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Motivating coherence-driven inference via sheaves

Steve Huntsman

CC-BY-SA, by Ché Lydia Xyang (Cybergothiché)

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Disclaimers

Presenting here in my personal capacity

“Argument” examples (incl. code) here done in my personal capacity

Much is joint work with Michael Robinson, Ludmilla Huntsman; Jewell Thomas: thanks to many others (see refs)

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Concept

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LLMs evaluate consistency of propositions

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LLMs evaluate consistency of propositions

Lack of hallucinations presumably due to interpolation vs extrapolation

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Ironically, LLMs can promote information integrity

Original goal: use LLMs in a principled way to automatically handle {mis, dis}information and bullshit

Frankfurt: bullshit is an attempt to manipulate opinions and attitudes with indifference to truth or falsity
Brandolini’s law: “the amount of energy needed to refute bullshit is an order of magnitude bigger than that needed to produce it”

“When LLMs get things wrong they aren't hallucinating. They are bullshitting.”

“Bullshit is a greater enemy of the truth than lies.”

Lack of hallucinations presumably due to interpolation vs extrapolation

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MAX-SAT evaluates coherence of propositions

First step: LLMs evaluate local relatedness/consistency of propositions
Next step: lifting local data for judgments about global data

Natural to use a sheaf: consistency is local; coherence is global

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MAX-SAT evaluates coherence of propositions

First step: LLMs evaluate local relatedness/consistency of propositions
Next step: lifting local data for judgments about global data

Natural to use a sheaf: consistency is local; coherence is global

Key insight: asking logical questions about propositions ⇒ satisfiability

Example: CNF formula for ≥ 1 true and ≥ 1 false proposition (n-lemma) is

(p₁ ∨ p₂ ∨ … ∨ p_n) ∧ (¬p₁ ∨ ¬p₂ ∨ … ∨ ¬p_n)

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MAX-SAT evaluates coherence of propositions

First step: LLMs evaluate local relatedness/consistency of propositions
Next step: lifting local data for judgments about global data

Natural to use a sheaf: consistency is local; coherence is global

Key insight: asking logical questions about propositions ⇒ satisfiability

Example: CNF formula for ≥ 1 true and ≥ 1 false proposition (n-lemma) is

(p₁ ∨ p₂ ∨ … ∨ p_n) ∧ (¬p₁ ∨ ¬p₂ ∨ … ∨ ¬p_n)

MAX-SAT gives a general framework for inference

Start with AND of weighted formulas encoding local (in)consistencies
Convert to weighted CNF required for solvers (highly nontrivial!)
Produce optimal assignment of truth values using a solver
Near-optimal assignments are also of interest as in Rashomon

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MAX-SAT evaluates coherence of propositions

First step: LLMs evaluate local relatedness/consistency of propositions
Next step: lifting local data for judgments about global data

Natural to use a sheaf: consistency is local; coherence is global

Key insight: asking logical questions about propositions ⇒ satisfiability

Example: CNF formula for ≥ 1 true and ≥ 1 false proposition (n-lemma) is

(p₁ ∨ p₂ ∨ … ∨ p_n) ∧ (¬p₁ ∨ ¬p₂ ∨ … ∨ ¬p_n)

MAX-SAT gives a general framework for inference

Start with AND of weighted formulas encoding local (in)consistencies
Convert to weighted CNF required for solvers (highly nontrivial!)
Produce optimal assignment of truth values using a solver
Near-optimal assignments are also of interest as in Rashomon

This is a plausible instantiation of a coherence theory of truth

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MAX-SAT evaluates coherence of propositions

First step: LLMs evaluate local relatedness/consistency of propositions
Next step: lifting local data for judgments about global data

Natural to use a sheaf: consistency is local; coherence is global

Key insight: asking logical questions about propositions ⇒ satisfiability

Example: CNF formula for ≥ 1 true and ≥ 1 false proposition (n-lemma) is

(p₁ ∨ p₂ ∨ … ∨ p_n) ∧ (¬p₁ ∨ ¬p₂ ∨ … ∨ ¬p_n)

MAX-SAT gives a general framework for inference

Start with AND of weighted formulas encoding local (in)consistencies
Convert to weighted CNF required for solvers (highly nontrivial!)
Produce optimal assignment of truth values using a solver
Near-optimal assignments are also of interest as in Rashomon

This is a plausible instantiation of a coherence theory of truth

“Clamping” yields a correspondence theory

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Motivation from sheaves

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Topological sheaves are simple and useful

Local data (“sections”) over open sets collectively define a sheaf if

Local sections defined over any open set can be consistently restricted to smaller open sets
If local sections agree over open sets, then each is a restriction of a common section over the union

Recent uses are morally quite similar to ours

Modeling opinion expression, lying, consensus (Ghrist, Hansen, Riess, …)
Computing maximal consistent data (Praggastis, Ambrose et al., …)
Measuring consistency of local data in sheaves (Robinson, …)

