The word and geodesic problems in free solvable groups
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Bibliographic record
Abstract
We study the computational complexity of the Word Problem (WP) in free solvable groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript r comma d"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">S_{r,d}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">r \geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the rank and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d \geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the solvability class of the group. It is known that the Magnus embedding of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript r comma d"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">S_{r,d}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into matrices provides a polynomial time decision algorithm for WP in a fixed group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript r comma d"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">S_{r,d}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Unfortunately, the degree of the polynomial grows together with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , so the uniform algorithm is not polynomial in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In this paper we show that WP has time complexity <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis r n log Subscript 2 Baseline n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>r</mml:mi> <mml:mi>n</mml:mi> <mml:msub> <mml:mi>log</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo> </mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(r n \log _2 n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript r comma 2"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">S_{r,2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis n cubed r d right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mi>r</mml:mi> <mml:mi>d</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(n^3 r d)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript r comma d"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">S_{r,d}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d greater-than-or-equal-to 3"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d \geq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it