ksopyla/cogito-probe-arith

CogitoProbe-Arith: nested arithmetic with mixed brackets

Part of Cogito Capability Checks — does your model actually use what it read? Small synthetic tests with exact answers, guessing floors and a length ladder (1k → 128k tokens). Project page: ai.ksopyla.com/projects/cogito-capability-datasets

TL;DR. Several nested + - * expressions with mixed brackets ()[]{} up front, filler, then a question about the first (farthest) one: an inner node's value or which bracket closes which. Tests whether a model stored the expression tree, not just the final number.

Lengths	1k, 4k, 8k, 16k, 32k tokens (seq_len)
Size	10,240 rows: train 8,448 · validation 896 · test 896
Tests	structure memory: sub-expression values, bracket matching
Tasks	subexpr 5,175 · match 3,028 (primary) · eval 2,037 (control)
Guessing	depends on the task; see prize_bits
Format	text + ready ids for HuggingFaceTB/SmolLM3-3B (Llama-3 vocab)
License	Apache-2.0

Example

( 3 + [ 2 * 4 ] ) - { 5 - 1 } . …more expressions…
… filler …
Q sub 0 . 2 .   →   values of internal nodes          (subexpr, primary)
Q match 2       →   index of the matching closer      (match, primary)
Q eval          →   the root value                    (eval, control only)

Quick start

from datasets import load_dataset

ds = load_dataset("ksopyla/cogito-probe-arith")            # train / validation / test
row = ds["validation"][0]
print(row["seq_len"], row["task"], "|", row["query"], "->", row["answer"])

# Already tokenized for HuggingFaceTB/SmolLM3-3B (Llama-3 vocab); loss only where labels != -100.
# Or ignore the ids and use the text fields: context / query / answer.
small = ds["train"].filter(lambda r: r["seq_len"] == 1024 and r["variant"] == "fixed")  # laptop-sized start

How to score

Teacher-forced token accuracy on the answer span, per task. Report subexpr and match; eval is a single number a calculator shortcut can produce, so treat it as a control, not as evidence that the model understands arithmetic.

The family: Cogito Capability Checks

One goal: check whether a model stores and uses facts from its context, not just predicts nearby words. Each dataset plants facts with known answers, adds filler, asks a question at the end, and labels how hard guessing is. Length is a ladder, so you see where a model stops working.

dataset	input	the question it answers	lengths
cogito-probe-bits	symbol keys and values	How many key→value facts did it keep? (memory capacity in bits)	1k–32k
cogito-probe-bind	people with attributes	Does it keep who has what together, and follow one link?	1k–32k
cogito-probe-arith	nested bracket arithmetic	Did it store the expression tree, not only the result?	1k–32k
cogito-probe-props	short facts in fluent filler	Does it keep the facts and ignore the padding?	1k–32k
cogito-text-world	simple English stories	Can a small LM trained from scratch quote, look up, track, chain, count and deduce facts in a story?	1k–128k

Start with a probe if you test a memory or compression module on a pretrained tokenizer; start with cogito-text-world if you compare architectures trained from scratch on language.

Related work and inspirations

• Needle in a Haystack (Kamradt, 2023) and passkey retrieval (Mohtashami & Jaggi, 2023, arXiv:2305.16300) — one fact in long filler; here: many facts, labelled bits, decoys.
• RULER (Hsieh et al., 2024, arXiv:2404.06654) — synthetic long-context tasks (multi-key, variable tracking) for pretrained LMs; these sets add from-scratch training data and evidence-removed controls.
• BABILong (Kuratov et al., 2024, arXiv:2406.10149) and bAbI (Weston et al., 2015, arXiv:1502.05698) — facts hidden in long natural text.
• Zoology / MQAR (Arora et al., 2023, arXiv:2312.04927) — multi-query associative recall as an architecture test.
• Lost in the Middle (Liu et al., 2023, arXiv:2307.03172) — answers depend on where the evidence sits.
• Binding IDs (Feng & Steinhardt, 2023, arXiv:2310.17191), ListOps (Nangia & Bowman, 2018, arXiv:1804.06028), PrOntoQA (Saparov & He, 2022, arXiv:2210.01240) — binding, tree structure and rule chains.
• TinyStories (Eldan & Li, 2023, arXiv:2305.07759) and SimpleStories (Finke et al., 2025, arXiv:2504.09184) — the simple language under cogito-text-world.

Schema, build statistics, rebuild, limitations

Columns

column	meaning
text, context, query, answer	readable surfaces; text is the full padded row
input_ids, attention_mask, labels	SmolLM3-3B ids, length = seq_len; labels = -100 except on the answer
seq_len, rung	1024, 4096, 8192, 16384 or 32768
variant	fixed: same facts as at 1k, longer filler (a drop = length failure); scaled: more facts with length (a drop = capacity failure)
task	question type (see above)
prize_bits	information in the gold answer (counting lower bound)
gap	tokens from the last evidence token to the answer
answer_start, answer_end, evidence_end	token indices into input_ids
meta	JSON: fact table, fingerprints, node values

Build (seed 20260916, 2026-09-16): 10,240 rows = train 8,448 / validation 896 / test 896. Rows per length: 1k 4,608 · 4k 2,560 · 8k 1,280 · 16k 1,024 · 32k 768; half fixed, half scaled. Mean prize_bits 12.9 (3.3–30.5). Splits use disjoint random streams; fact fingerprints, input_ids, text and answer strings have zero overlap between train, validation and test (shared answer values, such as colours, are expected).

Rebuild (deterministic):

uv run python scripts/build_concept_probe_datasets.py --scale full --seed 20260916 \
  --tokenizer HuggingFaceTB/SmolLM3-3B --families arith --out_dir Cache/concept_probes/full

Generator: data/concept_probes/ in github.com/ksopyla/MrCogito.

Limitations

• Not natural language: symbols are single tokens of the SmolLM3 / Llama-3 vocabulary (real rows use pieces like gonzalez, oslo, validators; the examples above use toy names).
• Answers are packed (several values per answer), so a small memory gets a dense learning signal.
• prize_bits is a counting lower bound, not a cross-entropy floor.
• The filler is a repeating cycle, so gzip of the whole row is tiny; compare prize_bits, not compressibility.
• Bracket shapes colour the tree; they do not change the value. Digits and brackets are injected as single ids, not BPE-encoded from a glued string.

Changelog

date	version	change
2026-10-10	card	Card rewritten: one-line goal, quick start, family table, related work, citations. Data unchanged.
2026-09-16	v0	First public release, seed 20260916, 10,240 rows.

Citation

Please cite the family, and the dataset you used:

@misc{sopyla2026cogitochecks,
  title        = {Cogito Capability Checks: synthetic tests of whether models store and use facts from long context},
  author       = {Sopyła, Krzysztof},
  year         = {2026},
  howpublished = {\url{https://ai.ksopyla.com/projects/cogito-capability-datasets/}},
  note         = {Datasets: cogito-probe-bits, -bind, -arith, -props; cogito-text-world}
}

@misc{cogitoprobe_arith,
  title        = {CogitoProbe-Arith: nested arithmetic with mixed brackets},
  author       = {Sopyła, Krzysztof},
  year         = {2026},
  howpublished = {\url{https://huggingface.co/datasets/ksopyla/cogito-probe-arith}},
  note         = {Cogito Capability Checks; seed 20260916, data v0}
}

Author: Krzysztof Sopyła · code: github.com/ksopyla/MrCogito.