Our use is basically implicit but very powerful

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Finite topological sheaves are combinatorial

arXiv:1509.02975 gives a formula for the discrete cyclic case

There are many global sections associated with “untethered” local data

These sections form an intricate subsheaf of a constant sheaf

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Finite topological sheaves are combinatorial

arXiv:1509.02975 gives a formula for the discrete cyclic case

There are many global sections associated with “untethered” local data

These sections form an intricate subsheaf of a constant sheaf

Exercise: detail in terms of sheaf cohomology

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Finite topological sheaves are combinatorial

There are many global sections associated with “untethered” local data

These sections form an intricate subsheaf of a constant sheaf

https://www.reddit.com/r/Jigsawpuzzles/comments/15gk50k/

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Finite topological sheaves are combinatorial

There are many global sections associated with “untethered” local data

These sections form an intricate subsheaf of a constant sheaf

https://www.reddit.com/r/Jigsawpuzzles/comments/pemyh9/

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Sheaves inform satisfiability problems and CDI

If local (in)consistencies can be encoded by logical formulas then global coherence can be encoded by AND of weighted local formulas

Meanwhile, the premise is trivial: recall the CNF n-lemma formula

(p₁ ∨ p₂ ∨ … ∨ p_n) ∧ (¬p₁ ∨ ¬p₂ ∨ … ∨ ¬p_n)

Coherence-driven inference (CDI) ≡ solving weighted MAX-SAT

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Sheaves inform satisfiability problems and CDI

Sections ≡ solutions to constraints

Srinivas (1993): “the solutions to a constraint satisfaction problem form a sheaf: any consistent assignment must be assembled from consistent parts. Constraint satisfaction algorithms search for consistent assignments of values to variables.” https://apps.dtic.mil/sti/citations/ADA272724

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Sheaves inform satisfiability problems and CDI

Sections ≡ solutions to constraints

Srinivas (1993): “the solutions to a constraint satisfaction problem form a sheaf: any consistent assignment must be assembled from consistent parts. Constraint satisfaction algorithms search for consistent assignments of values to variables.” https://apps.dtic.mil/sti/citations/ADA272724

Sheaf cohomology ≡ counting obstructions to gluing local sections

#SAT ≡ a certain computation of H¹DOI:10.1006/jsco.1998.0261

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Problems in propositional logic can be efficiently transformed into SAT

A recently developed transformation enables modern MAX-SAT solvers
Also enables reasoning about > 2 propositions at a time (trilemmas!)

Sheaves inform satisfiability problems and CDI

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Problems in propositional logic can be efficiently transformed into SAT

A recently developed transformation enables modern MAX-SAT solvers
Also enables reasoning about > 2 propositions at a time (trilemmas!)

Classical CDI is a special case

Local consistency of proposition pairs compiles to IFF
Local inconsistency of proposition pairs compiles to XOR
Express gradations with numerical weights
Combine local pair/edge formulae into a global graph formula using AND
Only reasons about two propositions at a time: no native trilemma handling

Sheaves inform satisfiability problems and CDI

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Problems in propositional logic can be efficiently transformed into SAT

A recently developed transformation enables modern MAX-SAT solvers
Also enables reasoning about > 2 propositions at a time (trilemmas!)

Classical CDI is a special case

Local consistency of proposition pairs compiles to IFF
Local inconsistency of proposition pairs compiles to XOR
Express gradations with numerical weights
Combine local pair/edge formulae into a global graph formula using AND
Only reasons about two propositions at a time: no native trilemma handling

Equivalent formulations of classical CDI:

MAX-CUT is the formulation in literature
MAX-2-XOR-SAT is more natural

Sheaves inform satisfiability problems and CDI

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Abstract simplicial complex corresponding to the SAT instance

(w ∨ ¬x) ∧ (w ∨ y)

∧ (x ∨ ¬y) ∧ (x ∨ y ∨ ¬z)

We use cellular sheaves

Adapted from Rosiak’s book

^ implicitly

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Abstract simplicial complex corresponding to the SAT instance

(w ∨ ¬x) ∧ (w ∨ y)

∧ (x ∨ ¬y) ∧ (x ∨ y ∨ ¬z)

Inclusion of simplices defines a poset and (Alexandrov) topology: open sets are unions of “up-sets” such as ↑y, which is highlighted

We use cellular sheaves

Adapted from Rosiak’s book

^ implicitly

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Abstract simplicial complex corresponding to the SAT instance

(w ∨ ¬x) ∧ (w ∨ y)

∧ (x ∨ ¬y) ∧ (x ∨ y ∨ ¬z)

Inclusion of simplices defines a poset and (Alexandrov) topology: open sets are unions of “up-sets” such as ↑y, which is highlighted

Redrawing poset with ∅ omitted indicates how open sets encode a sensible notion of locality in the simplicial complex

We use cellular sheaves

Adapted from Rosiak’s book

^ implicitly

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Abstract simplicial complex corresponding to the SAT instance

(w ∨ ¬x) ∧ (w ∨ y)

∧ (x ∨ ¬y) ∧ (x ∨ y ∨ ¬z)

Inclusion of simplices defines a poset and (Alexandrov) topology: open sets are unions of “up-sets” such as ↑y, which is highlighted

Redrawing poset with ∅ omitted indicates how open sets encode a sensible notion of locality in the simplicial complex

We use cellular sheaves

TL;DR – nothing fancy here, just finite topological sheaves

Adapted from Rosiak’s book

^ implicitly

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We generalized classical CDI

