Pith Number

pith:7HJO7SD7

pith:2023:7HJO7SD7DKWUR254JDPIC23PQA

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Instance complexity of Boolean functions

Alison Hsiang-Hsuan Liu, Nikhil S. Mande

Symmetric Boolean functions have instance complexity fully characterized, with only Parity and its complement achieving value 1.

arxiv:2309.15026 v4 · 2023-09-26 · cs.CC

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Claims

C1strongest claim

We complement the above-mentioned result of Grossman et al. by completely characterizing the instance complexity of symmetric Boolean functions. As a corollary we conclude that the only symmetric Boolean functions with instance complexity 1 are the Parity function and its complement. We show that the above-mentioned ratio is linear in the input size for both of these functions, while we exhibit algorithms for which the instance complexity is a constant.

C2weakest assumption

The measure of instance complexity is taken directly from the definition introduced by Grossman et al. (the max over inputs of algorithm cost divided by optimal known-input cost) and the standard decision-tree model for Boolean functions; this premise enters when the paper invokes the prior definition to derive the symmetric characterization and the separation for Greater-Than and Odd-Max-Bit.

C3one line summary

Instance complexity of symmetric Boolean functions is completely characterized with only Parity and complement at value 1; Greater-Than and Odd-Max-Bit have constant IC while QC/C_min is linear.

Receipt and verification

First computed	2026-06-24T14:15:49.440618Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

f9d2efc87f1aad48ebbc48de816b6f803abad7b5b875ace223382f19bc562b79

Aliases

arxiv: 2309.15026 · arxiv_version: 2309.15026v4 · doi: 10.48550/arxiv.2309.15026 · pith_short_12: 7HJO7SD7DKWU · pith_short_16: 7HJO7SD7DKWUR254 · pith_short_8: 7HJO7SD7

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/7HJO7SD7DKWUR254JDPIC23PQA \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: f9d2efc87f1aad48ebbc48de816b6f803abad7b5b875ace223382f19bc562b79

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "acdb715f5e831d63ad557dfa9b4cd663254b753f6a3c890461d05f8bbbb934e5",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.CC",
    "submitted_at": "2023-09-26T15:56:14Z",
    "title_canon_sha256": "bfc65f203104222fa67f3fab0511b2f8e5f7e57153d058cf696442aa9c850d95"
  },
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  "source": {
    "id": "2309.15026",
    "kind": "arxiv",
    "version": 4
  }
}