The other is GRL-SVO(NUP), a fast heuristic providing a total order directly.

Cylindrical Algebraic Decomposition (CAD) is one of the pillar algorithms of symbolic computation, and its worst-case complexity is double exponential to the number of variables. Researchers found that variable order dramatically affects efficiency and proposed various heuristics. The existing learning-based methods are all supervised learning methods that cannot cope with diverse polynomial sets.This paper proposes two Reinforcement Learning (RL) approaches combined with Graph Neural Networks (GNN) for Suggesting Variable Order (SVO). One is GRL-SVO(UP), a branching heuristic integrated with CAD. The other is GRL-SVO(NUP), a fast heuristic providing a total order directly. We generate a random dataset and collect a real-world dataset from SMT-LIB. The experiments show that our approaches outperform state-of-the-art learning-based heuristics and are competitive with the best expert-based heuristics. Interestingly, our models show a strong generalization ability, working well on various datasets even if they are only trained on a 3-var random dataset.

The other is GRL-SVO(NUP), a fast heuristic providing a total order directly.

Cylindrical Algebraic Decomposition (CAD) is one of the pillar algorithms of symbolic computation, and its worst-case complexity is double exponential to the number of variables. Researchers found that variable order dramatically affects efficiency and proposed various heuristics. The existing learning-based methods are all supervised learning methods that cannot cope with diverse polynomial sets.This paper proposes two Reinforcement Learning (RL) approaches combined with Graph Neural Networks (GNN) for Suggesting Variable Order (SVO). One is GRL-SVO(UP), a branching heuristic integrated with CAD. The other is GRL-SVO(NUP), a fast heuristic providing a total order directly.

We present a new methodology for utilising machine learning technology in symbolic computation research. We explain how a well known human-designed heuristic to make the choice of variable ordering in cylindrical algebraic decomposition may be represented as a constrained neural network. This allows us to then use machine learning methods to further optimise the heuristic, leading to new networks of similar size, representing new heuristics of similar complexity as the original human-designed one. We present this as a form of ante-hoc explainability for use in computer algebra development.

Computer Algebra Systems (e.g. Maple) are used in research, education, and industrial settings. One of their key functionalities is symbolic integration, where there are many sub-algorithms to choose from that can affect the form of the output integral, and the runtime. Choosing the right sub-algorithm for a given problem is challenging: we hypothesise that Machine Learning can guide this sub-algorithm choice. A key consideration of our methodology is how to represent the mathematics to the ML model: we hypothesise that a representation which encodes the tree structure of mathematical expressions would be well suited. We trained both an LSTM and a TreeLSTM model for sub-algorithm prediction and compared them to Maple's existing approach. Our TreeLSTM performs much better than the LSTM, highlighting the benefit of using an informed representation of mathematical expressions. It is able to produce better outputs than Maple's current state-of-the-art meta-algorithm, giving a strong basis for further research.

Symbolic Computation algorithms and their implementation in computer algebra systems often contain choices which do not affect the correctness of the output but can significantly impact the resources required: such choices can benefit from having them made separately for each problem via a machine learning model. This study reports lessons on such use of machine learning in symbolic computation, in particular on the importance of analysing datasets prior to machine learning and on the different machine learning paradigms that may be utilised. We present results for a particular case study, the selection of variable ordering for cylindrical algebraic decomposition, but expect that the lessons learned are applicable to other decisions in symbolic computation. We utilise an existing dataset of examples derived from applications which was found to be imbalanced with respect to the variable ordering decision. We introduce an augmentation technique for polynomial systems problems that allows us to balance and further augment the dataset, improving the machine learning results by 28\% and 38\% on average, respectively. We then demonstrate how the existing machine learning methodology used for the problem $-$ classification $-$ might be recast into the regression paradigm. While this does not have a radical change on the performance, it does widen the scope in which the methodology can be applied to make choices.

In recent years there has been increased use of machine learning (ML) techniques within mathematics, including symbolic computation where it may be applied safely to optimise or select algorithms. This paper explores whether using explainable AI (XAI) techniques on such ML models can offer new insight for symbolic computation, inspiring new implementations within computer algebra systems that do not directly call upon AI tools. We present a case study on the use of ML to select the variable ordering for cylindrical algebraic decomposition. It has already been demonstrated that ML can make the choice well, but here we show how the SHAP tool for explainability can be used to inform new heuristics of a size and complexity similar to those human-designed heuristics currently commonly used in symbolic computation.

This paper discusses and evaluates ideas of data balancing and data augmentation in the context of mathematical objects: an important topic for both the symbolic computation and satisfiability checking communities, when they are making use of machine learning techniques to optimise their tools. We consider a dataset of non-linear polynomial problems and the problem of selecting a variable ordering for cylindrical algebraic decomposition to tackle these with. By swapping the variable names in already labelled problems, we generate new problem instances that do not require any further labelling when viewing the selection as a classification problem. We find this augmentation increases the accuracy of ML models by 63% on average. We study what part of this improvement is due to the balancing of the dataset and what is achieved thanks to further increasing the size of the dataset, concluding that both have a very significant effect. We finish the paper by reflecting on how this idea could be applied in other uses of machine learning in mathematics.

Cylindrical Algebraic Decomposition (CAD) is a key proof technique for formal verification of cyber-physical systems. CAD is computationally expensive, with worst-case doubly-exponential complexity. Selecting an optimal variable ordering is paramount to efficient use of CAD. Prior work has demonstrated that machine learning can be useful in determining efficient variable orderings. Much of this work has been driven by CAD problems extracted from applications of the MetiTarski theorem prover. In this paper, we revisit this prior work and consider issues of bias in existing training and test data. We observe that the classical MetiTarski benchmarks are heavily biased towards particular variable orderings. To address this, we apply symmetries to create a new dataset containing more than 41K MetiTarski challenges designed to remove bias. Furthermore, we evaluate issues of information leakage, and test the generalizability of our models on the new dataset.