---
license: apache-2.0
pretty_name: ultraviolet param-object pools
tags: [math, reinforcement-learning, synthetic, verifiable]
configs:
- config_name: default
  data_files:
  - split: train
    path: train.jsonl
  - split: eval
    path: eval.jsonl
  - split: eval_tier0
    path: eval_tier0.jsonl
---
# ultraviolet param-object pools

Synthetic, exactly verifiable math construction problems for RL. Each item asks for a short integer tuple that
parameterises a mathematical object (polynomial coefficients, matrix entries, a lattice point, a permutation, ...)
with stated properties. Every item has a finite answer box, so all of its valid answers are enumerated exactly and
stored in the row; **any one valid answer is correct**. Each item has 2-64 valid answers and a blind-guess success
rate of at most 0.02.

| split | file | rows | md5 |
|---|---|---|---|
| train | `train.jsonl` | 239,584 | `d59f16328d437c68d6b96caf0b0c44a8` |
| eval | `eval.jsonl` | 2,241 | `6a8c2cf0f50a53ac5ba0223ca74da4c8` |
| eval_tier0 (tier 0 only) | `eval_tier0.jsonl` | 324 | `d4fb551a4c6afd10198c78f0f1460f76` |

The eval splits share no problems with train.

## Example

> Find integers a, d with -10 <= a, d <= 10 such that the arithmetic progression a, a + d, a + 2d, ... has sum of its
> first 6 terms equal to 81. Several answers are possible; any one correct answer is accepted. Give one solution. End
> with the answer in the form \boxed{a = ..., d = ...}, with the dots replaced by the values.

Valid answers (`meta.valid_answers`): `-9, 9`, `-4, 7`, `1, 5`, `6, 3`.

## Fields

`problem` (statement), `prompt` (`Problem:\n{problem}\n\n<think>\n`), `answer` (one valid answer), `family`, `domain`,
`tier` (0-3, difficulty), `n_valid`, `guess_rate`, `meta.params` / `meta.seed` (the generator's inputs) and
`meta.valid_answers`.

## Usage

```python
from datasets import load_dataset
ds = load_dataset("harribo11/ultraviolet-paramobj-pools", split="train")
```

Generator, verifier and scorer: `workbench/envs/paramobj/` in
[github.com/hliuson/ultraviolet](https://github.com/hliuson/ultraviolet). There the tasks are registered as
`wb_po_<name>`, and the rows here say `cx_po_<name>`, so rename the `task` field before scoring:

```python
from ultraviolet.tasks.score import score
row = dict(row, task="wb_" + row["task"][3:])     # cx_po_ap_sum -> wb_po_ap_sum
score(row["task"], completion, row)               # 1.0 if the last \boxed{} after </think> is a valid answer
```

## Families (53) and tiers

Train rows per tier, and eval rows per family:

