A comparison of three heuristics to choose the variable ordering for CAD

Cylindrical algebraic decomposition (CAD) is a key tool for problems in real algebraic geometry and beyond. When using CAD there is often a choice over the variable ordering to use, with some problems infeasible in one ordering but simple in another. Here we discuss a recent experiment comparing three heuristics for making this choice on thousands of examples

Using Machine Learning to Decide When to Precondition Cylindrical Algebraic Decomposition With Groebner Bases

Cylindrical Algebraic Decomposition (CAD) is a key tool in computational algebraic geometry, particularly for quantifier elimination over real-closed fields. However, it can be expensive, with worst case complexity doubly exponential in the size of the input. Hence it is important to formulate the problem in the best manner for the CAD algorithm. One possibility is to precondition the input polynomials using Groebner Basis (GB) theory. Previous experiments have shown that while this can often be very beneficial to the CAD algorithm, for some problems it can significantly worsen the CAD performance. In the present paper we investigate whether machine learning, specifically a support vector machine (SVM), may be used to identify those CAD problems which benefit from GB preconditioning. We run experiments with over 1000 problems (many times larger than previous studies) and find that the machine learned choice does better than the human-made heuristic

Using Machine Learning to Improve Cylindrical Algebraic Decomposition

Cylindrical Algebraic Decomposition (CAD) is a key tool in computational algebraic geometry, best known as a procedure to enable Quantifier Elimination over real-closed fields. However, it has a worst case complexity doubly exponential in the size of the input, which is often encountered in practice. It has been observed that for many problems a change in algorithm settings or problem formulation can cause huge differences in runtime costs, changing problem instances from intractable to easy. A number of heuristics have been developed to help with such choices, but the complicated nature of the geometric relationships involved means these are imperfect and can sometimes make poor choices. We investigate the use of machine learning (specifically support vector machines) to make such choices instead. Machine learning is the process of fitting a computer model to a complex function based on properties learned from measured data. In this paper we apply it in two case studies: the first to select between heuristics for choosing a CAD variable ordering; the second to identify when a CAD problem instance would benefit from Groebner Basis preconditioning. These appear to be the first such applications of machine learning to Symbolic Computation. We demonstrate in both cases that the machine learned choice outperforms human developed heuristics.This work was supported by EPSRC grant EP/J003247/1; the European Union’s Horizon 2020 research and innovation programme under grant agreement No 712689 (SC2); and the China Scholarship Council (CSC)

Performance indicators of three kinds of WER.

Waterborne epoxy resin (WER), a cleaning material with exceptional high-temperature resistance, has attracted much attention to modify emulsified asphalt in the pavement material field. Epoxy value is the critical characteristic index of WER. In this research, three WER with the epoxy values of 0.20 eq/100g, 0.44 eq/100g, and 0.51 eq/100g were utilized as asphalt modifiers. The influence of epoxy value on WER-EA was investigated by comparing the rheological properties of three kinds of WER emulsified asphalt (WER-EA). The modification mechanism of WER-EA has been analyzed using FTIR and SEM. The results demonstrate that different WER-EA resulted in significantly different rheological properties. WER-EA with the epoxy value of 0.20 eq/100g (E20) performed best at high temperatures, with a maximum increase of 17477% in G*/sinδ compared to the neat asphalt and a maximum increase of 66.3% in G*/sinδ compared to the other two WER-EA. WER-EA with 0.44 eq/100g epoxy value (E44) performed best at low temperatures, with a maximum increase in m value of 39.4% and a maximum decrease in S value of 33.3% compared to the other two WER-EA. In addition, the interpenetrating polymer network (IPN) in E20 was observed to be more solid and stable, and IPN in E44 was more uniform. To summarize, lower epoxy value led to a higher degree of WER reaction and higher content of rigid groups, which is more conducive to optimizing the high-temperature property of WER-EA. WER with moderate epoxy value resulted in a low content of polar bonds and thus high content of flexible segments, which helps emulsified asphalt to form a more uniform IPN.</div

The reaction of the WER system in emulsified asphalt.

The reaction of the WER system in emulsified asphalt.</p

Sample sizes of TS/MSCR test (a) and BBR test (b).

Sample sizes of TS/MSCR test (a) and BBR test (b).</p

Test procedures of rheological property.

Waterborne epoxy resin (WER), a cleaning material with exceptional high-temperature resistance, has attracted much attention to modify emulsified asphalt in the pavement material field. Epoxy value is the critical characteristic index of WER. In this research, three WER with the epoxy values of 0.20 eq/100g, 0.44 eq/100g, and 0.51 eq/100g were utilized as asphalt modifiers. The influence of epoxy value on WER-EA was investigated by comparing the rheological properties of three kinds of WER emulsified asphalt (WER-EA). The modification mechanism of WER-EA has been analyzed using FTIR and SEM. The results demonstrate that different WER-EA resulted in significantly different rheological properties. WER-EA with the epoxy value of 0.20 eq/100g (E20) performed best at high temperatures, with a maximum increase of 17477% in G*/sinδ compared to the neat asphalt and a maximum increase of 66.3% in G*/sinδ compared to the other two WER-EA. WER-EA with 0.44 eq/100g epoxy value (E44) performed best at low temperatures, with a maximum increase in m value of 39.4% and a maximum decrease in S value of 33.3% compared to the other two WER-EA. In addition, the interpenetrating polymer network (IPN) in E20 was observed to be more solid and stable, and IPN in E44 was more uniform. To summarize, lower epoxy value led to a higher degree of WER reaction and higher content of rigid groups, which is more conducive to optimizing the high-temperature property of WER-EA. WER with moderate epoxy value resulted in a low content of polar bonds and thus high content of flexible segments, which helps emulsified asphalt to form a more uniform IPN.</div

<i>m</i> and <i>S</i> at different temperatures and dosages.

