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First-Order Tests for Toricity

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Computer Algebra in Scientific Computing (CASC 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12291))

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Abstract

Motivated by problems arising with the symbolic analysis of steady state ideals in Chemical Reaction Network Theory, we consider the problem of testing whether the points in a complex or real variety with non-zero coordinates form a coset of a multiplicative group. That property corresponds to Shifted Toricity, a recent generalization of toricity of the corresponding polynomial ideal. The key idea is to take a geometric view on varieties rather than an algebraic view on ideals. Recently, corresponding coset tests have been proposed for complex and for real varieties. The former combine numerous techniques from commutative algorithmic algebra with Gröbner bases as the central algorithmic tool. The latter are based on interpreted first-order logic in real closed fields with real quantifier elimination techniques on the algorithmic side. Here we take a new logic approach to both theories, complex and real, and beyond. Besides alternative algorithms, our approach provides a unified view on theories of fields and helps to understand the relevance and interconnection of the rich existing literature in the area, which has been focusing on complex numbers, while from a scientific point of view the (positive) real numbers are clearly the relevant domain in chemical reaction network theory. We apply prototypical implementations of our new approach to a set of 129 models from the BioModels repository.

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Notes

  1. 1.

    https://www.ebi.ac.uk/compneur-srv/biomodels-main/publ-model.do?mid=BIOMD0000000092.

  2. 2.

    https://www.ebi.ac.uk/biomodels/.

  3. 3.

    http://odebase.cs.uni-bonn.de/.

  4. 4.

    SMT technically aims at the existential fragment, which in our context is equivalent to the universal fragment via logical negation.

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Acknowledgments

This work has been supported by the interdisciplinary bilateral project ANR-17-CE40-0036/DFG-391322026 SYMBIONT [5, 6]. We are grateful to Dima Grigoriev for numerous inspiring and very constructive discussions around toricity.

Author information

Authors and Affiliations

  1. CNRS, Inria, and the University of Lorraine, Nancy, France

    Hamid Rahkooy & Thomas Sturm

  2. MPI Informatics and Saarland University, Saarbrücken, Germany

    Thomas Sturm

Authors
  1. Hamid Rahkooy
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  2. Thomas Sturm
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Corresponding author

Correspondence to Thomas Sturm .

Editor information

Editors and Affiliations

  1. University of Lille, Villeneuve d’Ascq, France

    François Boulier

  2. Coventry University, Coventry, UK

    Matthew England

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. Institute of Theoretical and Applied Mechanics, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

Appendices

A Computation Results

We present results and computation times (in seconds) of our computations on models from the BioModels database [9].

Model

m

n

\(\iota \)

\(t_\iota \)

\(\mu \)

\(t_\mu \)

\(\eta \)

\(t_\eta \)

\(\gamma \)

\(t_\gamma \)

Coset

Group

\(t_\Sigma \)