Classical CDI is an approach to autonomous, explainable, and ethical decision-making
We were ignorant of this preexisting work until reviewing legal literature featuring words like “consistency” and “coherence” revealed a link

We thought attention + LLMs could compile weighted MAX-SAT instances
This turns out to be very finicky
Easier to just ask the LLM several times to produce a weighted graph

^ unknowingly

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We generalized classical CDI

Classical CDI is an approach to autonomous, explainable, and ethical decision-making
We were ignorant of this preexisting work until reviewing legal literature featuring words like “consistency” and “coherence” revealed a link

We thought attention + LLMs could compile weighted MAX-SAT instances
This turns out to be very finicky
Easier to just ask the LLM several times to produce a weighted graph

LLM + CDI = versatile neurosymbolic AI

LLM accurately and reproducibly compiles local information into a natural data structure (fast/system 1 reasoning)
CDI computes maximally coherent substructure to enable useful forms of machine cognition (slow/system 2)

^ unknowingly

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Related work focusing on sheaves and CDI

Consistency of language

Zadrozny and Garbayo: arXiv:1801.09036
Wang and Sadrzadeh: DOI:10.1098/rsta.2023.0013

Decision making

Tohmé: DOI:10.1080/03081079.2017.1355913

Extensive literature inaugurated by Thagard

In ignorance of CDI

Robinson, Ludmilla Huntsman, and SH: arXiv:2401.16713

With awareness of CDI

Thomas and SH: arXiv:2502.13953
Robinson, Ludmilla Huntsman, and SH: DOI:10.31235/osf.io/4tjz5_v1

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Classical CDI in detail

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Datum = vertex in a graph

Data consistency = edge weights

Accepted vs. rejected data = graph partition

Coherence(partition) = -∑(crossing weights)

Maximizing coherence = computing a maximum cut

LLMs can accurately and reproducibly construct coherence graphs

Coherence is a combinatorial problem

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Datum = vertex in a graph

Data consistency = edge weights

Accepted vs. rejected data = graph partition

Coherence(partition) = -∑(crossing weights)

Maximizing coherence = computing a maximum cut

LLMs can accurately and reproducibly construct coherence graphs

Coherence is a combinatorial problem

-1

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Datum = vertex in a graph

Data consistency = edge weights

Accepted vs. rejected data = graph partition

Coherence(partition) = -∑(crossing weights)

Maximizing coherence = computing a maximum cut

LLMs can accurately and reproducibly construct coherence graphs

Coherence is a combinatorial problem

-1

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Datum = vertex in a graph

Data consistency = edge weights

Accepted vs. rejected data = graph partition

Coherence(partition) = -∑(crossing weights)

Maximizing coherence = computing a maximum cut

LLMs can accurately and reproducibly construct coherence graphs

Coherence is a combinatorial problem

-1

Coherence = -(+1 + -1) = 0

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Datum = vertex in a graph

Data consistency = edge weights

Accepted vs. rejected data = graph partition

Coherence(partition) = -∑(crossing weights)

Maximizing coherence = computing a maximum cut

LLMs can accurately and reproducibly construct coherence graphs

Coherence is a combinatorial problem

-1

Coherence = -(-1 + -1) = 2

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Datum = vertex in a graph

Data consistency = edge weights

Accepted vs. rejected data = graph partition

Coherence(partition) = -∑(crossing weights)

Maximizing coherence = computing a maximum cut

LLMs can accurately and reproducibly construct coherence graphs

Coherence is a combinatorial problem

-1

Coherence = -(-1 + -1) = 2

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Example of perfect coherence graph reconstruction with o1-mini from arXiv:2502.13953

Reconstruct graphs w/ various LLMs

Scale: largest examples in literature

Several models reconstruct sparse coherence graphs with high fidelity from a single prompt

Taking medians and monitoring convergence improves matters

An algorithmic benchmark validates results

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Example of perfect coherence graph reconstruction with o1-mini from arXiv:2502.13953

Reconstruct graphs w/ various LLMs

Scale: largest examples in literature

Several models reconstruct sparse coherence graphs with high fidelity from a single prompt

Taking medians and monitoring convergence improves matters

An algorithmic benchmark validates results

CDI is robust to perturbations that are good cut sparsifications

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Example of perfect coherence graph reconstruction with o1-mini from arXiv:2502.13953

Reconstruct graphs w/ various LLMs

Scale: largest examples in literature

Several models reconstruct sparse coherence graphs with high fidelity from a single prompt

Taking medians and monitoring convergence improves matters

An algorithmic benchmark validates results

CDI is robust to perturbations that are good cut sparsifications

Plausible that LLMs and humans construct these

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https://computationalcognitivescience.github.io/lovelace/part_ii/coherence

CDI is a versatile cognitive framework

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Maximizing coherence ⇒ aspects of cognition

Data = observations ⇒ fusing/perceiving
Data = evidence & hypotheses ⇒ judging
Data = tasks & goals ⇒ planning
Generating/optimizing data ⇒ creating

CDI now is artisanal, not automated/scaled

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Maximizing coherence ⇒ aspects of cognition

