Datasets:

PranathReddy
/

solveall-literature-priors

	{
	"schema_version": "2.0",
	"seed_id": "solveall_capacity_binary_deletion_channel",
	"slug": "capacity-binary-deletion-channel",
	"split": "train",
	"title": "Capacity of the Binary Deletion Channel",
	"url": "https://solveall.org/problem/capacity-binary-deletion-channel",
	"categories": [
	"Information Theory",
	"Theoretical Computer Science"
	],
	"statement": "Fix a deletion probability $d\\in[0,1)$. The binary deletion channel $\\mathrm{BDC}(d)$ is defined as follows: for input blocklength $n$, the transmitter sends $X^n=(X_1,\\dots,X_n)\\in\\{0,1\\}^n$. Independently for each $i\\in\\{1,\\dots,n\\}$, a deletion indicator $D_i\\sim\\mathrm{Bernoulli}(d)$ is drawn, with $\\mathbb P(D_i=1)=d$ and $\\mathbb P(D_i=0)=1-d$, and $(D_i)$ is independent of $X^n$. The output is the random subsequence $$\nY=(X_i:\\,D_i=0)\\in\\{0,1\\}^{L_n},\n$$ where $L_n=\\sum_{i=1}^n(1-D_i)$ is random. Thus deleted symbols are removed, undeleted symbols keep their original order, and the receiver is not told the deleted positions.\n\nA block code of length $n$ and size $M_n$ consists of an encoder $f_n:\\{1,\\dots,M_n\\}\\to\\{0,1\\}^n$ and a decoder $g_n:\\{0,1\\}^\\to\\{1,\\dots,M_n\\}$, where $\\{0,1\\}^=\\bigcup_{\\ell\\ge 0}\\{0,1\\}^\\ell$. With uniformly distributed message $W\\in\\{1,\\dots,M_n\\}$, average error probability is $$\nP_e^{(n)}=\\mathbb P\\!\\big[g_n(Y)\\neq W\\big].\n$$ A rate $R\\ge 0$ is achievable if there exists a sequence of codes with $\\lim_{n\\to\\infty}P_e^{(n)}=0$ and $\\liminf_{n\\to\\infty}\\frac{1}{n}\\log_2 M_n\\ge R$. The Shannon capacity is $$\nC(d)=\\sup\\{R:\\,R\\ \\text{is achievable}\\}.\n$$\n\nOpen problem: determine $C(d)$ exactly as a function of $d$ for the binary deletion channel. In particular, determine the exact asymptotic behavior of $C(d)$ as $d\\to 1$, i.e., find an explicit asymptotic equivalent $a(d)$ such that $C(d)/a(d)\\to 1$ as $d\\to 1$ (and, ideally, further terms of the asymptotic expansion).",
	"unsolved_target": "Determine $C(d)$ exactly as a function of $d$ for the binary deletion channel. In particular, determine the exact asymptotic behavior of $C(d)$ as $d\\to 1$, i.e., find an explicit asymptotic equivalent $a(d)$ such that $C(d)/a(d)\\to 1$ as $d\\to 1$ (and, ideally, further terms of the asymptotic expansion).",
	"sources": [
	{
	"source_id": "cheraghchi-2020",
	"title": "Capacity Upper Bounds for Deletion-Type Channels",
	"authors": [
	"Mahdi Cheraghchi"
	],
	"year": 2019,
	"venue": "Journal of the ACM",
	"doi": "10.1145/3281275",
	"arxiv_id": "1711.01630",
	"url": "https://arxiv.org/abs/1711.01630",
	"version": "v4",
	"source_status": "verified",
	"source_note": "Main upper-bound theorems and numerical upper bounds for the binary deletion channel (global in $d$).",
	"verification_notes": "ArXiv identity and the JACM publication record are verified. The arXiv record also lists the related STOC DOI 10.1145/3188745.3188768; 10.1145/3281275 is the JACM DOI."
	},
	{
	"source_id": "dalai-2011-high-deletion",
	"title": "A new bound for the capacity of the deletion channel with high deletion probabilities",
	"authors": [
	"Marco Dalai"
	],
	"year": 2011,
	"venue": "2011 IEEE International Symposium on Information Theory Proceedings",
	"doi": null,
	"arxiv_id": "1004.0400",
	"url": "https://arxiv.org/abs/1004.0400",
	"version": "v2",
	"source_status": "verified",
	"source_note": "Primary arXiv record and author-hosted PDF inspected; proves existence of the high-deletion limit and records the 0.4143 upper bound.",
	"verification_notes": "ArXiv record verifies title, author, version history, and the abstract claims. The author-hosted ISIT PDF has closely equivalent title wording ('A new bound on the capacity...') and contains the cited introduction and equations."