Waterborne epoxy resin (WER), a cleaning material with exceptional high-temperature resistance, has attracted much attention to modify emulsified asphalt in the pavement material field. Epoxy value is the critical characteristic index of WER. In this research, three WER with the epoxy values of 0.20 eq/100g, 0.44 eq/100g, and 0.51 eq/100g were utilized as asphalt modifiers. The influence of epoxy value on WER-EA was investigated by comparing the rheological properties of three kinds of WER emulsified asphalt (WER-EA). The modification mechanism of WER-EA has been analyzed using FTIR and SEM. The results demonstrate that different WER-EA resulted in significantly different rheological properties. WER-EA with the epoxy value of 0.20 eq/100g (E20) performed best at high temperatures, with a maximum increase of 17477% in G*/sinδ compared to the neat asphalt and a maximum increase of 66.3% in G*/sinδ compared to the other two WER-EA. WER-EA with 0.44 eq/100g epoxy value (E44) performed best at low temperatures, with a maximum increase in m value of 39.4% and a maximum decrease in S value of 33.3% compared to the other two WER-EA. In addition, the interpenetrating polymer network (IPN) in E20 was observed to be more solid and stable, and IPN in E44 was more uniform. To summarize, lower epoxy value led to a higher degree of WER reaction and higher content of rigid groups, which is more conducive to optimizing the high-temperature property of WER-EA. WER with moderate epoxy value resulted in a low content of polar bonds and thus high content of flexible segments, which helps emulsified asphalt to form a more uniform IPN.</div

<i>G*</i>, <i>δ</i> and <i>G*</i>/sin<i>δ</i> at different temperatures.

Waterborne epoxy resin (WER), a cleaning material with exceptional high-temperature resistance, has attracted much attention to modify emulsified asphalt in the pavement material field. Epoxy value is the critical characteristic index of WER. In this research, three WER with the epoxy values of 0.20 eq/100g, 0.44 eq/100g, and 0.51 eq/100g were utilized as asphalt modifiers. The influence of epoxy value on WER-EA was investigated by comparing the rheological properties of three kinds of WER emulsified asphalt (WER-EA). The modification mechanism of WER-EA has been analyzed using FTIR and SEM. The results demonstrate that different WER-EA resulted in significantly different rheological properties. WER-EA with the epoxy value of 0.20 eq/100g (E20) performed best at high temperatures, with a maximum increase of 17477% in G*/sinδ compared to the neat asphalt and a maximum increase of 66.3% in G*/sinδ compared to the other two WER-EA. WER-EA with 0.44 eq/100g epoxy value (E44) performed best at low temperatures, with a maximum increase in m value of 39.4% and a maximum decrease in S value of 33.3% compared to the other two WER-EA. In addition, the interpenetrating polymer network (IPN) in E20 was observed to be more solid and stable, and IPN in E44 was more uniform. To summarize, lower epoxy value led to a higher degree of WER reaction and higher content of rigid groups, which is more conducive to optimizing the high-temperature property of WER-EA. WER with moderate epoxy value resulted in a low content of polar bonds and thus high content of flexible segments, which helps emulsified asphalt to form a more uniform IPN.</div

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Waterborne epoxy resin (WER), a cleaning material with exceptional high-temperature resistance, has attracted much attention to modify emulsified asphalt in the pavement material field. Epoxy value is the critical characteristic index of WER. In this research, three WER with the epoxy values of 0.20 eq/100g, 0.44 eq/100g, and 0.51 eq/100g were utilized as asphalt modifiers. The influence of epoxy value on WER-EA was investigated by comparing the rheological properties of three kinds of WER emulsified asphalt (WER-EA). The modification mechanism of WER-EA has been analyzed using FTIR and SEM. The results demonstrate that different WER-EA resulted in significantly different rheological properties. WER-EA with the epoxy value of 0.20 eq/100g (E20) performed best at high temperatures, with a maximum increase of 17477% in G*/sinδ compared to the neat asphalt and a maximum increase of 66.3% in G*/sinδ compared to the other two WER-EA. WER-EA with 0.44 eq/100g epoxy value (E44) performed best at low temperatures, with a maximum increase in m value of 39.4% and a maximum decrease in S value of 33.3% compared to the other two WER-EA. In addition, the interpenetrating polymer network (IPN) in E20 was observed to be more solid and stable, and IPN in E44 was more uniform. To summarize, lower epoxy value led to a higher degree of WER reaction and higher content of rigid groups, which is more conducive to optimizing the high-temperature property of WER-EA. WER with moderate epoxy value resulted in a low content of polar bonds and thus high content of flexible segments, which helps emulsified asphalt to form a more uniform IPN.</div