture and calculating a loss function, which will be used by the backpropagation algorithm to adjust the model weights.
In Figure 1a we show a simplified representation of the model architecture performing the planning language modeling task. The actions are encoded into vectors by several encoding blocks and, in the last step, the vectorial representation of [MASK] is passed to a feed-forward layer with softmax activation function that predicts a 1 at the index 703 and 0 for all the other indexes.
In order to properly train the model, there are some necessary choices that the user must make. Although typically BERT models treat a specific language, there are several multi-language models. Given that a planning domain has its own set of actions, predicates, and rules, we can consider it as a sort of an independent language. Therefore, the user should evaluate whether training a BERT model for each domain, perhaps specialising and improving its predictive capabilities, or building a multi-domain model, selecting which domains to consider and their quantity.
Next, an important constraint of the BERT model is that it must have a predefined vocabulary of actions. Therefore, during the training process the user has to define a predefined set of objects for each domain and generate all the possible actions that an agent could perform with them. Although the predefined vocabulary is sensitive to the object names, this operation only requires to set a maximum number of objects. In fact, a mapping algorithm can be easily implemented for translating new names into predefined ones.
As with all deep learning techniques, a crucial aspect for obtaining good performance is the number of training instances, i.e. the number of plans necessary to train the BERT model. The choices of how many domains to consider for training and how many objects to include have a Domains Logistics Blocksworld Satellite serious impact on the number of plans which have to be collected or generated for the training of the model, the amount of time required to this process, and its dificulty.
In our model, we have selected three well-known planning domains: Logistics, Blocksworld and Satellite. In Table 1 we report the characteristics of our dataset. Please note that while in Blocksworld the only objects present in the domain are the blocks, in the other two domains there are several types of objects (for instance, Logistics includes airplanes, airports, locations, cities, trucks and packages) and we report the overall sum of them for simplicity’s sake.
In order to train the multi-domain model, the typical approach is simply considering a single training set composed by the plans for all the three domains. However, we have also experimentally verified that it is possible to add a domain to a pre-trained model extending its vocabulary and continuing its training with additional epochs using the plans generated from the new domain. In this phase, however, we have noticed that it is useful to include also a percentage (such as 25%) of plans belonging to the already processed domains.
The overall dimension of our data set is about 350 MB of data; therefore, it is definitively
smaller than standard BERT architectures [
Another technique that can be useful in training a BERT model is data augmentation. In this procedure, we divide the longest plans in several subsequences with random dimension and add them to the dataset. With this technique, the training set can contain not only entire plans but also smaller sequences of actions. In order to avoid including a large number of very similar instances, which could create overfitting issues to the model, we assure that plans which are too similar to the ones already present in the dataset are not added. In our context, for Logistics and Blocksworld we divided the plans with more than 20 actions, randomly selecting a subsequence of the plan. The subsequence is added to the training set if the similarity calculated by the Ratclif-Obershelp algorithm [ 24] with the other plans is lower than 0.4, otherwise it isn’t. While this operation increases the predictive performance of the model for these two domains, in Satellite data augmentation did not improve the results. We claim that this is due to the smaller size of the Satellite plans, with respect to the other two domains. In Table 1 we report the dimension of the original dataset (with complete plans) and the augmented dataset, with the smaller subsequences generated by the data augmentation process.
In this section, we present the set of learning tasks we designed for our model. The model was trained on 2 NVIDIA V100, for 120 epochs, with a training time of almost 130 hours.
First of all, we test the capabilities of our model to perform the planning language modeling using a test set composed by plans which were not used during the training of the algorithm. In this experiment, we check if the predictions of the masked input actions are correct. Given that the softmax activation function of the output layers actually predicts a score between 0 and 1 for all the possible actions of our three domains, we evaluate the planning language masked in two configurations. In the first one ( Top-1 accuracy), we consider only the action predicted with the highest score. Given the very high number of possible classes in output (one for each action, i.e. more than 40 thousands in our multidomain model), we also use as an evaluation metric the Top-5 accuracy [25] into which we consider the set of the 5 actions with the highest score, and, if the original action belongs to the set, we consider that the model correctly predicts the masked action. The planning language modeling is evaluated using the following percentages of masked actions: 10%, 15%, 25% and 50%.
