Lean statement

EXPTIME-completeness of Weisfeiler–Leman graph identification

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Theorem selected for verification: OAI.WLIdentification.main

Lean statement file: WLIdentification.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 133 link it.

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The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem main : EXPTIMEComplete dimensionLanguage :=

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Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

structure OAI.WLIdentification.Graph line 7

structure Graph where  order : ℕ  edge : Fin order → Fin order → Bool  symmetric : ∀ u v, edge u v = edge v u  loopless : ∀ u, edge u u = false

definition OAI.WLIdentification.Isomorphic line 13

def Isomorphic (G H : Graph) : Prop :=  ∃ e : Fin G.order ≃ Fin H.order, ∀ u v, G.edge u v = H.edge (e u) (e v)

definition OAI.WLIdentification.NeighborColor line 17

def NeighborColor : ℕ → Type  | 0 => Unit  | s + 1 => NeighborColor s × Multiset (NeighborColor s)

definition OAI.WLIdentification.neighborColor line 21

def neighborColor (G : Graph) : (s : ℕ) → Fin G.order → NeighborColor s  | 0, _ => ()  | s + 1, v => (neighborColor G s v,      (Finset.univ.filter (fun w => G.edge v w = true)).val.map (neighborColor G s))

definition OAI.WLIdentification.neighborHistogram line 26

def neighborHistogram (G : Graph) (s : ℕ) : Multiset (NeighborColor s) :=  Finset.univ.val.map (neighborColor G s)

definition OAI.WLIdentification.TupleColor line 29

def TupleColor (k : ℕ) : ℕ → Type  | 0 => (Fin k → Fin k → Bool) × (Fin k → Fin k → Bool)  | s + 1 => TupleColor k s × Multiset (Fin k → TupleColor k s)

definition OAI.WLIdentification.tupleColor line 34

def tupleColor (G : Graph) (k : ℕ) : (s : ℕ) → (Fin k → Fin G.order) → TupleColor k s  | 0, v => (fun i j => decide (v i = v j), fun i j => G.edge (v i) (v j))  | s + 1, v => (tupleColor G k s v,      Finset.univ.val.map (fun z => fun i => tupleColor G k s (Function.update v i z)))

definition OAI.WLIdentification.tupleHistogram line 39

def tupleHistogram (G : Graph) (k s : ℕ) : Multiset (TupleColor k s) :=  Finset.univ.val.map (tupleColor G k s)

definition OAI.WLIdentification.Equivalent line 42

def Equivalent (k : ℕ) (G H : Graph) : Prop :=  if k = 1 then ∀ s, neighborHistogram G s = neighborHistogram H s  else ∀ s, tupleHistogram G k s = tupleHistogram H k s

definition OAI.WLIdentification.Identifies line 47

def Identifies (k : ℕ) (G : Graph) : Prop :=  ∀ H : Graph, Equivalent k G H → Isomorphic G H

definition OAI.WLIdentification.WLdim line 51

noncomputable def WLdim (G : Graph) : ℕ :=  sInf {k : ℕ | 0 < k ∧ Identifies k G}

abbreviation OAI.WLIdentification.Word line 54
```
abbrev Word := List Bool
```
abbreviation OAI.WLIdentification.BinaryLanguage line 55
```
abbrev BinaryLanguage := Set Word
```

definition OAI.WLIdentification.binaryValue line 57

def binaryValue (bits : Word) : ℕ :=  bits.foldl (fun z b => 2 * z + if b then 1 else 0) 0

definition OAI.WLIdentification.matrixWord line 60

def matrixWord (G : Graph) : Word :=  (List.ofFn (fun i : Fin G.order => List.ofFn (G.edge i))).flatten

definition OAI.WLIdentification.inputWord line 64

def inputWord (G : Graph) (bits : Word) : Word :=  List.replicate G.order true ++ [false] ++ matrixWord G ++ bits

definition OAI.WLIdentification.dimensionLanguage line 68

def dimensionLanguage : BinaryLanguage :=  {w | ∃ (G : Graph) (bits : Word),    0 < G.order ∧ (∃ tail, bits = true :: tail) ∧ 0 < binaryValue bits ∧    w = inputWord G bits ∧ WLdim G ≤ binaryValue bits}

inductive type OAI.WLIdentification.Move line 76

inductive Move where  | left | stay | right  deriving DecidableEq

definition OAI.WLIdentification.Move.displacement line 80

def Move.displacement : Move → ℤ  | .left => -1  | .stay => 0  | .right => 1

structure OAI.WLIdentification.Machine line 85

structure Machine where  extraSymbols : ℕ  extraStates : ℕ  transition : Fin (extraStates + 3) → Fin (extraSymbols + 3) →    Fin (extraStates + 3) × Fin (extraSymbols + 3) × Move