001

12

12

True

7.826

True

7.86

False

4.267

False

0.053

True

False

20.007

040

5

3

False

1.415

False

0.173

False

0.114

False

0.043

False

False

1.746

050

14

9

True

1.051

True

2.458

True

0.113

False

0.05

False

False

3.673

052

11

6

True

3.605

True

1.635

True

0.096

False

0.059

False

False

5.396

057

6

6

True

0.271

True

0.263

False

0.858

False

0.045

True

False

1.438

072

7

7

True

0.763

True

0.496

True

0.08

False

0.06

False

False

1.4

077

8

7

True

0.296

True

0.356

False

0.097

False

0.051

True

False

0.801

080

10

10

True

0.714

True

1.341

True

0.103

False

0.06

False

False

2.219

082

10

10

True

0.384

True

0.39

True

0.086

False

0.041

False

False

0.902

091

16

14

True

0.031

True

0.045

True

0.003

False

0.062

False

False

0.142

092

4

3

True

0.293

True

0.244

False

0.104

False

1.03

True

False

1.671

099

7

7

True

0.298

True

0.698

False

0.087

False

0.036

True

False

1.119

101

6

6

False

4.028

False

10.343

False

0.917

False

0.073

False

False

15.361

104

6

4

True

0.667

True

0.146

True

0.084

False

0.039

False

False

0.937

105

39

26

True

0.455

True

0.367

True

0.043

False

0.038

False

False

0.905

125

5

5

False

0.193

False

0.098

False

0.078

False

0.038

False

False

0.408

150

4

4

True

0.173

True

0.153

False

0.094

False

0.043

True

False

0.464

156

3

3

True

2.638

True

0.248

False

0.86

False

0.052

True

False

3.8

158

3

3

False

0.148

False

0.149

False

0.16

False

0.045

False

False

0.503

159

3

3

True

0.959

True

0.175

False

0.083

False

0.04

True

False

1.257

178

6

4

True

0.52

True

1.71

True

0.877

False

1.201

False

False

4.308

186

11

10

True

31.785

True

1026.464

True

1.956

False

0.095

False

False

1060.301

187

11

10

True

27.734

True

1023.648

True

0.103

False

0.062

False

False

1051.548

188

20

10

True

0.075

True

0.079

True

0.04

False

0.047

False

False

0.242

189

18

7

True

0.035

True

0.02

True

0.002

False

0.062

False

False

0.12

194

5

5

False

2.338

False

1.922

False

0.612

False

0.05

False

False

4.922

197

7

5

False

7.562

False

71.864

False

0.485

False

0.05

False

False

79.962

198

12

9

True

0.397

True

0.793

True

0.077

False

0.042

False

False

1.31

199

15

8

True

1.404

True

1.531

False

0.215

False

0.054

True

False

3.205

220

58

56

True

146.146

True

534.832

True

6.921

False

0.964

False

False

688.866

227

60

39

True

0.273

True

0.485

True

0.01

False

0.077

False

False

0.847

229

7

7

True

1.917

True

3.348

False

0.131

False

0.062

True

False

5.458

233

4

2

False

0.16

False

0.44

False

0.17

False

0.557

False

False

1.328

243

23

19

True

8.598

True

1171.687

True

2.512

False

0.171

False

False

1182.97

259

17

16

True

1.334

True

1.913

True

0.092

False

0.045

False

False

3.385

260

17

16

True

2.182

True

0.748

True

0.079

False

0.047

False

False

3.057

261

17

16

True

3.359

True

2.872

True

0.113

False

0.095

False

False

6.44

262

11

9

True

0.402

True

0.41

True

0.091

False

0.071

False

False

0.975

263

11

9

True

0.379

True

0.403

True

0.085

False

0.066

False

False

0.934

264

14

11

True

1.031

True

2.036

True

0.136

False

0.063

False

False

3.268

267

4

3

True

1.084

True

0.246

True

0.095

False

0.049

False

False

1.475

271

6

4

True

0.286

True

0.283

True

0.746

False

0.045

False

False

1.361

272

6

4

True

0.361

True

0.323

True

0.086

False

0.055

False

False

0.826

281

32

32

True

20.987

True

29.791

True

0.602

False

0.055

False

False

51.437

282

6

3

True

0.205

True

0.19

True

0.087

False

0.046

False

False

0.528

283

4

3

True

0.294

True

0.211

True

0.087

False

0.412

False

False

1.005

289

5

4

False

2.291

False

1.118

False

0.165

False

0.044

False

False

3.619

292

6

2

True

0.06

True

0.048

True

0.063

False

0.046

False

False

0.218

306

5

2

True

0.149

True

0.121

False

0.079

False

0.041

True

False

0.391

307

5

2

True

0.129

True

0.121

True

0.043

False

0.148

False

False

0.441

310

4

1

True

0.053

True

0.369

True

0.047

False

0.04

False

False

0.509

311

4

1

True

0.076

True

0.048

True

0.224

False

0.048

False

False

0.397

312

3

2

True

0.098

True

0.512

True

0.043

False

0.043

False

False

0.697

314

12

10

True

0.515

True

1.789

True

0.1

False

0.059

False

False

2.464

321

3

3

True

0.163

True

0.148

True

0.042

False

0.039

False

False

0.393

357

9

8

True

0.353

True

1.517

True

0.07

False

0.045

False

False

1.986

359

9

8

True

1.677

True

3.605

True

0.11

False

0.055

False

False

5.448

360

9

8

True

0.479

True

0.47

True

0.096

False

0.05

False

False

1.096

361

8

8

True

1.069

True

2.746

True

0.156

False

0.045

False

False

4.017

363

4

3

True

0.244

True

0.199

True

0.077

False

0.041

False

False

0.561

364

14

12

True

2.483

True

7.296

True

0.55

False

0.064

False

False

10.394

413

5

5

False

1.55

False

22.323

False

0.117

False

0.053

False

False

24.044

459

4

3

True

0.542

True

0.224

False

0.18

False

0.068

True

False

1.014

460

4

3

False

1.025

False

0.936

False

0.143

False

0.216

False

False

2.321

475

23

22

True

97.876

True

3377.021

True

0.231

False

0.062

False

False

3475.192

484

2

1

True

0.384

True

0.143

False

0.099

False

0.048

True

False

0.674

485

2

1

False

0.564

False

0.354

False

0.209

False

0.042

False

False

1.169

486

2

2

True

0.119

True

0.106

False

0.073

False

0.041

True

False

0.339

487

6

6

True

0.475

True

1.008

False

0.099

False

0.045

True

False

1.628

491

57

57

True

123.138

True

536.865

False

2.067

True

0.007

True

True

662.08

492

52

52

True

85.606

True

284.753

False

1.123

True

0.003

True

True

371.489

519

3

3

True

1.357

True

2.367

False

5.142

False

0.097

True

False

8.964

546

7

3

True

0.327

True

0.338

True

0.109

False

0.042

False

False

0.817

559

90

71

True

4.742

True

7.525

True

0.19

False

0.053

False

False

12.515

584

35

9

True

0.4

True

0.655

False

0.095

False

0.043

True

False

1.194

619

10

8

True

0.411

True

0.443

True

0.087

False

0.052

False

False

0.994

629

5

5

True

0.209

True

0.197

False

0.079

False

0.046

True

False

0.532

647

11

11

False

0.854

False

16.436

False

0.165

False

0.051

False

False

17.507

B Program Code Used for Our Computations

The following is Maple code for computing one row of the table in Appendix A.

figure g
figure h

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Rahkooy, H., Sturm, T. (2020). First-Order Tests for Toricity. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2020. Lecture Notes in Computer Science(), vol 12291. Springer, Cham. https://doi.org/10.1007/978-3-030-60026-6_30

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