| family | domain | t0 | t1 | t2 | t3 | eval |
|---|---|---|---|---|---|---|
| po_ap_sum | sequence | 186 | 842 | 1,979 | 1,998 | 48 |
| po_binary_runs | combinatorics | 15 | 43 | 73 | 64 | 32 |
| po_circle_through | geometry | 498 | 1,853 | 2,015 | 2,002 | 48 |
| po_consecutive_sum | number | 18 | 117 | 211 | 1,471 | 35 |
| po_crt_range | number | 1,987 | 2,037 | 1,989 | 2,033 | 48 |
| po_diff_squares | number | 26 | 30 | 52 | 119 | 14 |
| po_egyptian | number | 23 | 97 | 74 | 141 | 29 |
| po_fraction_between | number | 1,564 | 1,971 | 1,985 | 2,014 | 48 |
| po_gauss_factor | number | 33 | 166 | 453 | 1,926 | 36 |
| po_gcd_lcm | number | 280 | 774 | 1,170 | 91 | 48 |
| po_gp_term | sequence | 110 | 113 | 433 | 191 | 48 |
| po_int_triangle | geometry | 118 | 503 | 1,981 | 1,988 | 48 |
| po_lattice_tri_area | geometry | 1,611 | 1,956 | 2,001 | 2,009 | 48 |
| po_lin2_box | number | 1,995 | 2,011 | 2,029 | 2,008 | 48 |
| po_lin_commute | function | 756 | 1,599 | 2,008 | 2,018 | 48 |
| po_lin_congruence | number | 1,997 | 1,995 | 2,012 | 1,971 | 48 |
| po_magic3 | combinatorics | 1,990 | 2,007 | 2,002 | 2,013 | 48 |
| po_mat2_commute | linalg | 2,017 | 2,032 | 1,999 | 2,014 | 48 |
| po_mat2_eigvec | linalg | 679 | 963 | 1,149 | 1,293 | 48 |
| po_mat2_idempotent | linalg | 47 | 715 | 1,364 | 2,023 | 37 |
| po_mat2_involution | linalg | 42 | 609 | 1,318 | 1,999 | 44 |
| po_mat2_nilpotent | linalg | 48 | 456 | 846 | 2,011 | 36 |
| po_mat2_trdet | linalg | 181 | 735 | 1,887 | 2,032 | 48 |
| po_mat3_det | linalg | 2,027 | 1,990 | 1,993 | 2,043 | 48 |
| po_mobius | function | 1,014 | 1,466 | 2,000 | 2,007 | 48 |
| po_moment_weights | combinatorics | 107 | 328 | 675 | 928 | 48 |
| po_order_mod | number | 22 | 154 | 528 | 1,680 | 36 |
| po_parallelogram | geometry | 1,987 | 2,021 | 1,987 | 2,007 | 48 |
| po_partition_sq | combinatorics | 366 | 1,623 | 2,004 | 1,981 | 48 |
| po_perm_cycles | combinatorics | 19 | 45 | 45 | 517 | 35 |
| po_perm_descents | combinatorics | 76 | 248 | 248 | 424 | 48 |
| po_perm_inversions | combinatorics | 15 | 40 | 40 | 354 | 45 |
| po_poly_critical | polynomial | 258 | 1,998 | 1,997 | 1,942 | 48 |
| po_poly_integral | polynomial | 1,993 | 2,002 | 2,016 | 1,999 | 48 |
| po_poly_mixed | polynomial | 2,006 | 2,020 | 1,987 | 1,977 | 48 |
| po_poly_roots | polynomial | 32 | 62 | 143 | 68 | 19 |
| po_poly_slope | polynomial | 767 | 1,963 | 2,007 | 2,015 | 48 |
| po_poly_values | polynomial | 1,003 | 2,018 | 2,000 | 1,994 | 48 |
| po_pyth_leg | number | 25 | 80 | 242 | 867 | 29 |
| po_pyth_perim | geometry | 28 | 68 | 189 | 491 | 41 |
| po_quad_rootdiff | polynomial | 52 | 294 | 518 | 882 | 37 |
| po_recur_coef | sequence | 338 | 1,829 | 700 | 394 | 48 |
| po_recur_init | sequence | 1,008 | 1,915 | 2,019 | 1,982 | 48 |
| po_sfft | number | 1,461 | 1,987 | 1,986 | 1,991 | 48 |
| po_sigma_inv | number | 25 | 126 | 357 | 1,248 | 36 |
| po_sqrt_mod | number | 414 | 2,029 | 2,018 | 1,988 | 48 |
| po_stats_multiset | combinatorics | 32 | 103 | 193 | 1,610 | 36 |
| po_tau_range | number | 2,005 | 1,975 | 2,052 | 1,980 | 48 |
| po_totient_inv | number | 14 | 25 | 48 | 88 | 4 |
| po_two_sq_comp | number | 87 | 727 | 1,997 | 2,039 | 48 |
| po_vec_cross | linalg | 2,007 | 1,980 | 2,002 | 1,991 | 48 |
| po_vec_dot | linalg | 989 | 1,980 | 1,992 | 1,997 | 48 |
| po_vieta3 | polynomial | 14 | 84 | 467 | 1,975 | 28 |
| **total** | | 36,412 | 56,804 | 67,480 | 78,888 | |