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CS-361 · computer science · mathematics · Computational complexity

Karchmer–Raz–Wigderson Composition Conjecture

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

For $f:\{0,1\}^a\to\{0,1\}$ and $g:\{0,1\}^b\to\{0,1\}$, block composition applies $g$ to each of $a$ disjoint blocks and then applies $f$ to the resulting bits. The standard depth measure uses fan-in-two AND/OR gates and NOT gates; it minimizes the longest gate path of a Boolean circuit computing the function.

A Boolean function maps $\{0,1\}^n$ to $\{0,1\}$. A Boolean circuit is an acyclic network of logical gates; a formula is a circuit with no sharing of intermediate gate outputs. Size and depth measure its number of gates and longest gate path.

Statement recorded in the supplied source.
For nonconstant Boolean functions $f$ and $g$, is the circuit-depth complexity of their composition equal, up to lower-order terms, to the sum of their individual depth complexities? [S2]
Scope qualification.
The supplied phrase up to lower-order terms is retained. Its precise asymptotic comparison must come from the source; no unrestricted-fan-in gate convention is substituted.
Source status.
The supplied catalogue labels this record partially solved. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.

Short English statement

Determine whether composing Boolean functions adds their depth complexities up to the stated lower-order terms.

Sources

Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30004010 and source code OWR-16634-003 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Catalogue entry

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

Contributions 1

CS-361-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-361

Karchmer–Raz–Wigderson Composition Conjecture

Definitions and mathematical statement

Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

For $f:\{0,1\}^a\to\{0,1\}$ and $g:\{0,1\}^b\to\{0,1\}$, block composition applies $g$ to each of $a$ disjoint blocks and then applies $f$ to the resulting bits. The standard depth measure uses fan-in-two AND/OR gates and NOT gates; it minimizes the longest gate path of a Boolean circuit computing the function.

A Boolean function maps $\{0,1\}^n$ to $\{0,1\}$. A Boolean circuit is an acyclic network of logical gates; a formula is a circuit with no sharing of intermediate gate outputs. Size and depth measure its number of gates and longest gate path.

Statement recorded in the supplied source.
For nonconstant Boolean functions $f$ and $g$, is the circuit-depth complexity of their composition equal, up to lower-order terms, to the sum of their individual depth complexities? [S2]
Scope qualification.
The supplied phrase up to lower-order terms is retained. Its precise asymptotic comparison must come from the source; no unrestricted-fan-in gate convention is substituted.
Source status.
The supplied catalogue labels this record partially solved. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.

Short English statement

Determine whether composing Boolean functions adds their depth complexities up to the stated lower-order terms.

Sources

Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30004010 and source code OWR-16634-003 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not 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.

Composition works exactly for conjunctions up to rounding

The source retains the qualitative phrase about lower-order terms; we do not assign it a new universal asymptotic meaning. Work with fan-in-two AND/OR and unary NOT gates, counting every gate on a longest input-output path. A single input variable has depth zero. Let $\operatorname{AND}_a$ be conjunction of $a\ge1$ inputs.
Proposition: For arbitrary Boolean functions, substitution of circuits gives $D(f\circ g)\le D(f)+D(g)$. For conjunctions, \[ D(\operatorname{AND}_a\circ\operatorname{AND}_b)=\lceil\log_2(ab)\rceil, \] which differs from $D(\operatorname{AND}_a)+D(\operatorname{AND}_b)$ by at most one.
Proof: Replace each input occurrence of a circuit for $f$ by a copy of a circuit for $g$ on its corresponding disjoint block. Every resulting path passes through at most $D(g)$ gates followed by at most $D(f)$, proving the upper bound. Any output at depth $h$ can depend on at most $2^h$ distinct input variables. This follows by induction: a binary gate takes the union of at most two sets each of size $2^{h-1}$; a unary NOT gate cannot increase the number. Conjunction of $r$ inputs depends on all $r$, because changing one input while the others are 1 changes the output. Its depth is thus at least $\lceil\log_2 r\rceil$. A balanced binary AND tree achieves that depth. Block composition of the two conjunctions is conjunction of all $ab$ variables. Finally, for real $x,y$, $\lceil x\rceil+\lceil y\rceil-1\le\lceil x+y\rceil\le\lceil x\rceil+\lceil y\rceil$, proving the rounding assertion. ∎
The exact loss can occur: $a=3,b=5$ gives depths 2 and 3 separately, but the composed conjunction has depth 4. This does not refute a conjecture allowing lower-order terms. The attempted general lower-bound route stalls because dependence on all $ab$ variables gives only $\log_2(ab)$, not a lower bound involving the actual potentially larger complexities $D(f)$ and $D(g)$. The substitution upper bound has no corresponding reverse-substitution proof here.

Verification and remaining gaps

Balanced-tree depths and the rounding difference were enumerated for $1\le a,b\le64$. The proof uses the stated bounded-fan-in convention, and counts unary NOT gates although conjunctions need none. No unrestricted KRW lower bound, formalization of its unspecified lower-order term, or claim about later restricted composition theorems is obtained.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2018/51. Relevant passages read: 2018/51 p. 3033, KRW composition survey and qualitative formulation.

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, reviewed the written mathematical arguments, and contributed formatting/repository review. This additional model review is not independent human verification. 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 batch022/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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