structure OAI.WLIdentification.Configuration line 91

structure Configuration (M : Machine) where  state : Fin (M.extraStates + 3)  head : ℤ  tape : ℤ → Fin (M.extraSymbols + 3)

definition OAI.WLIdentification.inputSymbol line 96

def inputSymbol (M : Machine) (b : Bool) : Fin (M.extraSymbols + 3) :=  if b then 2 else 1

definition OAI.WLIdentification.initial line 99

def initial (M : Machine) (w : Word) : Configuration M where  state := 0  head := 0  tape z := if 0 ≤ z then (w[z.toNat]?).elim 0 (inputSymbol M) else 0

definition OAI.WLIdentification.Halted line 104

def Halted {M : Machine} (c : Configuration M) : Prop :=  c.state = 1 ∨ c.state = 2

8 more definitions are reached the same way; the full file lists them.

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: neighbor, ldim, color, matrix, update, complete, many, joint and ten more. Found in the paper’s title, which its main theorem states: identifying. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: refinement, adjacency, reductions, polynomial, uncolored and time.

Statements in The complexity of identifying a graph by Weisfeiler–Leman refinement come first because the AI draft reading of OpenAI’s note shares words with that paper’s title.

Theorem 1.1 of The complexity of identifying a graph by Weisfeiler–Leman refinement

Theorem

The following problem is EXPTIME-complete under deterministic polynomial-time many-one reductions: given a nonempty finite simple undirected uncolored graph G by its adjacency matrix and a positive integer k in binary, decide whether \WLdim(G)\le k. Here dimension one uses ordinary neighbor color refinement, and dimensions at least two use the joint-update rule described above.

Section 1, Introduction Read it in the paper TeX source, line 36

Show the TeX

The following problem is EXPTIME-complete under deterministic
polynomial-time many-one reductions: given a nonempty finite simple
undirected uncolored graph $G$ by its adjacency matrix and a positive
integer $k$ in binary, decide whether $\WLdim(G)\le k$.
Here dimension one uses ordinary neighbor color refinement, and
dimensions at least two use the joint-update rule described above.

Other possible paper matches, closest wording first

Proposition 2.4 of The complexity of identifying a graph by Weisfeiler–Leman refinement. Words of the Lean statement, its names split into words, that also appear in the theorem: matrix, deciding, exptime, binary, positive, integer and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: weisfeiler, leman and adjacency.
Corollary 7.3 of The complexity of identifying a graph by Weisfeiler–Leman refinement. Words of the Lean statement, its names split into words, that also appear in the theorem: unary, complete, exptime, positive and dimension. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: identification and encoded.
Theorem 1.1 of Unconditional time lower bounds for Weisfeiler–Leman equivalence. Words of the Lean statement, its names split into words, that also appear in the theorem: required, replacement, multiset, many, joint, word and eight more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: leman, refinement, weisfeiler, uncolored, polynomial and time.
Proposition 8.2 of Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: output, word, uniform, input, computation, binary and eight more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: polynomial, uncolored and time.
Corollary 8.3 of Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: unary, complete, many, exptime, binary, dimension and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: supplied, encoded, reductions, polynomial and time.
Theorem 5.2 of Unconditional time lower bounds for Weisfeiler–Leman equivalence. Words of the Lean statement, its names split into words, that also appear in the theorem: rejecting, configurations, language, tape, update, input and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: time.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

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Carlos Toledo’s cert-machine

How they ran it, in their words

Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 133’s Lean statements express the advertised result?

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MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: None; all three headline claims are stated. The conditional running-time exclusions are not (docs).
Definitions the group asks readers to compare with the paper: Hand-rolled computation models: TM (multi-tape, fixed finite tables, halting states 1/2), a RAM with logarithmic word width (WordWidth.logarithmic), memory Word w → Word w, compute (op : WordOperation R) where a whole word operation (any map implemented by a TM within constant * (w+1)^exponent) counts as one instruction and one step runs a finite instruction tree; InEXPTIME (2^(c (n+1)^d) steps), PolytimeReduces/PolytimeManyOne, EXPTIMEComplete; WL colors are canonical nested multisets (Color c k t) with Equivalent := ∀ t, histogram G t = histogram H t
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves that deciding whether Weisfeiler–Leman refinement of a supplied dimension identifies a graph is EXPTIME-complete under polynomial-time many-one reductions. The input is a nonempty finite simple uncolored graph encoded by its adjacency matrix together with a positive binary-encoded dimension. Identification quantifies over every comparison graph, and the statement includes both the exponential-time upper bound and hardness.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

146 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 93 to its own declarations, 50 to Mathlib’s documentation. 33 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves main exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 32 other declarations of the file; with the theorem itself they span 93 lines. 2 of them are named in the statement itself.

