Trajectory Planning and Certification for 3-DOF Robot Manipulators Using Real Quantifier Elimination Based on Comprehensive Gr\"obner Systems

Akira Terui; Masahiko Mikawa; Yu Nakai

REVIEW 2 major objections 4 minor 17 references

A precomputed comprehensive Gröbner system turns inverse-kinematics checks along a robot path into a single quantifier-free formula whose true set is exactly the reachable parameter range.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-07-14 04:03 UTC pith:37JFJRVC

load-bearing objection Solid algebraic extension of the authors’ CGS-QE robotics line to spline paths and exact feasible ranges; math is clean, scope is deliberately 3-DOF fixed-wrist, experiments promised but missing from the text. the 2 major comments →

arxiv 2607.11657 v1 pith:37JFJRVC submitted 2026-07-13 cs.RO cs.SCmath.AC

Trajectory Planning and Certification for 3-DOF Robot Manipulators Using Real Quantifier Elimination Based on Comprehensive Gr\"obner Systems

Yu Nakai , Akira Terui , Masahiko Mikawa This is my paper

classification cs.RO cs.SCmath.AC MSC 68W3013P1070B15

keywords comprehensive Gröbner systemsreal quantifier eliminationinverse kinematicstrajectory planning3-DOF manipulatorcubic natural splineRisa/Asir

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to certify, before any motion begins, which parts of a continuous end-effector path a three-joint robot can actually follow. By treating the path parameter as a free variable and computing a comprehensive Gröbner system once, the authors convert the existence of joint angles into a quantifier-free formula in that single parameter. An isolating-interval procedure then returns the precise subintervals of the path on which solutions exist. The same algebraic object also supplies the Gröbner bases needed for ordinary inverse-kinematics evaluation at sample points, so planning and certification share one precomputation. The method covers both straight-line segments and cubic natural splines and is implemented entirely inside free computer-algebra software.

Core claim

For a 3-DOF myCobot whose end-effector path is given by a line segment or cubic natural spline C(r), r∈[0,1], the CGS-QE pipeline produces a quantifier-free formula ψ(r) that is equivalent to the existence of joint variables satisfying the inverse-kinematics polynomials, and Algorithm 1 returns the exact subset R⊂[0,1] on which those solutions exist.

What carries the argument

The CGS-QE method: a comprehensive Gröbner system of the parametric inverse-kinematics ideal yields, via the signature of a multiplication-map matrix, a quantifier-free first-order formula whose truth set is precisely the parameter values for which real joint solutions exist.

Load-bearing premise

The robot is treated as a pure three-joint positioner by locking the wrist joints at zero and ignoring end-effector orientation; any motion that needs wrist reorientation lies outside the certified claim.

What would settle it

Pick a concrete cubic spline whose intermediate points place part of the path outside the geometric workspace of the three-joint arm; run Algorithm 1 and check whether the returned set R correctly excludes exactly those parameter intervals that numerical inverse-kinematics solvers report as unreachable.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Solid algebraic extension of the authors’ CGS-QE robotics line to spline paths and exact feasible ranges; math is clean, scope is deliberately 3-DOF fixed-wrist, experiments promised but missing from the text.

read the letter

This is a clean methods paper that does exactly what it claims inside a narrow algebraic-robotics niche. The new pieces are real: they lift the earlier line-segment yes/no certificates (Yoshizawa et al.) to cubic natural splines, return the precise parameter intervals R ⊂ [0,1] on which real IK solutions exist, and ship the whole pipeline in free Risa/Asir rather than mixing in Mathematica. Algorithm 1 (isolating-interval walk over the sign conditions of the signature criterion) is carefully written and correctly specialized from Fukasaku–Iwane–Sato.

What works: the reduction of the myCobot position IK to the six-polynomial system (2), the parametric ideal with end-effector path C(r) substituted, the conversion of rational-function sign conditions into polynomial ones via non-vanishing denominators on each CGS segment, and the appeal to the signature theorems are all standard and correctly applied. Self-citation is heavy but the load-bearing results sit on independent published foundations (Weispfenning, Suzuki–Sato, Fukasaku et al.). No circularity or algebraic gap appears.