We also design two additional tasks to test the capabilities of the model to handle sequences of actions and to infer some of the domain knowledge such as actions’ preconditions and efects. For both of these tasks, we use the same plans used for the planning language modeling. In the first task, called Next Action Prediction, we ask our model to predict the action that follows an input action sequence. In the second, called Previous Action Prediction, we ask our model to predict the action that precedes an input action sequence. In detail, given an action sequence of length , for Next Action Prediction, we pass the first − 1 actions to the model and ask the model to predict the -th action. Similarly, for Previous Action Prediction the model should predict the ifrst action of the sequence while the remaining actions are passed in input. We evaluate Next Action Prediction and Previous Action Prediction using diferent input sequence lengths (i.e. 3, 5, 10, 20) and with the entire plan without the last and first action, respectively (Tot).
For the evaluation of these tasks, in our opinion, verifying whether the predicted action matches the one in the input plan is not enough. In fact, the input action sequence filled with the predicted action might be valid even though the predicted action does not match the one in the label. Moreover, we are not providing the model the initial state or the goal (see Section 5), therefore the model cannot always predict the best action based only on an incomplete set of actions. For this reason we use VAL [26] to check if, starting from the initial state of the problem, the action sequence, which also comprehends the action predicted by the model, can be executed or not, providing insight into how the model is able to learn the inner or working of a planning domain1. 1Please note that VAL does not always validate our plan as the goal of the problem is often not reached; nonetheless, we use VAL errors to determine whether the sequence is valid or not
10 %
Considering all three domains, in Table 2 we show the results, in terms of Top-1 and Top-5, of the planning language modeling task, using 3000 plans, 1000 for each domain, as a test set. In this experiment, we masked diferent percentages of actions: 10%, 15%, 25% and 50%. As it should be expected, the accuracy decreases proportionally to the increasing of the percentage of masked actions (i.e. less input data). The overall performances are more than acceptable, in particular for the 50% in Top-5, where the model, analyzing just half of the plan, is able to correctly predict almost half of the actions, highlighting a deep comprehension of the context. With less masked actions, the accuracy is very high, with more than 80% in Top-5 for 10% and 15% and good performance also in terms of Top-1.
The results for the Next Action Prediction task in terms of Top-1, Top-5 are shown in Table 3. In each column, we consider a diferent length for the input sequence. For instance, if the length is 5 we have 5 consecutive actions in input and we want to predict the 6-th.
In each row we can notice the same trend: the accuracy increases with the length of the plan, except for the plans with length 10, where there is a drop of the performances. However, in terms of Top-5 we have very good results even with small action sequences, underlining a deep understanding of the inner workings of the planning domain. Focusing only on the Top-1 correct, the results are much lower. That could be due to the fact that our model does not receive any information about the agent’s goal and therefore it can only predict a set of executable actions, into which it is possible to find the right one.
To better understand these results, we study the performance for each considered domain for the prediction of valid actions. In Figure 2, we analyze the accuracy in terms of Top-1 and Top-5 accuracy. On the left, we can see how Top-1 accuracy increases proportionally to the length of the plans in the Logistics domain, from 51% to 92%, while it’s almost constant in Satellite domain. In the Blocksworld domain the results are excellent with shorter plans, with 95%, while they drop with length equals to 10, to 22%. Then they recover with longer plans, arriving to 84%. As it can be seen in Figure 2, the performance drop for length 10 regards almost only the Blocksworld domain. We will address this issue in Section 5. In terms of Top-5 accuracy, on the right of Figure 2 we can see the same trend as the previous task on the performances, with good results in predictions, on Logistics up to to 99% and on Satellite up to 92% .
In Table 4 we report the results for the Previous Action Prediction task using the same metrics and format used in the evaluation of the Next Action Prediction task. Similarly to the Next Action Prediction results, we can notice that, for each row, the accuracy increases as the input length increases. These results are very promising, even if they are slightly lower than the ones obtained in Next Action Prediction. The performances increase with the length of the plan and the collapse on length 10, seen in the previous task, is not present.
As we did previously, in Figure 3 we show the results for each domain in terms of Top-1 and Top-5 accuracy (considering also the valid actions) on the Previous Action Prediction task. In terms of Top-1, we can notice that the performances on the Logistics and the Blocksworld domains remains quite constant, (with a maximum of 47% and 57% respectively), while on the Satellite domain they increase proportionally with the length of the plans, from 33% to 84%. In terms of Top-5, we have higher results for all three domains. Howevever, while the performance in the Logistics domain remains almost the same for all lengths, we can notice a small increase in Blocksworld and a strong increase in Satellite. This domain in particular has some issues with actions sequences of length 3 (36%) but reaches 86% at length 10, showing that for this domain we need an adequate number of actions in order to have significant predictive results.
The results presented in Section 4.2 show that a deep learning model such as BERT, trained on a set of action sequences from well-known planning domains, can efectively accomplish three predictive tasks: the planning language modeling, i.e. predicting actions from context, the prediction of the next action given sequences of diferent length and, in the same conditions, the prediction of the previous action.