Names in the namespace OAI.WLIdentification are listed without it.

structure Graphline 7library
def Isomorphicline 13library
def NeighborColorline 17library
def neighborColorline 21library
def neighborHistogramline 26library
def TupleColorline 29library
def tupleColorline 34library
def tupleHistogramline 39library
def Equivalentline 42library
def Identifiesline 47library
def WLdimline 51library
abbrev Wordline 54library
abbrev BinaryLanguageline 55library
def binaryValueline 57library
def matrixWordline 60library
def inputWordline 64library
def dimensionLanguageline 68in the statementlibrary
inductive Moveline 76library
def Move.displacementline 80library
structure Machineline 85library
structure Configurationline 91library
def inputSymbolline 96library
def initialline 99library
def Haltedline 104library
def stepline 108library
def runline 114library
def Outputline 118library
def DecidesWithinline 123library
def InEXPTIMEline 126library
def ComputesWithinline 130library
def PolytimeManyOneline 133library
def EXPTIMECompleteline 137in the statementlibrary
theorem mainline 140selected theoremlibrary

import Mathlibnamespace OAInamespace WLIdentificationstructure Graph where  order : ℕ  edge : Fin order → Fin order → Bool  symmetric : ∀ u v, edge u v = edge v u  loopless : ∀ u, edge u u = falsedef Isomorphic (G H : Graph) : Prop :=  ∃ e : Fin G.order ≃ Fin H.order, ∀ u v, G.edge u v = H.edge (e u) (e v)/-- Colors retain their full history, and hence are shared between graphs. -/def NeighborColor : ℕ → Type  | 0 => Unit  | s + 1 => NeighborColor s × Multiset (NeighborColor s)def neighborColor (G : Graph) : (s : ℕ) → Fin G.order → NeighborColor s  | 0, _ => ()  | s + 1, v => (neighborColor G s v,      (Finset.univ.filter (fun w => G.edge v w = true)).val.map (neighborColor G s))def neighborHistogram (G : Graph) (s : ℕ) : Multiset (NeighborColor s) :=  Finset.univ.val.map (neighborColor G s)def TupleColor (k : ℕ) : ℕ → Type  | 0 => (Fin k → Fin k → Bool) × (Fin k → Fin k → Bool)  | s + 1 => TupleColor k s × Multiset (Fin k → TupleColor k s)/-- Joint update: one multiset of whole replacement vectors, not independent marginals. -/def tupleColor (G : Graph) (k : ℕ) : (s : ℕ) → (Fin k → Fin G.order) → TupleColor k s  | 0, v => (fun i j => decide (v i = v j), fun i j => G.edge (v i) (v j))  | s + 1, v => (tupleColor G k s v,      Finset.univ.val.map (fun z => fun i => tupleColor G k s (Function.update v i z)))def tupleHistogram (G : Graph) (k s : ℕ) : Multiset (TupleColor k s) :=  Finset.univ.val.map (tupleColor G k s)def Equivalent (k : ℕ) (G H : Graph) : Prop :=  if k = 1 then ∀ s, neighborHistogram G s = neighborHistogram H s  else ∀ s, tupleHistogram G k s = tupleHistogram H k s/-- There is no restriction to mates from a graph construction or to a selected order. -/def Identifies (k : ℕ) (G : Graph) : Prop :=  ∀ H : Graph, Equivalent k G H → Isomorphic G H/-- The infimum of the positive identifying dimensions; this set is nonempty for finite G. -/noncomputable def WLdim (G : Graph) : ℕ :=  sInf {k : ℕ | 0 < k ∧ Identifies k G}abbrev Word := List Boolabbrev BinaryLanguage := Set Worddef binaryValue (bits : Word) : ℕ :=  bits.foldl (fun z b => 2 * z + if b then 1 else 0) 0def matrixWord (G : Graph) : Word :=  (List.ofFn (fun i : Fin G.order => List.ofFn (G.edge i))).flatten/-- Unary order header, row-major binary matrix, then most-significant-first binary k. -/def inputWord (G : Graph) (bits : Word) : Word :=  List.replicate G.order true ++ [false] ++ matrixWord G ++ bits/-- A malformed word is rejected: membership requires a valid nonempty simple matrix and k>0. -/def dimensionLanguage : BinaryLanguage :=  {w | ∃ (G : Graph) (bits : Word),    0 < G.order ∧ (∃ tail, bits = true :: tail) ∧ 0 < binaryValue bits ∧    w = inputWord G bits ∧ WLdim G ≤ binaryValue bits}/- The deterministic local machine model is explicit; no oracle or arbitrary   state-transition on whole inputs is included. Both alphabets are fixed finite   types for each machine. Extra symbols and states allow arbitrary finite tables. -/inductive Move where  | left | stay | right  deriving DecidableEqdef Move.displacement : Move → ℤ  | .left => -1  | .stay => 0  | .right => 1structure Machine where  extraSymbols : ℕ  extraStates : ℕ  transition : Fin (extraStates + 3) → Fin (extraSymbols + 3) →    Fin (extraStates + 3) × Fin (extraSymbols + 3) × Movestructure Configuration (M : Machine) where  state : Fin (M.extraStates + 3)  head : ℤ  tape : ℤ → Fin (M.extraSymbols + 3)def inputSymbol (M : Machine) (b : Bool) : Fin (M.extraSymbols + 3) :=  if b then 2 else 1def initial (M : Machine) (w : Word) : Configuration M where  state := 0  head := 0  tape z := if 0 ≤ z then (w[z.toNat]?).elim 0 (inputSymbol M) else 0def Halted {M : Machine} (c : Configuration M) : Prop :=  c.state = 1 ∨ c.state = 2/-- One read/write/head-movement transition. A halted configuration is frozen. -/def step (M : Machine) (c : Configuration M) : Configuration M :=  if c.state = 1 ∨ c.state = 2 then c else    let t := M.transition c.state (c.tape c.head)    { state := t.1, head := c.head + t.2.2.displacement,      tape := Function.update c.tape c.head t.2.1 }def run (M : Machine) (w : Word) (t : ℕ) : Configuration M :=  (step M)^[t] (initial M w)/-- Output is the binary word beginning at origin and ending at its first blank. -/def Output {M : Machine} (c : Configuration M) (w : Word) : Prop :=  (∀ i : Fin w.length, c.tape (i.val : ℤ) = inputSymbol M w[i]) ∧    c.tape (w.length : ℤ) = 0/-- Uniform genuine executions on every bit word, including malformed graph words. -/def DecidesWithin (M : Machine) (L : BinaryLanguage) (bound : ℕ → ℕ) : Prop :=  ∀ w, ∃ t ≤ bound w.length, Halted (run M w t) ∧ ((run M w t).state = 1 ↔ w ∈ L)def InEXPTIME (L : BinaryLanguage) : Prop :=  ∃ (M : Machine) (C d : ℕ),    DecidesWithin M L (fun n => 2 ^ (C * (n + 1) ^ d))def ComputesWithin (M : Machine) (f : Word → Word) (bound : ℕ → ℕ) : Prop :=  ∀ w, ∃ t ≤ bound w.length, (run M w t).state = 1 ∧ Output (run M w t) (f w)def PolytimeManyOne (L S : BinaryLanguage) : Prop :=  ∃ (f : Word → Word) (M : Machine) (C d : ℕ),    ComputesWithin M f (fun n => C * (n + 1) ^ d) ∧ ∀ w, w ∈ L ↔ f w ∈ Sdef EXPTIMEComplete (S : BinaryLanguage) : Prop :=  InEXPTIME S ∧ ∀ L : BinaryLanguage, InEXPTIME L → PolytimeManyOne L Stheorem main : EXPTIMEComplete dimensionLanguage :=  by    sorryend WLIdentificationend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

Parity lifts and bounded-treewidth witnesses for Weisfeiler–Leman equivalence

Theorem (Parity reduction)

Fix k\ge4, either refinement convention, and the corresponding t in (eq:bag-parameter). From every choice system on t domains one can explicitly construct two uncolored simple graphs X_0,X_\star of equal order such that

X_0\equiv_k X_\star \quad\Longleftrightarrow\quad \text{the choice system has no successful choice}.

The assertion holds both with common color names on separate graphs and with pure-tuple comparison in their disjoint union.

If every domain and label set has size at most A, where A\ge1, their common order is at most A M_t, where

M_t=t\,2^{\,t-2+3\binom{t-1}{2}}+12\binom t3.

When a successful choice exists, the two graphs already have different histograms after at most two joint rounds, or one separate round.