Soft spots, in proportion: the abstract advertises “experimental results” that are simply not in the supplied manuscript; that is a presentational hole, not a mathematical one. The modeling choice θ4=θ5=θ6=0 (orientation discarded) is stated explicitly after (1) and is therefore not a hidden flaw, but it does mean the certificates only cover position reachability under a fixed-wrist model. Anyone who needs wrist reorientation is outside the claim. Epsilon in the isolation shift is a free parameter of no theoretical consequence.

Who it is for: people already working with CGS or real QE for kinematics who want a free-software reference implementation and a clean range-extraction algorithm. It will not move industrial 6-DOF practice. It deserves a serious referee once the missing experiments and code artifacts are supplied and the 3-DOF scope is stated up front. I would accept it for peer review as a methods contribution; I would not desk-reject it.

Referee Report

2 major / 4 minor

Summary. The paper reduces inverse kinematics for the myCobot manipulator (treated as 3-DOF by fixing θ4=θ5=θ6=0 and discarding orientation) to a parametric polynomial system (Eq. 2) in cosines/sines of the first three joint angles with end-effector coordinates as parameters. It precomputes a Comprehensive Gröbner System so that Gröbner bases for discrete points on a trajectory C(r) (line segment or cubic natural spline) are obtained by specialization rather than recomputation. For continuous certification it specializes the CGS-QE method of Fukasaku–Iwane–Sato: the signature criterion on the multiplication map yields a quantifier-free formula ψ(r) equivalent to existence of real joint solutions, which Algorithm 1 converts, via real-root isolation of the appearing polynomials, into the exact feasible subset R ⊂ [0,1]. The whole pipeline is claimed to be implemented in Risa/Asir.

Significance. If the claims hold, the work supplies an exact, non-numerical certificate of position reachability along entire continuous trajectories (including obstacle-avoiding cubic splines) for a concrete 3-DOF industrial arm, together with a free-software realization. Precomputing the CGS once and then specializing is a genuine efficiency gain over repeated Gröbner-basis calls; the isolating-interval walk of Algorithm 1 is a clean, reproducible way to extract the feasible parameter intervals. These features are valuable for certified motion planning and complement purely numerical IK solvers. The algebraic core rests on published theorems (Weispfenning, Suzuki–Sato, Fukasaku et al.) and is assembled correctly for the zero-dimensional case.

major comments (2)

[Abstract / Section 1] Abstract and Section 1 (final paragraph of the introduction) explicitly state that the algorithm is implemented in Risa/Asir and that experimental results are presented. The supplied manuscript contains neither an experimental section, nor any timing tables, nor any concrete example of a computed CGS segment, nor any output of Algorithm 1 on a line segment or cubic spline for the myCobot geometry. The central efficiency claim and the claim of a working implementation are therefore unsupported by the text; these results must be added or the claims withdrawn.
[Section 2, Eq. (2)] Section 2 states that the IK problem is formulated as the system f1=···=f7=0, yet the displayed equations (2) contain only six polynomials f1–f6. The subsequent ideal is written ⟨f1,…,f6⟩. This inconsistency must be resolved (either a seventh polynomial is missing or the count is a typographical error) before the zero-dimensionality arguments of Section 4 can be trusted.

minor comments (4)

[Abstract / Section 2] The modeling restriction θ4=θ5=θ6=0 is stated clearly after Eq. (1), yet the title and abstract speak of “3-DOF robot manipulators” without immediately reminding the reader that orientation is discarded. A single clarifying sentence in the abstract would prevent misreading.
[Algorithm 1] Algorithm 1 uses an ad-hoc rational shift ε (default 1/100) when a root lands exactly on an endpoint. The text never discusses how the choice of ε interacts with the isolating-interval precision or with floating-point evaluation of the sign conditions; a short remark would improve reproducibility.
[Section 1] Several self-citations to the authors’ prior CGS-QE robotics papers appear; while the present contribution is distinct, a brief comparison paragraph locating the new range-extraction algorithm relative to those works would help the reader.
[Throughout] Typographical issues: missing spaces after periods and commas in several places (e.g., “equationsf 1”, “parametersp 1”), inconsistent use of boldface for vectors, and the arXiv identifier printed as 2607.11657 (future date) should be cleaned.