	},
	{
	"source_id": "kanoria-montanari-2013",
	"title": "Optimal coding for the binary deletion channel with small deletion probability",
	"authors": [
	"Yashodhan Kanoria",
	"Andrea Montanari"
	],
	"year": 2013,
	"venue": "IEEE Transactions on Information Theory",
	"doi": "10.1109/TIT.2013.2262020",
	"arxiv_id": "1104.5546",
	"url": "https://arxiv.org/abs/1104.5546",
	"version": null,
	"source_status": "verified",
	"source_note": "Introduction and Theorem 1: asymptotic expansion of capacity for small deletion probability ($d\\to 0$).",
	"verification_notes": "Title, authors, arXiv identity, and publication metadata are consistent. This source is not directly used by any submitted evidence card."
	},
	{
	"source_id": "mitzenmacher-2009",
	"title": "A survey of results for deletion channels and related synchronization channels",
	"authors": [
	"Michael Mitzenmacher"
	],
	"year": 2009,
	"venue": "Probability Surveys",
	"doi": "10.1214/08-PS141",
	"arxiv_id": null,
	"url": "https://doi.org/10.1214/08-PS141",
	"version": null,
	"source_status": "verified",
	"source_note": "Section 7 (upper bounds) and Open Questions (asymptotics as deletion probability goes to 0 and to 1), p. 22.",
	"verification_notes": "Bibliographic identity verified from the Probability Surveys record and the author-hosted PDF. This source is not directly used by any of the submitted evidence cards."
	},
	{
	"source_id": "rubinstein-con-2023",
	"title": "Improved Upper and Lower Bounds on the Capacity of the Binary Deletion Channel",
	"authors": [
	"Ittai Rubinstein",
	"Roni Con"
	],
	"year": 2023,
	"venue": "arXiv preprint",
	"doi": "10.48550/arXiv.2305.07156",
	"arxiv_id": "2305.07156",
	"url": "https://arxiv.org/abs/2305.07156",
	"version": "v1",
	"source_status": "verified",
	"source_note": "Primary arXiv abstract inspected; reports improved high-deletion upper and lower bounds.",
	"verification_notes": "ArXiv record verifies title, authors, submission date, and the complete abstract passage used by the card."
	}
	],
	"evidence_cards": [
	{
	"evidence_id": "dalai-limit-exists",
	"reference_statement": "For the binary deletion channel, the normalized capacity C(d)/(1-d) has a limit as d→1, and this limit equals inf_{d∈(0,1)} C(d)/(1-d).",
	"scope_conditions": [
	"The infimum should be understood over the admissible deletion-probability range, conventionally d in (0,1) or 0<=d<1 depending on the paper's notation; this does not affect the stated d->1 result."
	],
	"passages": [
	{
	"source_id": "dalai-2011-high-deletion",
	"locator": "Introduction, equations (2) and surrounding text, author-hosted PDF p. 1 / arXiv HTML lines 39–50",
	"text": "In this paper, it is proved that the limit exists and, in particular, that\n\nlim\n d→1\n C(d)\n (1 − d)\n = inf\n d\n C(d)\n (1 − d)\n .",
	"content_sha256": "febdb59c0d13f59fb4681e1f52a9c6dd7753a9673673cd711ca765d4de3a3bdb"
	}
	],
	"verification_status": "machine_verified",
	"verification_notes": "Verified. This is a genuine theorem/result of Dalai and is appropriately scoped as an existence/equality statement rather than an exact evaluation."
	},
	{
	"evidence_id": "dalai-high-deletion-upper-bound",
	"reference_statement": "Dalai records the then-best numerical upper bound lim_{d→1} C(d)/(1-d)≤0.4143.",
	"scope_conditions": [
	"Binary deletion channel.",
	"Asymptotic regime d→1.",
	"The numerical value is a bound, not an exact evaluation of the limit."
	],
	"passages": [
	{
	"source_id": "dalai-2011-high-deletion",
	"locator": "Introduction, equation (3), author-hosted PDF p. 1 / arXiv HTML lines 51–57",
	"text": "The best currently known upper bound for C(d), when used in the right hand side of (2), leads to the upper bound\n\nlim\n d→1\n C(d)\n (1 − d)\n ≤ 0.4143,",
	"content_sha256": "f4cb9c151a49faf17df475857ae8ee52d51881346b6ebc0fe5a5bd5656c46dc4"
	}
	],
	"verification_status": "machine_verified",
	"verification_notes": "Verified. The card correctly presents 0.4143 as an upper bound, not as the limiting constant itself."