Data = observations ⇒ fusing/perceiving
Data = evidence & hypotheses ⇒ judging
Data = tasks & goals ⇒ planning
Generating/optimizing data ⇒ creating

But at present (and since the 1990s):

Graph edges and weights are produced by hand
There are no good ways to encode trilemmas or to reason over them
Algorithms in common use are inextensible and resist introspection

CDI now is artisanal, not automated/scaled

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Maximizing coherence ⇒ aspects of cognition

Data = observations ⇒ fusing/perceiving
Data = evidence & hypotheses ⇒ judging
Data = tasks & goals ⇒ planning
Generating/optimizing data ⇒ creating

But at present (and since the 1990s):

Graph edges and weights are produced by hand
There are no good ways to encode trilemmas or to reason over them
Algorithms in common use are inextensible and resist introspection

Recent developments (LLMs, new algorithms, and new data structures) address these problems and enable automated CDI

CDI now is artisanal, not automated/scaled

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Examples

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Recent event motivated a small example

First presented within two weeks of the event

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Recent event motivated a small example

First presented within two weeks of the event

Later examples fully automatic on transcripts

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Produce a non-redundant list of the substantive claims made by the participants in the following transcript.

…

Can you consolidate any of these claims?

…

Please replace pronouns with their antecedents.

…

Label these propositions a-s.

…

GPT-4o produces a labeled list of substantive claims/ propositions in the meeting transcript

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buildCoherence:

Effective diplomacy requires engaging both Russia and Ukraine, and overt hostility toward Vladimir Putin would hinder negotiations.
Donald Trump prioritizes the interests of the United States and the world, not personal alignment with Vladimir Putin.
Ukraine is in a weak negotiating position and heavily dependent on military and financial support from the United States.
The United States has provided significant military aid to Ukraine, without which the Ukrainian military would have lost the war quickly.
A ceasefire between Ukraine and Russia should be pursued first, as a ceasefire is easier to achieve than a full peace agreement.
Vladimir Putin respected Donald Trump and did not invade Ukraine during Donald Trump’s presidency, unlike during the presidencies of Barack Obama and Joe Biden.
Previous United States administrations, including those led by Barack Obama and Joe Biden, were weak, which allowed Russia to act aggressively.
If Ukraine does not negotiate a deal with Russia, the United States may withdraw support, leaving Ukraine to fight against Russia alone.
Volodymyr Zelenskyy has not expressed sufficient gratitude for the military and financial support provided by the United States.
Joe Biden’s public tough stance on Vladimir Putin has been ineffective in preventing Russia’s invasion of Ukraine.
Diplomacy between the United States, Ukraine, and Russia is the best way to achieve peace.
Ukraine has manpower shortages and is forcing Ukrainian citizens to fight against Russia as conscripts.
Volodymyr Zelenskyy should not publicly criticize United States policy while speaking in the Oval Office.
Ukraine should acknowledge military and financial support from the United States and express more gratitude.
Russia has occupied Ukrainian territory, including Crimea and eastern Ukraine, since 2014 and has consistently broken agreements with Ukraine.
Diplomacy between Ukraine and Russia has been attempted through ceasefires and prisoner exchanges but has repeatedly failed due to Russian violations.
The war between Ukraine and Russia will eventually affect the United States, even though the United States does not currently feel its full impact.
A ceasefire between Ukraine and Russia without guarantees is meaningless because Russia has a history of breaking agreements with Ukraine.
Ukraine has expressed gratitude for military and financial support from the United States multiple times.

Produce a non-redundant list of the substantive claims made by the participants in the following transcript.

…

Can you consolidate any of these claims?

…

Please replace pronouns with their antecedents.

…

Label these propositions a-s.

…

GPT-4o produces a labeled list of substantive claims/ propositions in the meeting transcript

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First, determine which pairs of propositions are substantively related.

Second, for each related pair of propositions, determine their logical relationship, assuming that at least one is true, whether or not either actually is. I want you to ignore the truth, falsity or basis in fact of either claim.

Third, based on your determination just above, numerically rate the relative consistency of the two propositions. Do not pay attention to or comment on the truth or basis in fact of either proposition independent of the other. Your rating of relative consistency should be on a scale from 0 to 10, with a value of 0 for a pair of propositions that are not at all consistent and a value of 10 for a pair of propositions that are totally consistent. I cannot emphasize enough that for your rating, I want you to ignore the truth or basis in fact of either proposition, since anything that is not consistent with reality cannot be true. If you determine that propositions are unrelated despite previously determining otherwise, omit that pair. To be clear, a pair of false but consistent claims should also be rated a 10. Meanwhile, a pair of propositions of which one is true and the other is false, should be rated a 0.