Circularity Check

1 steps flagged

Minor self-citation of the authors' prior CGS-QE robotics applications; the load-bearing algebraic certification (signature criterion, CGS segments, Algorithm 1) rests on independent external theorems and is not circular by construction.

specific steps

self citation load bearing [Section 1, paragraph on prior group work and contributions 1–2]
"Our research group has proposed methods for solving inverse kinematics and trajectory planning problems for manipulators using Gröbner bases [8–11,15]. … Our previous studies applied CGS-QE to inverse kinematics and trajectory planning for 3-DOF manipulators [10,15], guaranteeing solution existence along a trajectory."

The narrative of novelty leans on the authors' own earlier applications of CGS-QE to 3-DOF arms and line-segment trajectories. While the present algebraic steps are independently grounded in Fukasaku et al., the claim that the method 'guarantees' existence for the new spline case inherits its credibility partly from those self-citations rather than from a fresh external verification of the same pipeline.

full rationale

The paper's derivation chain for trajectory certification is self-contained: the parametric IK ideal Ī is formed by substituting the trajectory polynomials C(r) into the fixed 3-DOF polynomials (2), the existence test reduces to the signature of the multiplication map via Corollaries 1–2 and Definition 1 taken verbatim from the independent Fukasaku–Iwane–Sato ISSAC 2015 paper, rational-function sign conditions are rewritten as polynomial conditions using non-vanishing denominators on CGS segments, and Algorithm 1 walks isolating intervals of the resulting univariate factors. None of these steps equals its own input by definition, fits a free parameter to data then re-labels the fit as a prediction, or imports a uniqueness theorem from the authors. Self-citations ([8–11,15], [10,15]) appear for prior applications of the same toolkit to related manipulators and for the modeling reduction θ4=θ5=θ6=0, but they supply context and engineering choices rather than the mathematical justification of the quantifier-free formula ψ(r) or of the feasible set R. The modeling restriction itself is stated explicitly and does not create an internal algebraic loop. Consequently the circularity score is at most 1.

Axiom & Free-Parameter Ledger

1 free parameters · 4 axioms · 0 invented entities

The work rests on standard real-algebraic geometry (signature of multiplication maps, real-root isolation) and classical robot kinematics (D-H parameters). The only paper-specific modeling choice is the 3-DOF reduction of a 6-DOF arm. No new physical entities are postulated; free parameters are limited to a numerical isolation tolerance.

free parameters (1)

epsilon (Alg. 1 isolation shift) = 1/100 (example)
Hand-chosen small rational (example 1/100) used to nudge interval endpoints off roots; affects only numerical isolation, not the algebraic certificate.

axioms (4)

standard math Signature of the multiplication-map matrix M_I^1 equals the number of real roots of a zero-dimensional ideal (Fukasaku et al. Thm 1 / Cor 2).
Invoked in §4.1 as the foundation of CGS-QE root counting.
domain assumption The specialized inverse-kinematics ideals arising from the CGS segments are zero-dimensional so that the residue-class ring is finite-dimensional.
Required for the matrix M and characteristic polynomial construction; not proved for every segment in the paper.
domain assumption Denavit–Hartenberg kinematic model with fixed wrist angles θ4=θ5=θ6=0 correctly describes reachable end-effector positions of myCobot for the intended tasks.
Section 2 modeling step that turns a 6-DOF arm into the 3-DOF polynomial system (2).
domain assumption Path coordinate functions C_i(r) lie in Q[r], so substitution yields a parametric ideal over Q(r).
Stated at the opening of §3; excludes transcendental or floating-point paths.

pith-pipeline@v1.1.0-grok45 · 14108 in / 2619 out tokens · 30145 ms · 2026-07-14T04:03:17.151010+00:00 · methodology

0 comments

read the original abstract

We propose an algorithm and its implementation for trajectory planning and certification for 3-DOF robot manipulators. The method uses Real Quantifier Elimination (QE) based on Comprehensive Gr\"obner Systems (CGS), also known as the CGS-QE method. The main advantage of the proposed method is its efficiency in trajectory planning and solution certification. This efficiency comes from the effective use of the CGS. First, for trajectory planning, we solve the inverse kinematics problem at each point along the trajectory via Gr\"obner basis computation. This usually requires recalculating the Gr\"obner basis at every point, which is time-consuming. We avoid this by computing the CGS for a parametric system. Here, the end-effector coordinates are parameters. This approach streamlines the algorithm. Second, for solution certification, the CGS-QE method certifies that an inverse kinematics solution exists at any point along the end-effector's trajectory. Our method also certifies solutions for trajectories composed of line segments and cubic natural splines. The algorithm is implemented within the computer algebra system Risa/Asir.

