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CS-766 · computer science · mathematics · Computational complexity · Information and coding theory

Restriction Condensation of Deterministic Communication Complexity

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Statement

Definitions and mathematical statement

Shared notation.
Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

Communication complexity measures the number of transmitted bits needed for parties holding separate inputs to compute a joint function. Randomized protocols allow random choices, with their error probability specified by the model.

Record scope and qualifications.
The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. Source-record limitation: The allowed restrictions and definition of condensation, together with the parameters in the proposed polynomial loss, are absent.
Statement recorded in the supplied source.
Can deterministic communication complexity be condensed by restriction, and is some polynomial loss necessary? [S2]
Source status.
The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

The source asks about restriction Condensation of Deterministic Communication Complexity.

Sources

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The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30005830 and code OWR-14298166-006 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

CS-766-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-766

Restriction Condensation of Deterministic Communication Complexity

Definitions and mathematical statement

Shared notation.
Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

Communication complexity measures the number of transmitted bits needed for parties holding separate inputs to compute a joint function. Randomized protocols allow random choices, with their error probability specified by the model.

Record scope and qualifications.
The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. Source-record limitation: The allowed restrictions and definition of condensation, together with the parameters in the proposed polynomial loss, are absent.
Statement recorded in the supplied source.
Can deterministic communication complexity be condensed by restriction, and is some polynomial loss necessary? [S2]
Source status.
The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

The source asks about restriction Condensation of Deterministic Communication Complexity.

Sources

Preparation and review.
The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30005830 and code OWR-14298166-006 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Research attempt

Literature, source context and scope

The full original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems.

A later attributed result and an independently bounded test case

The primary workshop describes a polynomial loss in deterministic communication condensation and asks whether it is necessary. A later primary preprint by Simon Mackenzie, dated September 2026, claims lossless condensation. Its Theorem1.1 states: for a finite total Boolean matrix of deterministic cost $c\ge4$ and fixed $0<\varepsilon<1$, there is an original $2^k$-by-$2^k$ submatrix with $1\le k\le c-1$, $k\ge(\varepsilon c-4)/3$, and cost at least $(1-\varepsilon)(k+1)$. We read its statement and conventions, not its entire proof. This attributed claimed resolution postdates the preserved catalogue assessment; it is not our new result or an independently certified solution. We use deterministic binary protocols with fixed Boolean outputs at leaves. Let $\mathrm{EQ}_N(i,j)$ be equality on $[N]^2$, $N\ge2$.
Proposition: Its cost lies between $\lceil\log_2N\rceil$ and $\lceil\log_2N\rceil+1$. More generally, on a restriction $A\times B$, put $r=|A\cap B|$. If $r\ge1$, the restricted cost lies between $\lceil\log_2r\rceil$ and $\lceil\log_2(r+1)\rceil+1$; if $r=0$, it is zero.
Proof: Each protocol transcript defines a rectangle: at an Alice node its branch restricts only the row set, and at a Bob node only the column set, so induction from the full rectangle proves the claim. A monochromatic one-rectangle cannot contain two distinct diagonal entries $(i,i),(j,j)$, since it would also contain the zero entry $(i,j)$. Thus there are at least $r$ leaves, and a binary depth-$d$ tree has at most $2^d$ leaves, yielding the lower bound. For the unrestricted upper bound, Alice sends her index using $\lceil\log_2N\rceil$ bits and Bob sends the answer. For the restriction, Alice sends either the index of her element in $A\cap B$ or a special symbol for an element outside that intersection, using $\lceil\log_2(r+1)\rceil$ bits. Bob can then test equality and announce one bit. If the intersection is empty the output is constantly zero. ∎
In particular an $m$-by-$m$ equality square retains cost of order $\log m$. Retaining a constant fraction of $\log N$ needs $m$ polynomial in $N$, which is exponential in the original cost. This is consistent with the source's $2^{O(c)}$ target, not a counterexample. Confusing matrix dimension with its logarithm would give an erroneous obstruction. A basic cover route also shows the remaining step. If the rows are covered by $h$ sets, each supporting a protocol of cost at most $d$ against all columns, Alice can announce the first covering set containing her row and run that protocol. Hence total cost is at most $\lceil\log_2h\rceil+d$. Without a quantitative bound on $h$, cheap restrictions do not give a cheap global protocol. We do not supply the rank-sensitive covering analysis needed by the later preprint.

Verification and remaining gaps

Exact dynamic programming over all nontrivial row/column bipartitions computed protocol depth for small equality matrices and restrictions, checking the stated intervals. The cover protocol was checked directly on finite covers. These auxiliary facts do not independently prove general condensation. No polynomial-time extraction claim follows from the cited paper's doubly exponential dependence on its target depth, and no catalogue status is overwritten.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2024/15. Relevant passages read: 2024/15 pp.899--900, deterministic restrictions, known polynomial loss and lossless-condensation question.

https://arxiv.org/html/2609.28691. Read primary Introduction, Section1.1 Theorems1.1--1.3 and protocol conventions; lossless-condensation assertion attributed, full proof not verified. Reported algorithm is doubly exponential in target depth.

All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.

Integration review.
Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch, read and reviewed its written mathematical arguments, and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch025/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.

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