Finally, construct a networkx graph where propositions are vertices and edges correspond to substantively related pairs of propositions, with weights given by the consistency ratings just above. Only return the edge list with proposition labels for vertices. i.e., return responses in this format(here 'b', 'c', 'd', and 'e' are labels):

[(’b’, ’c’, 0),(’b’, ’e’, 10),(’c’, ’d’, 9),(’c’, ’e’, 2)]

buildCoherence:

Effective diplomacy requires engaging both Russia and Ukraine, and overt hostility toward Vladimir Putin would hinder negotiations.
Donald Trump prioritizes the interests of the United States and the world, not personal alignment with Vladimir Putin.
Ukraine is in a weak negotiating position and heavily dependent on military and financial support from the United States.
The United States has provided significant military aid to Ukraine, without which the Ukrainian military would have lost the war quickly.
A ceasefire between Ukraine and Russia should be pursued first, as a ceasefire is easier to achieve than a full peace agreement.
Vladimir Putin respected Donald Trump and did not invade Ukraine during Donald Trump’s presidency, unlike during the presidencies of Barack Obama and Joe Biden.
Previous United States administrations, including those led by Barack Obama and Joe Biden, were weak, which allowed Russia to act aggressively.
If Ukraine does not negotiate a deal with Russia, the United States may withdraw support, leaving Ukraine to fight against Russia alone.
Volodymyr Zelenskyy has not expressed sufficient gratitude for the military and financial support provided by the United States.
Joe Biden’s public tough stance on Vladimir Putin has been ineffective in preventing Russia’s invasion of Ukraine.
Diplomacy between the United States, Ukraine, and Russia is the best way to achieve peace.
Ukraine has manpower shortages and is forcing Ukrainian citizens to fight against Russia as conscripts.
Volodymyr Zelenskyy should not publicly criticize United States policy while speaking in the Oval Office.
Ukraine should acknowledge military and financial support from the United States and express more gratitude.
Russia has occupied Ukrainian territory, including Crimea and eastern Ukraine, since 2014 and has consistently broken agreements with Ukraine.
Diplomacy between Ukraine and Russia has been attempted through ceasefires and prisoner exchanges but has repeatedly failed due to Russian violations.
The war between Ukraine and Russia will eventually affect the United States, even though the United States does not currently feel its full impact.
A ceasefire between Ukraine and Russia without guarantees is meaningless because Russia has a history of breaking agreements with Ukraine.
Ukraine has expressed gratitude for military and financial support from the United States multiple times.

Produce a non-redundant list of the substantive claims made by the participants in the following transcript.

…

Can you consolidate any of these claims?

…

Please replace pronouns with their antecedents.

…

Label these propositions a-s.

…

GPT-4o produces a labeled list of substantive claims/ propositions in the meeting transcript

GPT-o3-mini builds graphs

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('a', 'e', 7),	('a', 'e', 10),	( 'a', 'e', 8),	( 'a', 'e', 8),	('a', 'e', 7),	8
('a', 'k', 10),	('a', 'k', 10),	( 'a', 'k', 10),	( 'a', 'k', 10),	('a', 'k', 10),	10
('a', 'p', 7),	('a', 'p', 8),	( 'a', 'p', 5),	( 'a', 'p', 7),	('a', 'p', 6),	6.67
	('a', 'r', 10),		( 'a', 'r', 5),	('a', 'r', 6),	7
('b', 'f', 8),	('b', 'f', 10),	( 'b', 'f', 9),	( 'b', 'f', 8),	('b', 'f', 10),	9
('b', 'g', 7),	('b', 'g', 10),		( 'b', 'g', 9),	('b', 'g', 7),	8.25
('b', 'j', 6),	('b', 'j', 10),	( 'b', 'j', 8),	( 'b', 'j', 8),		8
('c', 'd', 10),	('c', 'd', 10),	( 'c', 'd', 10),	( 'c', 'd', 10),	('c', 'd', 10),	10
('c', 'h', 10),	('c', 'h', 10),	( 'c', 'h', 9),	( 'c', 'h', 10),	('c', 'h', 10),	9.8
('c', 'i', 8),	('c', 'i', 10),	( 'c', 'l', 8),	( 'c', 'i', 7),	('c', 'i', 8),	8.2
('c', 'l', 10),	('c', 'l', 10),		( 'c', 'l', 8),	('c', 'l', 8),	9
('c', 'n', 10),	('c', 'n', 10),		( 'c', 'n', 8),	('c', 'n', 8),	9
	('c', 's', 10),		( 'c', 's', 10),	('c', 's', 8),	9.33
	('d', 'h', 10),	( 'd', 'h', 9),		('d', 'h', 10),	9.67
('d', 'i', 10),	('d', 'i', 10),		( 'd', 'i', 9),	('d', 'i', 8),	9.25
('d', 'n', 10),	('d', 'n', 10),		( 'd', 'n', 10),	('d', 'n', 8),	9.5
('d', 's', 10),	('d', 's', 10),		( 'd', 's', 10),	('d', 's', 8),	9.5
('e', 'k', 7),	('e', 'k', 10),	( 'e', 'k', 9),	( 'e', 'k', 9),	('e', 'k', 8),	8.6
('e', 'p', 2),	('e', 'p', 0),		( 'e', 'p', 4),	('e', 'p', 0),	1.5
('e', 'r', 3),	('e', 'r', 0),	( 'e', 'r', 0),	( 'e', 'r', 0),	('e', 'r', 0),	0.6
('f', 'g', 10),	('f', 'g', 10),	( 'f', 'g', 10),	( 'f', 'g', 9),	('f', 'g', 10),	9.8
('f', 'j', 10),	('f', 'j', 10),		( 'f', 'j', 9),	('f', 'j', 10),	9.75
('g', 'j', 10),	('g', 'j', 10),	( 'g', 'j', 10),	( 'g', 'j', 9),	('g', 'j', 10),	9.8
('h', 'l', 8),	('h', 'i', 8),		( 'h', 'l', 8),	('h', 'i', 7),	7.75
('h', 'q', 7),		( 'h', 'q', 7),		('h', 'q', 7),	7
('i', 'm', 7),	('i', 'm', 10),	( 'i', 'm', 6),	( 'i', 'm', 8),	('i', 'm', 8),	7.8
('i', 'n', 10),	('i', 'n', 10),	( 'i', 'n', 10),	( 'i', 'n', 10),	('i', 'n', 10),	10
('i', 's', 0),	('i', 's', 0),	( 'i', 's', 0),	( 'i', 's', 0),	('i', 's', 0),	0
('k', 'p', 3),	('k', 'p', 0),	( 'k', 'p', 0),	( 'k', 'p', 0),	('k', 'p', 5),	1.6
	('k', 'r', 8),		( 'k', 'r', 7),	('k', 'r', 5),	6.67
('m', 'n', 10),	('m', 'n', 10),	( 'm', 'n', 8),	( 'm', 'n', 10),	('m', 'n', 8),	9.2
('m', 's', 7),		( 'm', 's', 6),	( 'm', 's', 10),	('m', 's', 8),	7.75
('o', 'p', 10),	('o', 'p', 10),	( 'o', 'p', 10),	( 'o', 'p', 10),	('o', 'p', 10),	10
('o', 'r', 10),	('o', 'r', 10),	( 'o', 'r', 10),	( 'o', 'r', 10),	('o', 'r', 10),	10
('p', 'r', 10)	('p', 'r', 10)	( 'p', 'r', 10)	( 'p', 'r', 10)	('p', 'r', 10)	10