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 7 canonical work pages

[1]

Springer, Berlin, Heidelberg (2006) 8 Yu Nakai, Akira Terui, and Masahiko Mikawa

Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry (Algo- rithms and Computation in Mathematics). Springer, Berlin, Heidelberg (2006) 8 Yu Nakai, Akira Terui, and Masahiko Mikawa

2006
[2]

Journal of Applied Mechanics22, 215–221 (1955).https: //doi.org/10.1115/1.4011045

Denavit, J., Hartenberg, R.: A Kinematic Notation for Lower-Pair Mechanisms Based on Matrices. Journal of Applied Mechanics22, 215–221 (1955).https: //doi.org/10.1115/1.4011045

doi:10.1115/1.4011045 1955
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elephantrobotics.com/en/mycobot-280-m5-2023-en, accessed 2026-04-04

Elephant Robotics Co., Ltd.: myCobot 280 M5 (2023),https://www. elephantrobotics.com/en/mycobot-280-m5-2023-en, accessed 2026-04-04

2023
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Research Report RR-5923, INRIA (2006),https://hal.inria

Faug` ere, J.C., Merlet, J.P., Rouillier, F.: On solving the direct kinematics problem for parallel robots. Research Report RR-5923, INRIA (2006),https://hal.inria. fr/inria-00072366

2006
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Fukasaku, R., Iwane, H., Sato, Y.: Real Quantifier Elimination by Computation of Comprehensive Gr¨ obner Systems. In: Proc. ISSAC 2015. pp. 173–180. ACM, New York, NY, USA (2015).https://doi.org/10.1145/2755996.2756646

work page doi:10.1145/2755996.2756646 2015
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Kalker-Kalkman, C.M.: An implementation of Buchbergers’ algorithm with ap- plications to robotics. Mech. Mach. Theory28(4), 523–537 (1993).https://doi. org/10.1016/0094-114X(93)90033-R

work page doi:10.1016/0094-114x(93)90033-r 1993
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Kim, C.H., H. R. Kam, S.H.L., Park, T.: RViz: a toolkit for real domain data vi- sualization. In: Telecommunication Systems, vol. 60, pp. 337–345. Springer (2015). https://doi.org/10.1007/s11235-015-0034-5

doi:10.1007/s11235-015-0034-5 2015
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Okazaki, T., Terui, A., Mikawa, M.: An Effective Trajectory Planning and an Optimized Path Planning for a 6-Degree-of-Freedom Robot Manipulator. In: CASC
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16235, pp

Lecture Notes in Computer Science, vol. 16235, pp. 312–335. Springer (2025), https://doi.org/10.1007/978-3-032-09645-6_17

work page doi:10.1007/978-3-032-09645-6_17 2025
[10]

In: CASC 2025

Okazaki, T., Terui, A., Mikawa, M.: Inverse Kinematics for a 6-Degree-of-Freedom Robot Manipulator Using Comprehensive Gr¨ obner Systems. In: CASC 2025. Lecture Notes in Computer Science, vol. 16235, pp. 336–359. Springer (2025), https://doi.org/10.1007/978-3-032-09645-6_18

work page doi:10.1007/978-3-032-09645-6_18 2025
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Preprint (2021).https://doi.org/10.48550/ arXiv.2111.00384

Otaki, S., Terui, A., Mikawa, M.: A Design and an Implementation of an Inverse Kinematics Computation in Robotics Using Real Quantifier Elimination based on Comprehensive Gr¨ obner Systems. Preprint (2021).https://doi.org/10.48550/ arXiv.2111.00384

Pith/arXiv arXiv 2021
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In: Proceedings of the SCSS 2024 Work in Progress Work- shop