Section 1, Introduction Read it in the paper TeX source, line 76

Show the TeX

Fix \(k\ge4\), either refinement convention, and the corresponding
\(t\) in \eqref{eq:bag-parameter}. From every choice system on \(t\)
domains one can explicitly construct two uncolored simple graphs
\(X_0,X_\star\) of equal order such that
\[
 X_0\equiv_k X_\star
 \quad\Longleftrightarrow\quad
 \text{the choice system has no successful choice}.
\]
The assertion holds both with common color names on separate graphs
and with pure-tuple comparison in their disjoint union.

If every domain and label set has size at most \(A\), where \(A\ge1\),
their common order is at most \(A M_t\), where
\[
 M_t=t\,2^{\,t-2+3\binom{t-1}{2}}+12\binom t3.
\]
When a successful choice exists, the two graphs already have different
histograms after at most two joint rounds, or one separate round.

Theorem (Fixed-dimension ETH consequence)

Assume positive exponential-rate ETH. There are constants c>0 and K\ge4 such that, for every fixed integer k\ge K, no deterministic algorithm decides k-WL equivalence of two n-vertex graphs in time O(n^{ck}). This holds for both joint and separate refinement. The inputs are explicit adjacency matrices of uncolored simple undirected graphs, and time is measured on a multitape Turing machine.

Section 1, Introduction Read it in the paper TeX source, line 197

Show the TeX

Assume positive exponential-rate ETH. There are constants \(c>0\)
and \(K\ge4\) such that, for every fixed integer \(k\ge K\), no
deterministic algorithm decides \(k\)-WL equivalence of two
\(n\)-vertex graphs in time \(O(n^{ck})\).
This holds for both joint and separate refinement. The inputs are
explicit adjacency matrices of uncolored simple undirected graphs,
and time is measured on a multitape Turing machine.

The complexity of identifying a graph by Weisfeiler–Leman refinement

Theorem

Section 1, Introduction Read it in the paper TeX source, line 36

Show the TeX

The following problem is EXPTIME-complete under deterministic
polynomial-time many-one reductions: given a nonempty finite simple
undirected uncolored graph $G$ by its adjacency matrix and a positive
integer $k$ in binary, decide whether $\WLdim(G)\le k$.
Here dimension one uses ordinary neighbor color refinement, and
dimensions at least two use the joint-update rule described above.

Unconditional time lower bounds for Weisfeiler–Leman equivalence

Theorem

There are absolute constants c>0 and k_0 with the following property. For every fixed integer k\ge k_0 and every correct deterministic decider A for k-Weisfeiler–Leman equivalence, there is an integer n_0=n_0(A,k) such that

T_A(n)\ge n^{ck} \qquad\text{for every integer }n\ge n_0.

This holds for multitape Turing machines and for sequential random-access machines with O(\log(n+2))-bit addresses and words, boundedly many memory accesses per instruction, and fixed instructions computable in time polynomial in the word length. The same bound holds if A is required to decide correctly only on pairs of simple connected uncolored graphs of diameter at most two, with T_A(n) maximized over that class. It applies both to the joint replacement-multiset convention for k-dimensional refinement and to the separate-multiset convention defined below.

Section 1, Introduction Read it in the paper TeX source, line 82

Show the TeX

There are absolute constants $c>0$ and $k_0$ with the following
property. For every fixed integer $k\ge k_0$ and every correct
deterministic decider $A$ for $k$-Weisfeiler--Leman equivalence, there
is an integer $n_0=n_0(A,k)$ such that
\[
                     T_A(n)\ge n^{ck}
             \qquad\text{for every integer }n\ge n_0.
\]
This holds for multitape Turing machines and for sequential
random-access machines with $O(\log(n+2))$-bit addresses and words,
boundedly many memory accesses per instruction, and fixed instructions
computable in time polynomial in the word length.
The same bound holds if $A$ is required to decide correctly only on
pairs of simple connected uncolored graphs of diameter at most two,
with $T_A(n)$ maximized over that class. It applies both to the
joint replacement-multiset convention for $k$-dimensional refinement
and to the separate-multiset convention defined below.

Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs

Theorem

Both \WL and \SubWL are \EXPTIME-complete under deterministic polynomial-time many-one reductions.

Section 1, Introduction Read it in the paper TeX source, line 28

Show the TeX

Both \(\WL\) and \(\SubWL\) are \(\EXPTIME\)-complete under deterministic
polynomial-time many-one reductions.

Configuration and file facts

File: lean/ComparatorChallenges/WLIdentification.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 133 link it.
Solution module: OAI.Computability.WLIdentification.Main
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 32 local vocabulary declarations
Lines of Lean, without comments or blank lines: 93
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

Plectis Atlas