('a', 'e', 2),
('a', 'k', 2),
('a', 'p', 1),
('a', 'r', 1),
('b', 'f', 2),
('b', 'g', 2),
( 'b', 'j', 2),
('c', 'd', 2),
('c', 'h', 2),
('c', 'i', 2),
('c', 'l', 2),
('c', 'n', 2),
('c', 's', 2),
('d', 'h', 2),
('d', 'i', 2),
('d', 'n', 2),
('d', 's', 2),
('e', 'k', 2),
('e', 'p', -2),
('e', 'r', -2),
('f', 'g', 2),
('f', 'j', 2),
('g', 'j', 2),
('h', 'i', 1),
('h', 'q', 1),
('i', 'm', 1),
('i', 'n', 2),
('i', 's', -2),
('k', 'p', -2),
('k', 'r', 1),
('m', 'n', 2),
('m', 's', 1),
('o', 'p', 2),
('o', 'r', 2),
('p', 'r', 2)

First five columns: results from running the buildCoherence graph prompt 5x; aligning graph edges across responses, and keeping only edges that occurred most of the time

Sixth column: averaged weights (rescaled for coherence graph)

Seventh column: rescaled and quantized weights in {-2, -1, 0, 1, 2}

In practice now we use 25-50x and monitor convergence to median

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presidents

gratitude

War termination

Thick {blue, red} weights = {+2, -2};

thin {blue, red} weights = {+1, -1}

Focus on the most salient connected component

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Not speculative

a. Effective diplomacy requires engaging both Russia and Ukraine, and overt hostility toward Vladimir Putin would hinder negotiations.

e. A ceasefire between Ukraine and Russia should be pursued first, as a ceasefire is easier to achieve than a full peace agreement.

k. Diplomacy between the United States, Ukraine, and Russia is the best way to achieve peace.

o. Russia has occupied Ukrainian territory, including Crimea and eastern Ukraine, since 2014 and has consistently broken agreements with Ukraine.

p. Diplomacy between Ukraine and Russia has been attempted through ceasefires and prisoner exchanges but has repeatedly failed due to Russian violations.

r. A ceasefire between Ukraine and Russia without guarantees is meaningless because Russia has a history of breaking agreements with Ukraine.

Speculative

Arguably speculative

Speculative

Arguably speculative

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Not speculative

Coherence({a, e, k}, {o, p, r})

= 2 + 2 + 2 - 1 - 1 - 1 = 3

Data priority weighs in favor of accepting {o, p, r} and rejecting {a, e, k}

a. Effective diplomacy requires engaging both Russia and Ukraine, and overt hostility toward Vladimir Putin would hinder negotiations.

e. A ceasefire between Ukraine and Russia should be pursued first, as a ceasefire is easier to achieve than a full peace agreement.

k. Diplomacy between the United States, Ukraine, and Russia is the best way to achieve peace.

o. Russia has occupied Ukrainian territory, including Crimea and eastern Ukraine, since 2014 and has consistently broken agreements with Ukraine.

p. Diplomacy between Ukraine and Russia has been attempted through ceasefires and prisoner exchanges but has repeatedly failed due to Russian violations.

r. A ceasefire between Ukraine and Russia without guarantees is meaningless because Russia has a history of breaking agreements with Ukraine.