Shirato, Y., Oka, N., Terui, A., Mikawa, M.: An optimized path planning of ma- nipulator with spline curves using real quantifier elimination based on comprehen- sive Gr¨ obner systems. In: Proceedings of the SCSS 2024 Work in Progress Work- shop. pp. 105–112. Sun SITE Central Europe, RWTH Aachen University (2024), https://ceur-ws.org/Vol-3754/

2024
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Mathematical Problems in Engineering2021, 6680687 (2021).https://doi.org/10.1155/2021/6680687

Ricardo Xavier da Silva, S., Schnitman, L., Cesca Filho, V.: A Solution of the Inverse Kinematics Problem for a 7-Degrees-of-Freedom Serial Redundant Manip- ulator Using Gr¨ obner Bases Theory. Mathematical Problems in Engineering2021, 6680687 (2021).https://doi.org/10.1155/2021/6680687

work page doi:10.1155/2021/6680687 2021
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doi:10.1016/0747-7171(92)90023-w 1992
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In: CASC 2023, Lecture Notes in Computer Science, vol

Yoshizawa, M., Terui, A., Mikawa, M.: Inverse Kinematics and Path Planning of Manipulator Using Real Quantifier Elimination Based on Comprehensive Gr¨ obner Systems. In: CASC 2023, Lecture Notes in Computer Science, vol. 14139, pp. 393–

2023
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,¯αl = ((γl, δl], fl)} ←(the set of isolating intervals satisfying (5) for the set of distinct real roots{α 1,

Springer (2023).https://doi.org/10.1007/978-3-031-41724-5_21 Trajectory Planning and Certification for 3-DOF Robot Manipulators 9 Algorithm 1Deriving of the range of variables satisfying a disjunctive normal form Input:ψ(r): a disjuncive normal form, [γ, δ]⊂Rwithγ, δ∈Qsuch thatγ < δ,ϵ >0: a small rational number (say,ϵ= 1/100) Output:R ⊂[γ, δ]: the subset...

work page doi:10.1007/978-3-031-41724-5_21 2023

[1] [1]

Springer, Berlin, Heidelberg (2006) 8 Yu Nakai, Akira Terui, and Masahiko Mikawa

Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry (Algo- rithms and Computation in Mathematics). Springer, Berlin, Heidelberg (2006) 8 Yu Nakai, Akira Terui, and Masahiko Mikawa

2006

[2] [2]

Journal of Applied Mechanics22, 215–221 (1955).https: //doi.org/10.1115/1.4011045

Denavit, J., Hartenberg, R.: A Kinematic Notation for Lower-Pair Mechanisms Based on Matrices. Journal of Applied Mechanics22, 215–221 (1955).https: //doi.org/10.1115/1.4011045

doi:10.1115/1.4011045 1955

[3] [3]

elephantrobotics.com/en/mycobot-280-m5-2023-en, accessed 2026-04-04

Elephant Robotics Co., Ltd.: myCobot 280 M5 (2023),https://www. elephantrobotics.com/en/mycobot-280-m5-2023-en, accessed 2026-04-04

2023

[4] [4]

Research Report RR-5923, INRIA (2006),https://hal.inria

Faug` ere, J.C., Merlet, J.P., Rouillier, F.: On solving the direct kinematics problem for parallel robots. Research Report RR-5923, INRIA (2006),https://hal.inria. fr/inria-00072366

2006

[5] [5]

In: Proc

Fukasaku, R., Iwane, H., Sato, Y.: Real Quantifier Elimination by Computation of Comprehensive Gr¨ obner Systems. In: Proc. ISSAC 2015. pp. 173–180. ACM, New York, NY, USA (2015).https://doi.org/10.1145/2755996.2756646

work page doi:10.1145/2755996.2756646 2015

[6] [6]

Kalker-Kalkman, C.M.: An implementation of Buchbergers’ algorithm with ap- plications to robotics. Mech. Mach. Theory28(4), 523–537 (1993).https://doi. org/10.1016/0094-114X(93)90033-R

work page doi:10.1016/0094-114x(93)90033-r 1993

[7] [7]

Kim, C.H., H. R. Kam, S.H.L., Park, T.: RViz: a toolkit for real domain data vi- sualization. In: Telecommunication Systems, vol. 60, pp. 337–345. Springer (2015). https://doi.org/10.1007/s11235-015-0034-5

doi:10.1007/s11235-015-0034-5 2015

[8] [8]