In our opinion, this demonstrates that BERT is able to understand how a planning domain works and which rules are followed by the action sequences. Despite a great number of possible actions to predict (especially for the Satellite domain, which has more than 33k actions in our configuration), our results in terms of Top-1 and Top-5 accuracy show that the model is able to identify which are the most probable ones. Although we have relatively bad results in terms of Top-1 for the Next Action Prediction and Previous Action Prediction tasks considering only if the action belonged to the original plan, the performance are much higher considering the valid actions. This can be also an indicator that our model does not simply memorize a lot of action sequences, but it is really capable to adapt its knowledge with new plans, avoiding overfitting issues.
A problem of this first implementation is that we do not provide any data on the initial state and the goal to our model. This could be the reason for the performance drop in Blocksworld with sequences of length 10 in the Next Action Prediction task. In fact, while the model seems capable of predicting the next action with smaller sequences, when the complexity increases it could be necessary to provide more data in order to direct its reasoning. However, it is interesting to note that for Logistics this does not happen. This may be due to the fact that information about initial state could be indirectly contained in the action sequences. While we will conduct a further analysis of these results, considering also other domains, we think that including the initial state and the goal in the model could improve the results and avoid this performance drop.
While we have verified that a BERT model has this learning capability, we are also conscious of several limits. For instance, as we show in Table 1, the number of plans necessary to train such a model is very high. In a real-world application, this can be a huge problem because it would require a long time to collect the necessary data to exploit this kind of model. While our data augmentation technique shows interesting results, perhaps similar results could be obtained with simpler models, which could require less training data. Another related problem with this kind of architectures is the training time and the computational resources required. An interesting line of research could be apply knowledge distillation techniques and build smaller versions of our model [27].
Moreover, the original BERT for natural language processing [
In the last few years, deep learning techniques started to surface across several contexts and tasks related with automated planning.
For instance, the work in [12] exploits variational autoencoders based on neural networks to create latent and symbolic representations of actions starting from pairs of images which describe a transition between two states. This representation can be further exploited for other purposes, such as Goal Recognition, which is defined as the task of recognising the goal that an agent is trying to achieve from observations about the agent’s behaviour in the environment [28]. Using only image-based domains such as MNIST or 8-Puzzle, the work [13] first computes a latent representation using the tool proposed by [12] and then analyses the sequence of actions using an LSTM Neural Network [29] for predicting the goal.
Goal Recognition has seen several other applications of deep learning techniques. For instance, the work [15] presents an LSTM network to infer player goals from the observation of their low-level actions in the context of digital games. However, more general works such as [30] and [31] used the classical domains of automated planning and, starting from partial observations of states or actions represented symbolically, trained neural networks or other machine learning algorithms in order to predict the agent’s goal.
Deep learning techniques are also used in many path planning applications with robots [32]. For example, in [33] a neural network is used to process data from robot’s sensor in order to get an approximate direction. This approximation is then used to build a navigation system. Furthermore, the controller for an autonomous mobile robot presented in [34] implements a feed forward neural network which predicts the steering angle given the obstacles distances from the left, right and front directions.
Another problem which has been addressed by deep learning techniques is the learning of heuristic functions for classical planning. For instance, the authors of [11] trained a neural network to compute the distance between two states and used the output measure as a heuristic. However, one of the drawbacks of their approach is the need to train a neural network for each planning task. The work in [35] solves this issue for a subset of planning domains by learning a strong heuristic function for PDDL domains with an underlying grid structure. The authors implement a convolutional neural network structure that takes the current state and the goal state as input and predicts both the next action and the heuristic value.
To the best of our knowledge, ours is the first implementation of a Transformer-based architecture such as BERT in the context of automated planning. While most of the works show slight variations or improvements [22, 36], versions for some specific contexts [ 37] or jargons [38], the work in [19] shows how these techniques can be useful even for artificial languages such as programming languages, proposing a BERT model for evaluating the semantic similarity between two portions of code. Their model is trained using publicly available open source code repositories from GitHub. In [39], a similar model for code summarization and generation is proposed. An interesting study about the capabilities of such models is shown in [40].
In this work, we trained a BERT model in the context of automated planning. While this type
of deep learning architecture has completely changed the state-of-the art for several NLP tasks
[
In our experimental evaluation, we designed two tasks in order to verify if the model has the capability of understanding the inner working a planning domain: Next Action Prediction and Previous Action Prediction, into which given a sequence of actions the model has to predict the action that immediately follows or precedes the sequence. Our models obtains very promising results and we show that it has a high accuracy in predicting action that can be actually executed by an agent, before or after an action sequence.
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