Most coherent partition

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Not speculative

Coherence({e, k}, {a, o, p, r})

= 2 + 2 + 2 - 2 - 2 - 1 = 1

Data priority weighs in favor of accepting {a, o, p, r} and rejecting {e, k}

a. Effective diplomacy requires engaging both Russia and Ukraine, and overt hostility toward Vladimir Putin would hinder negotiations.

e. A ceasefire between Ukraine and Russia should be pursued first, as a ceasefire is easier to achieve than a full peace agreement.

k. Diplomacy between the United States, Ukraine, and Russia is the best way to achieve peace.

o. Russia has occupied Ukrainian territory, including Crimea and eastern Ukraine, since 2014 and has consistently broken agreements with Ukraine.

p. Diplomacy between Ukraine and Russia has been attempted through ceasefires and prisoner exchanges but has repeatedly failed due to Russian violations.

r. A ceasefire between Ukraine and Russia without guarantees is meaningless because Russia has a history of breaking agreements with Ukraine.

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Not speculative

Coherence({a, k}, {e, o, p, r})

= 2 - 2 - 2 - 1 - 1 - 1 = -5

Data priority weighs in favor of accepting {e, o, p, r} and rejecting {a, k}

a. Effective diplomacy requires engaging both Russia and Ukraine, and overt hostility toward Vladimir Putin would hinder negotiations.

e. A ceasefire between Ukraine and Russia should be pursued first, as a ceasefire is easier to achieve than a full peace agreement.

k. Diplomacy between the United States, Ukraine, and Russia is the best way to achieve peace.

o. Russia has occupied Ukrainian territory, including Crimea and eastern Ukraine, since 2014 and has consistently broken agreements with Ukraine.

p. Diplomacy between Ukraine and Russia has been attempted through ceasefires and prisoner exchanges but has repeatedly failed due to Russian violations.

r. A ceasefire between Ukraine and Russia without guarantees is meaningless because Russia has a history of breaking agreements with Ukraine.

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Not speculative

Coherence({a, o}, {e, k, p, r})

= -2 - 2 - 2 - 2 - 1 - 1 = -10

Data priority weighs in favor of accepting {a, o} and rejecting {e, k, p, r}

a. Effective diplomacy requires engaging both Russia and Ukraine, and overt hostility toward Vladimir Putin would hinder negotiations.

e. A ceasefire between Ukraine and Russia should be pursued first, as a ceasefire is easier to achieve than a full peace agreement.

k. Diplomacy between the United States, Ukraine, and Russia is the best way to achieve peace.

o. Russia has occupied Ukrainian territory, including Crimea and eastern Ukraine, since 2014 and has consistently broken agreements with Ukraine.

p. Diplomacy between Ukraine and Russia has been attempted through ceasefires and prisoner exchanges but has repeatedly failed due to Russian violations.

r. A ceasefire between Ukraine and Russia without guarantees is meaningless because Russia has a history of breaking agreements with Ukraine.

Least coherent partition

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Accepted propositions

Coherence (quantized)

Coherence (real)

Symmetry between acceptance/rejection can be broken by prioritizing according to provenance

(e.g., direct observation vs speculation)

Accept	CohQ	CohR
ae	-2	-0.8733
aer	-8	-3.6867
aep	-5	-2.56
aepr	-7	-3.3733
aeo	-6	-2.8733
aeor	-8	-3.6867
aeop	-5	-2.56
aeopr	-3	-1.3733
aek	3	1.1933
aekr	-1	-0.9533
aekp	-4	-1.8533
aekpr	-4	-2
aeko	-1	-0.8067
aekor	-1	-0.9533
aekop	-4	-1.8533
aekopr	0	0

Accept	CohQ	CohR
a	-6	-2.3333
ar	-8	-3.3867
ap	-5	-2.62
apr	-3	-1.6733
ao	-10	-4.3333
aor	-8	-3.3867
aop	-5	-2.62
aopr	1	0.3267
ak	-5	-1.7067
akr	-5	-2.0933
akp	-8	-3.3533
akpr	-4	-1.74
ako	-9	-3.7067
akor	-5	-2.0933
akop	-8	-3.3533
akopr	0	0.26

Accept	CohQ	CohR
e	0	0.26
er	-8	-3.3533
ep	-5	-2.0933
epr	-9	-3.7067
eo	-4	-1.74
eor	-8	-3.3533
eop	-5	-2.0933
eopr	-5	-1.7067
ek	1	0.3267
ekr	-5	-2.62
ekp	-8	-3.3867
ekpr	-10	-4.3333
eko	-3	-1.6733
ekor	-5	-2.62
ekop	-8	-3.3867
ekopr	-6	-2.3333

Accept	CohQ	CohR
_	0	0
r	-4	-1.8533
p	-1	-0.9533
pr	-1	-0.8067
o	-4	-2
or	-4	-1.8533
op	-1	-0.9533
opr	3	1.1933
k	-3	-1.3733
kr	-5	-2.56
kp	-8	-3.6867
kpr	-6	-2.8733
ko	-7	-3.3733
kor	-5	-2.56
kop	-8	-3.6867
kopr	-2	-0.8733

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Accepted propositions

Coherence (quantized)

Coherence (real)

Symmetry between acceptance/rejection can be broken by prioritizing according to provenance