In: CASC

Okazaki, T., Terui, A., Mikawa, M.: An Effective Trajectory Planning and an Optimized Path Planning for a 6-Degree-of-Freedom Robot Manipulator. In: CASC

[9] [9]

16235, pp

Lecture Notes in Computer Science, vol. 16235, pp. 312–335. Springer (2025), https://doi.org/10.1007/978-3-032-09645-6_17

work page doi:10.1007/978-3-032-09645-6_17 2025

[10] [10]

In: CASC 2025

Okazaki, T., Terui, A., Mikawa, M.: Inverse Kinematics for a 6-Degree-of-Freedom Robot Manipulator Using Comprehensive Gr¨ obner Systems. In: CASC 2025. Lecture Notes in Computer Science, vol. 16235, pp. 336–359. Springer (2025), https://doi.org/10.1007/978-3-032-09645-6_18

work page doi:10.1007/978-3-032-09645-6_18 2025

[11] [11]

Preprint (2021).https://doi.org/10.48550/ arXiv.2111.00384

Otaki, S., Terui, A., Mikawa, M.: A Design and an Implementation of an Inverse Kinematics Computation in Robotics Using Real Quantifier Elimination based on Comprehensive Gr¨ obner Systems. Preprint (2021).https://doi.org/10.48550/ arXiv.2111.00384

Pith/arXiv arXiv 2021

[12] [12]

In: Proceedings of the SCSS 2024 Work in Progress Work- shop

Shirato, Y., Oka, N., Terui, A., Mikawa, M.: An optimized path planning of ma- nipulator with spline curves using real quantifier elimination based on comprehen- sive Gr¨ obner systems. In: Proceedings of the SCSS 2024 Work in Progress Work- shop. pp. 105–112. Sun SITE Central Europe, RWTH Aachen University (2024), https://ceur-ws.org/Vol-3754/

2024

[13] [13]

Mathematical Problems in Engineering2021, 6680687 (2021).https://doi.org/10.1155/2021/6680687

Ricardo Xavier da Silva, S., Schnitman, L., Cesca Filho, V.: A Solution of the Inverse Kinematics Problem for a 7-Degrees-of-Freedom Serial Redundant Manip- ulator Using Gr¨ obner Bases Theory. Mathematical Problems in Engineering2021, 6680687 (2021).https://doi.org/10.1155/2021/6680687

work page doi:10.1155/2021/6680687 2021

[14] [14]

In: Proc

Suzuki, A., Sato, Y.: A simple algorithm to compute comprehensive Gr¨ obner bases using Gr¨ obner bases. In: Proc. ISSAC 2006. pp. 326–331 (2006).https://doi. org/10.1145/1145768.1145821

work page doi:10.1145/1145768.1145821 2006

[15] [15]

Weispfenning, V.: Comprehensive Gr¨ obner Bases. J. Symbolic Comput.14(1), 1– 29 (1992).https://doi.org/10.1016/0747-7171(92)90023-W

doi:10.1016/0747-7171(92)90023-w 1992

[16] [16]

In: CASC 2023, Lecture Notes in Computer Science, vol

Yoshizawa, M., Terui, A., Mikawa, M.: Inverse Kinematics and Path Planning of Manipulator Using Real Quantifier Elimination Based on Comprehensive Gr¨ obner Systems. In: CASC 2023, Lecture Notes in Computer Science, vol. 14139, pp. 393–

2023

[17] [17]

,¯αl = ((γl, δl], fl)} ←(the set of isolating intervals satisfying (5) for the set of distinct real roots{α 1,

Springer (2023).https://doi.org/10.1007/978-3-031-41724-5_21 Trajectory Planning and Certification for 3-DOF Robot Manipulators 9 Algorithm 1Deriving of the range of variables satisfying a disjunctive normal form Input:ψ(r): a disjuncive normal form, [γ, δ]⊂Rwithγ, δ∈Qsuch thatγ < δ,ϵ >0: a small rational number (say,ϵ= 1/100) Output:R ⊂[γ, δ]: the subset...

work page doi:10.1007/978-3-031-41724-5_21 2023