	},
	{
	"evidence_id": "rubinstein-con-high-deletion-bounds",
	"reference_statement": "Rubinstein and Con report a lower bound C(BDC_d)>0.1221(1-d) and an upper bound C(BDC_d)<0.3745(1-d) for all d≥0.68.",
	"scope_conditions": [
	"Binary deletion channel with i.i.d. deletion probability d.",
	"Upper bound applies for every d≥0.68.",
	"The lower bound is reported without a narrower d-domain in the abstract passage.",
	"These are strict bounds, not an asymptotic equivalent or an identified limiting constant."
	],
	"passages": [
	{
	"source_id": "rubinstein-con-2023",
	"locator": "Abstract, arXiv record lines 16–19",
	"text": "In this paper, we show that the Blahut-Arimoto algorithm can be implemented with a lower space complexity, allowing us to extend the upper bound analyses, and prove an upper bound of $\\mathcal{C}(\\text{BDC}_d) < 0.3745 \\cdot(1-d)$ for all $d \\geq 0.68$. Furthermore, we show that an extension of the Blahut-Arimoto algorithm can also be used to select better run length distributions for Mitzenmacher and Drinea's construction, yielding a lower bound of $\\mathcal{C}(\\text{BDC}_d) > 0.1221 \\cdot (1-d)$.",
	"content_sha256": "2ae4966237bc40d6717c6fe26065dc214ae14481decad42d1c3ec538490e45f9"
	}
	],
	"verification_status": "machine_verified",
	"verification_notes": "Verified. Strict inequalities and domain qualification are preserved."
	},
	{
	"evidence_id": "cheraghchi-global-upper-bound",
	"reference_statement": "Cheraghchi proves C(d) <= (1-d) log_2(phi) for d >= 1/2, under the paper's bits-per-channel-use convention; phi=(1+sqrt(5))/2.",
	"scope_conditions": [
	"Binary deletion channel.",
	"Deletion probability d≥1/2.",
	"Entropy/capacity is measured in bits per channel use.",
	"The source also states a separate d<1/2 bound only under a convexity assumption; that conditional result is not included here."
	],
	"passages": [
	{
	"source_id": "cheraghchi-2020",
	"locator": "Abstract, arXiv record lines 5 and 83",
	"text": "The capacity of the binary deletion channel with deletion probability $d$ is at most $(1-d)\\log\\varphi$ for $d\\geq 1/2$, and, assuming the capacity function is convex, is at most $1-d\\log(4/\\varphi)$ for $d<1/2$, where $\\varphi=(1+\\sqrt{5})/2$ is the golden ratio.",
	"content_sha256": "56676c242bf351dd8ae270aa085146b82ae5e98f8a95a844c3a2df01d00a2175"
	}
	],
	"verification_status": "machine_verified",
	"verification_notes": "Verified. Writing log_2 explicitly is preferable because the card's scope specifies bits."
	},
	{
	"evidence_id": "dalai-no-closed-form-known",
	"reference_statement": "At the time of Dalai's paper, no closed-form expression for C(d) was known; only upper and lower bounds were available.",
	"scope_conditions": [
	"This is explicitly a 2011 literature-status statement and must not be read as a claim about all later literature."
	],
	"passages": [
	{
	"source_id": "dalai-2011-high-deletion",
	"locator": "Introduction, author-hosted PDF p. 1 / PDF lines 21–25",
	"text": "Dobrushin ([1]) showed that for such a channel it is possible to define a capacity C(d) and that a Shannon like theorem applies to this channel. However, no closed formula expression is known up to now for the capacity C(d), and only upper and lower bounds are currently available (see [2], [3], [4], [5], [6]).",
	"content_sha256": "4a43f7ccbcca39d2601bd5d79c0e1ed6ab97f61a6dd4620178f346db660e3422"
	}
	],
	"verification_status": "machine_verified",
	"verification_notes": "Verified with the candidate's historical scope restriction."
	}
	],
	"priors": [],
	"generation_metadata": {
	"source_dataset": "data/solveall_dataset/problems.jsonl",
	"source_sha256": "8c495a0059526e41559ffa4abb93bb84ba4f48eb244ee7d4c5ae9ffc2792243f",
	"split_seed": 20260721,
	"evidence_research_model": "gpt-5.6-luna",
	"evidence_verifier_model": "gpt-5.6-luna",
	"candidate_evidence_cards": 5,
	"verified_evidence_cards": 5
	}
	}