(e.g., direct observation vs speculation)

Accept	CohQ	CohR
ae	-2	-0.8733
aer	-8	-3.6867
aep	-5	-2.56
aepr	-7	-3.3733
aeo	-6	-2.8733
aeor	-8	-3.6867
aeop	-5	-2.56
aeopr	-3	-1.3733
aek	3	1.1933
aekr	-1	-0.9533
aekp	-4	-1.8533
aekpr	-4	-2
aeko	-1	-0.8067
aekor	-1	-0.9533
aekop	-4	-1.8533
aekopr	0	0

Accept	CohQ	CohR
a	-6	-2.3333
ar	-8	-3.3867
ap	-5	-2.62
apr	-3	-1.6733
ao	-10	-4.3333
aor	-8	-3.3867
aop	-5	-2.62
aopr	1	0.3267
ak	-5	-1.7067
akr	-5	-2.0933
akp	-8	-3.3533
akpr	-4	-1.74
ako	-9	-3.7067
akor	-5	-2.0933
akop	-8	-3.3533
akopr	0	0.26

Accept	CohQ	CohR
e	0	0.26
er	-8	-3.3533
ep	-5	-2.0933
epr	-9	-3.7067
eo	-4	-1.74
eor	-8	-3.3533
eop	-5	-2.0933
eopr	-5	-1.7067
ek	1	0.3267
ekr	-5	-2.62
ekp	-8	-3.3867
ekpr	-10	-4.3333
eko	-3	-1.6733
ekor	-5	-2.62
ekop	-8	-3.3867
ekopr	-6	-2.3333

Accept	CohQ	CohR
_	0	0
r	-4	-1.8533
p	-1	-0.9533
pr	-1	-0.8067
o	-4	-2
or	-4	-1.8533
op	-1	-0.9533
opr	3	1.1933
k	-3	-1.3733
kr	-5	-2.56
kp	-8	-3.6867
kpr	-6	-2.8733
ko	-7	-3.3733
kor	-5	-2.56
kop	-8	-3.6867
kopr	-2	-0.8733

In this example, coherence-driven inference leads to accepting propositions {o, p, r} and rejecting propositions {a, e, k}. Recall that r is:

A ceasefire between Ukraine and Russia without guarantees is meaningless because Russia has a history of breaking agreements with Ukraine.

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CDI informs interpretation of argument narratives

61 of 84

The Melian Dialogue

Propositions extracted with GPT-4o

# Athenians

- p1: Right is determined by power; the strong dominate, and the weak endure.

- p2: Internal rebellion is more concerning than external threats.

- p3: Preserving the Athenian empire and Melos's submission is beneficial.

- p4: Neutrality from Melos is unacceptable as it would show Athenian weakness.

- p5: Subjugating Melos would strengthen Athenian security and the Athenian empire.

- p6: Islanders like Melos are more threatening than continentals.

- p7: The conflict with Melos is about survival, not honor.

- p8: Hope is unreliable and can lead to ruin.

- p9: Athenian actions align with the natural order of ruling where possible.

- p10: Lacedaemonian support for Melos is doubtful due to the Lacedaemonians' cautious nature.

- p11: Submitting to Athens is a secure and honorable choice for Melos.

# Melians

- p12: Fairness and justice are crucial in conflict.

- p13: Destroying Melos would set a dangerous precedent for Athens.

- p14: Neutrality is preferable to choosing sides.

- p15: Attacking neutrals could create more enemies for Athens.

- p16: Trying all options before submitting is important.

- p17: War's unpredictability offers hope.

- p18: Divine support exists for the just cause.

- p19: Lacedaemonian aid is expected due to kinship and honor.

- p20: Proximity to Peloponnese makes Lacedaemonian help more feasible.

o1-mini separates key Athenian and Melian views; CDI suggests focusing on p16 to delay conflict

Athens

Melos

Using transparency vs thickness

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Inherit the Wind

Act II, Scene 2

propositions extracted with GPT-4o

# Drummond

- p1: Expert testimony on evolution should be included in the trial of Bertram Cates.

- p2: Understanding evolution is essential for the jury's judgment in the trial.

- p3: The exclusion of scientific experts from the trial is wrong.

- p4: Bertram Cates' teachings on evolution are not criminal.

- p5: Matthew Harrison Brady's literal interpretation of the Bible is flawed.

- p6: Reason and scientific progress are more important than strict adherence to religious texts.

- p7: The Bible is not the sole source of truth and should not be the only reference in matters of education and law.

# Brady

- p8: Testimony on evolution should not be included in the trial of Bertram Cates.

- p9: The law excludes evolution from being taught in schools.

- p10: Brady is an authority on the Bible but has not read Charles Darwin's "Origin of Species."

- p11: The Bible's literal truth is valid, and God can change natural law as described in the Bible.

- p12: Scientific evidence that contradicts the Bible's teachings is invalid.

- p13: Faith in the Bible is more important than scientific reasoning and evidence.

- p14: God guides Brady's actions, and opposing Brady is akin to opposing God.

CDI separates arguments and identifies the pivot for winning

Drummond

Brady

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Inherit the Wind

Inherit the Wind Act II, Scene 2